Smooth actions on complex projective spaces
Krzysztof M. Pawałowski
Adam Mickiewicz University · Poznań · Poland

43rd Symposium on Transformation Groups
November 17–19, 2016 · Himeji · Japan

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The goal

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The goal
The goal of the lecture is to present constructions of smoothactions of compact Lie groups G

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The goal
The goal of the lecture is to present constructions of smoothactions of compact Lie groups G on complex projective spaces
CP n ,

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The goal
The goal of the lecture is to present constructions of smoothactions of compact Lie groups G on complex projective spaces
CP n , such that the manifold
F (G  CP n ) = {x ∈ CP n : g · x = x for all g ∈ G }Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The goal
The goal of the lecture is to present constructions of smoothactions of compact Lie groups G on complex projective spaces
CP n , such that the manifold
F (G  CP n ) = {x ∈ CP n : g · x = x for all g ∈ G }consisting of points fixed under the action of G on CP n

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The goal
The goal of the lecture is to present constructions of smoothactions of compact Lie groups G on complex projective spaces
CP n , such that the manifold
F (G  CP n ) = {x ∈ CP n : g · x = x for all g ∈ G }consisting of points fixed under the action of G on CP n contains
a connected component M admitting some geometric structure
while missing another one.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The goal
The goal of the lecture is to present constructions of smoothactions of compact Lie groups G on complex projective spaces
CP n , such that the manifold
F (G  CP n ) = {x ∈ CP n : g · x = x for all g ∈ G }consisting of points fixed under the action of G on CP n contains
a connected component M admitting some geometric structure
while missing another one. We shall see how to construct smoothactions of G on CP n ,Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The goal
The goal of the lecture is to present constructions of smoothactions of compact Lie groups G on complex projective spaces
CP n , such that the manifold
F (G  CP n ) = {x ∈ CP n : g · x = x for all g ∈ G }consisting of points fixed under the action of G on CP n contains
a connected component M admitting some geometric structure
while missing another one. We shall see how to construct smoothactions of G on CP n , in such a way thatKrzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The goal
The goal of the lecture is to present constructions of smoothactions of compact Lie groups G on complex projective spaces
CP n , such that the manifold
F (G  CP n ) = {x ∈ CP n : g · x = x for all g ∈ G }consisting of points fixed under the action of G on CP n contains
a connected component M admitting some geometric structure
while missing another one. We shall see how to construct smoothactions of G on CP n , in such a way that
M is stable almost complex and M is not almost complex,

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The goal
The goal of the lecture is to present constructions of smoothactions of compact Lie groups G on complex projective spaces
CP n , such that the manifold
F (G  CP n ) = {x ∈ CP n : g · x = x for all g ∈ G }consisting of points fixed under the action of G on CP n contains
a connected component M admitting some geometric structure
while missing another one. We shall see how to construct smoothactions of G on CP n , in such a way that
M is stable almost complex and M is not almost complex,
M is almost complex and M is not homotopically symplectic,

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The goal
The goal of the lecture is to present constructions of smoothactions of compact Lie groups G on complex projective spaces
CP n , such that the manifold
F (G  CP n ) = {x ∈ CP n : g · x = x for all g ∈ G }consisting of points fixed under the action of G on CP n contains
a connected component M admitting some geometric structure
while missing another one. We shall see how to construct smoothactions of G on CP n , in such a way that
M is stable almost complex and M is not almost complex,
M is almost complex and M is not homotopically symplectic,
M is homotopically symplectic and M is not symplectic.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The goal
The goal of the lecture is to present constructions of smoothactions of compact Lie groups G on complex projective spaces
CP n , such that the manifold
F (G  CP n ) = {x ∈ CP n : g · x = x for all g ∈ G }consisting of points fixed under the action of G on CP n contains
a connected component M admitting some geometric structure
while missing another one. We shall see how to construct smoothactions of G on CP n , in such a way that
M is stable almost complex and M is not almost complex,
M is almost complex and M is not homotopically symplectic,
M is homotopically symplectic and M is not symplectic.
Almost complex actions,Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The goal
The goal of the lecture is to present constructions of smoothactions of compact Lie groups G on complex projective spaces
CP n , such that the manifold
F (G  CP n ) = {x ∈ CP n : g · x = x for all g ∈ G }consisting of points fixed under the action of G on CP n contains
a connected component M admitting some geometric structure
while missing another one. We shall see how to construct smoothactions of G on CP n , in such a way that
M is stable almost complex and M is not almost complex,
M is almost complex and M is not homotopically symplectic,
M is homotopically symplectic and M is not symplectic.
Almost complex actions, homotopically symplectic actions,Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The goal
The goal of the lecture is to present constructions of smoothactions of compact Lie groups G on complex projective spaces
CP n , such that the manifold
F (G  CP n ) = {x ∈ CP n : g · x = x for all g ∈ G }consisting of points fixed under the action of G on CP n contains
a connected component M admitting some geometric structure
while missing another one. We shall see how to construct smoothactions of G on CP n , in such a way that
M is stable almost complex and M is not almost complex,
M is almost complex and M is not homotopically symplectic,
M is homotopically symplectic and M is not symplectic.
Almost complex actions, homotopically symplectic actions, andsymplectic actions
Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The goal
The goal of the lecture is to present constructions of smoothactions of compact Lie groups G on complex projective spaces
CP n , such that the manifold
F (G  CP n ) = {x ∈ CP n : g · x = x for all g ∈ G }consisting of points fixed under the action of G on CP n contains
a connected component M admitting some geometric structure
while missing another one. We shall see how to construct smoothactions of G on CP n , in such a way that
M is stable almost complex and M is not almost complex,
M is almost complex and M is not homotopically symplectic,
M is homotopically symplectic and M is not symplectic.
Almost complex actions, homotopically symplectic actions, andsymplectic actions of compact Lie groups G on CP n
Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The goal
The goal of the lecture is to present constructions of smoothactions of compact Lie groups G on complex projective spaces
CP n , such that the manifold
F (G  CP n ) = {x ∈ CP n : g · x = x for all g ∈ G }consisting of points fixed under the action of G on CP n contains
a connected component M admitting some geometric structure
while missing another one. We shall see how to construct smoothactions of G on CP n , in such a way that
M is stable almost complex and M is not almost complex,
M is almost complex and M is not homotopically symplectic,
M is homotopically symplectic and M is not symplectic.
Almost complex actions, homotopically symplectic actions, andsymplectic actions of compact Lie groups G on CP n form threedifferent classes of transformation groups.
Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex and stable almost complex manifolds

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex and stable almost complex manifolds
By an almost complex manifold (M 2n , J)

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex and stable almost complex manifolds
By an almost complex manifold (M 2n , J) we mean a smoothmanifold M 2n with a vector bundle map J : T (M) → T (M)over idM

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex and stable almost complex manifolds
By an almost complex manifold (M 2n , J) we mean a smoothmanifold M 2n with a vector bundle map J : T (M) → T (M)over idM such that J 2 = −idT (M) .

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex and stable almost complex manifolds
By an almost complex manifold (M 2n , J) we mean a smoothmanifold M 2n with a vector bundle map J : T (M) → T (M)over idM such that J 2 = −idT (M) . In other words, the tangent
bundle T (M) admists the structure of a complex vector bundle.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex and stable almost complex manifolds
By an almost complex manifold (M 2n , J) we mean a smoothmanifold M 2n with a vector bundle map J : T (M) → T (M)over idM such that J 2 = −idT (M) . In other words, the tangent
bundle T (M) admists the structure of a complex vector bundle.
Existence of an almost complex structure J on M is equivalent toexistence of a differential 2-form ω on M

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex and stable almost complex manifolds
By an almost complex manifold (M 2n , J) we mean a smoothmanifold M 2n with a vector bundle map J : T (M) → T (M)over idM such that J 2 = −idT (M) . In other words, the tangent
bundle T (M) admists the structure of a complex vector bundle.
Existence of an almost complex structure J on M is equivalent toexistence of a differential 2-form ω on M which is non-degenerate
at each point x ∈ M,

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex and stable almost complex manifolds
By an almost complex manifold (M 2n , J) we mean a smoothmanifold M 2n with a vector bundle map J : T (M) → T (M)over idM such that J 2 = −idT (M) . In other words, the tangent
bundle T (M) admists the structure of a complex vector bundle.
Existence of an almost complex structure J on M is equivalent toexistence of a differential 2-form ω on M which is non-degenerate
at each point x ∈ M, i.e., for any non-zero vector v ∈ Tx (M),the 1-form {λx : Tx (M) → R}x∈M given by λx (w ) = ωx (v , w )do not vanish.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex and stable almost complex manifolds
By an almost complex manifold (M 2n , J) we mean a smoothmanifold M 2n with a vector bundle map J : T (M) → T (M)over idM such that J 2 = −idT (M) . In other words, the tangent
bundle T (M) admists the structure of a complex vector bundle.
Existence of an almost complex structure J on M is equivalent toexistence of a differential 2-form ω on M which is non-degenerate
at each point x ∈ M, i.e., for any non-zero vector v ∈ Tx (M),the 1-form {λx : Tx (M) → R}x∈M given by λx (w ) = ωx (v , w )do not vanish.
By a stable almost complex manifold (M, f )Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex and stable almost complex manifolds
By an almost complex manifold (M 2n , J) we mean a smoothmanifold M 2n with a vector bundle map J : T (M) → T (M)over idM such that J 2 = −idT (M) . In other words, the tangent
bundle T (M) admists the structure of a complex vector bundle.
Existence of an almost complex structure J on M is equivalent toexistence of a differential 2-form ω on M which is non-degenerate
at each point x ∈ M, i.e., for any non-zero vector v ∈ Tx (M),the 1-form {λx : Tx (M) → R}x∈M given by λx (w ) = ωx (v , w )do not vanish.
By a stable almost complex manifold (M, f ) we mean a smoothmanifold M with a smooth embedding f : M → E into some
Euclidean space E

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex and stable almost complex manifolds
By an almost complex manifold (M 2n , J) we mean a smoothmanifold M 2n with a vector bundle map J : T (M) → T (M)over idM such that J 2 = −idT (M) . In other words, the tangent
bundle T (M) admists the structure of a complex vector bundle.
Existence of an almost complex structure J on M is equivalent toexistence of a differential 2-form ω on M which is non-degenerate
at each point x ∈ M, i.e., for any non-zero vector v ∈ Tx (M),the 1-form {λx : Tx (M) → R}x∈M given by λx (w ) = ωx (v , w )do not vanish.
By a stable almost complex manifold (M, f ) we mean a smoothmanifold M with a smooth embedding f : M → E into some
Euclidean space E such that the normal bundle of the embedding
f admits a complex structure

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex and stable almost complex manifolds
By an almost complex manifold (M 2n , J) we mean a smoothmanifold M 2n with a vector bundle map J : T (M) → T (M)over idM such that J 2 = −idT (M) . In other words, the tangent
bundle T (M) admists the structure of a complex vector bundle.
Existence of an almost complex structure J on M is equivalent toexistence of a differential 2-form ω on M which is non-degenerate
at each point x ∈ M, i.e., for any non-zero vector v ∈ Tx (M),the 1-form {λx : Tx (M) → R}x∈M given by λx (w ) = ωx (v , w )do not vanish.
By a stable almost complex manifold (M, f ) we mean a smoothmanifold M with a smooth embedding f : M → E into some
Euclidean space E such that the normal bundle of the embedding
f admits a complex structure or, equivalently, the tangent bundle
T (M) to M admits the structure of a complex vector bundle,
Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex and stable almost complex manifolds
By an almost complex manifold (M 2n , J) we mean a smoothmanifold M 2n with a vector bundle map J : T (M) → T (M)over idM such that J 2 = −idT (M) . In other words, the tangent
bundle T (M) admists the structure of a complex vector bundle.
Existence of an almost complex structure J on M is equivalent toexistence of a differential 2-form ω on M which is non-degenerate
at each point x ∈ M, i.e., for any non-zero vector v ∈ Tx (M),the 1-form {λx : Tx (M) → R}x∈M given by λx (w ) = ωx (v , w )do not vanish.
By a stable almost complex manifold (M, f ) we mean a smoothmanifold M with a smooth embedding f : M → E into some
Euclidean space E such that the normal bundle of the embedding
f admits a complex structure or, equivalently, the tangent bundle
T (M) to M admits the structure of a complex vector bundle, after
adding the product bundle M × Rk over M for an integer k ­ 0.
Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex and stable almost complex manifolds
By an almost complex manifold (M 2n , J) we mean a smoothmanifold M 2n with a vector bundle map J : T (M) → T (M)over idM such that J 2 = −idT (M) . In other words, the tangent
bundle T (M) admists the structure of a complex vector bundle.
Existence of an almost complex structure J on M is equivalent toexistence of a differential 2-form ω on M which is non-degenerate
at each point x ∈ M, i.e., for any non-zero vector v ∈ Tx (M),the 1-form {λx : Tx (M) → R}x∈M given by λx (w ) = ωx (v , w )do not vanish.
By a stable almost complex manifold (M, f ) we mean a smoothmanifold M with a smooth embedding f : M → E into some
Euclidean space E such that the normal bundle of the embedding
f admits a complex structure or, equivalently, the tangent bundle
T (M) to M admits the structure of a complex vector bundle, after
adding the product bundle M × Rk over M for an integer k ­ 0.
Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex and stable almost complex manifolds
By an almost complex manifold (M 2n , J) we mean a smoothmanifold M 2n with a vector bundle map J : T (M) → T (M)over idM such that J 2 = −idT (M) . In other words, the tangent
bundle T (M) admists the structure of a complex vector bundle.
Existence of an almost complex structure J on M is equivalent toexistence of a differential 2-form ω on M which is non-degenerate
at each point x ∈ M, i.e., for any non-zero vector v ∈ Tx (M),the 1-form {λx : Tx (M) → R}x∈M given by λx (w ) = ωx (v , w )do not vanish.
By a stable almost complex manifold (M, f ) we mean a smoothmanifold M with a smooth embedding f : M → E into some
Euclidean space E such that the normal bundle of the embedding
f admits a complex structure or, equivalently, the tangent bundle
T (M) to M admits the structure of a complex vector bundle, after
adding the product bundle M × Rk over M for an integer k ­ 0.
Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Orientation of stable almost complex manifolds

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Orientation of stable almost complex manifolds
A way of saying that a real vector bundle E over a manifold M
is oriented

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Orientation of stable almost complex manifolds
A way of saying that a real vector bundle E over a manifold M
is oriented is to claim that the structure group of the principal
bundle associated to E

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Orientation of stable almost complex manifolds
A way of saying that a real vector bundle E over a manifold M
is oriented is to claim that the structure group of the principal
bundle associated to E is reduced from the group GLn (R) to
the subgroup GL+n (R) of matrices with positive determinant;Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Orientation of stable almost complex manifolds
A way of saying that a real vector bundle E over a manifold M
is oriented is to claim that the structure group of the principal
bundle associated to E is reduced from the group GLn (R) to
the subgroup GL+n (R) of matrices with positive determinant;i.e., the classifying map f : M → BGLn (R) of E

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Orientation of stable almost complex manifolds
A way of saying that a real vector bundle E over a manifold M
is oriented is to claim that the structure group of the principal
bundle associated to E is reduced from the group GLn (R) to
the subgroup GL+n (R) of matrices with positive determinant;i.e., the classifying map f : M → BGLn (R) of E lifts to a map
fe: M → BGL+n (R)Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Orientation of stable almost complex manifolds
A way of saying that a real vector bundle E over a manifold M
is oriented is to claim that the structure group of the principal
bundle associated to E is reduced from the group GLn (R) to
the subgroup GL+n (R) of matrices with positive determinant;i.e., the classifying map f : M → BGLn (R) of E lifts to a map
fe: M → BGL+n (R)with respect to the map BGL+n (R) → BGLn (R).

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Orientation of stable almost complex manifolds
A way of saying that a real vector bundle E over a manifold M
is oriented is to claim that the structure group of the principal
bundle associated to E is reduced from the group GLn (R) to
the subgroup GL+n (R) of matrices with positive determinant;i.e., the classifying map f : M → BGLn (R) of E lifts to a map
fe: M → BGL+n (R)with respect to the map BGL+n (R) → BGLn (R).
An orientation on two of the three vector bundles E1 , E2 , andE1 ⊕ E2 over M

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Orientation of stable almost complex manifolds
A way of saying that a real vector bundle E over a manifold M
is oriented is to claim that the structure group of the principal
bundle associated to E is reduced from the group GLn (R) to
the subgroup GL+n (R) of matrices with positive determinant;i.e., the classifying map f : M → BGLn (R) of E lifts to a map
fe: M → BGL+n (R)with respect to the map BGL+n (R) → BGLn (R).
An orientation on two of the three vector bundles E1 , E2 , andE1 ⊕ E2 over M determines an orientation on the third.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Orientation of stable almost complex manifolds
A way of saying that a real vector bundle E over a manifold M
is oriented is to claim that the structure group of the principal
bundle associated to E is reduced from the group GLn (R) to
the subgroup GL+n (R) of matrices with positive determinant;i.e., the classifying map f : M → BGLn (R) of E lifts to a map
fe: M → BGL+n (R)with respect to the map BGL+n (R) → BGLn (R).
An orientation on two of the three vector bundles E1 , E2 , andE1 ⊕ E2 over M determines an orientation on the third.
By definition, a smooth manifold M is oriented if the tangent
bundle T (M) is oriented,

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Orientation of stable almost complex manifolds
A way of saying that a real vector bundle E over a manifold M
is oriented is to claim that the structure group of the principal
bundle associated to E is reduced from the group GLn (R) to
the subgroup GL+n (R) of matrices with positive determinant;i.e., the classifying map f : M → BGLn (R) of E lifts to a map
fe: M → BGL+n (R)with respect to the map BGL+n (R) → BGLn (R).
An orientation on two of the three vector bundles E1 , E2 , andE1 ⊕ E2 over M determines an orientation on the third.
By definition, a smooth manifold M is oriented if the tangent
bundle T (M) is oriented, as a real vector bundle over M.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Orientation of stable almost complex manifolds
A way of saying that a real vector bundle E over a manifold M
is oriented is to claim that the structure group of the principal
bundle associated to E is reduced from the group GLn (R) to
the subgroup GL+n (R) of matrices with positive determinant;i.e., the classifying map f : M → BGLn (R) of E lifts to a map
fe: M → BGL+n (R)with respect to the map BGL+n (R) → BGLn (R).
An orientation on two of the three vector bundles E1 , E2 , andE1 ⊕ E2 over M determines an orientation on the third.
By definition, a smooth manifold M is oriented if the tangent
bundle T (M) is oriented, as a real vector bundle over M.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Orientation of stable almost complex manifolds
A way of saying that a real vector bundle E over a manifold M
is oriented is to claim that the structure group of the principal
bundle associated to E is reduced from the group GLn (R) to
the subgroup GL+n (R) of matrices with positive determinant;i.e., the classifying map f : M → BGLn (R) of E lifts to a map
fe: M → BGL+n (R)with respect to the map BGL+n (R) → BGLn (R).
An orientation on two of the three vector bundles E1 , E2 , andE1 ⊕ E2 over M determines an orientation on the third.
By definition, a smooth manifold M is oriented if the tangent
bundle T (M) is oriented, as a real vector bundle over M.
Since any complex vector bundle E over M is oriented,

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Orientation of stable almost complex manifolds
A way of saying that a real vector bundle E over a manifold M
is oriented is to claim that the structure group of the principal
bundle associated to E is reduced from the group GLn (R) to
the subgroup GL+n (R) of matrices with positive determinant;i.e., the classifying map f : M → BGLn (R) of E lifts to a map
fe: M → BGL+n (R)with respect to the map BGL+n (R) → BGLn (R).
An orientation on two of the three vector bundles E1 , E2 , andE1 ⊕ E2 over M determines an orientation on the third.
By definition, a smooth manifold M is oriented if the tangent
bundle T (M) is oriented, as a real vector bundle over M.
Since any complex vector bundle E over M is oriented, and the
product vector bundle M × Rk over M is oriented for any k ­ 1,
Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Orientation of stable almost complex manifolds
A way of saying that a real vector bundle E over a manifold M
is oriented is to claim that the structure group of the principal
bundle associated to E is reduced from the group GLn (R) to
the subgroup GL+n (R) of matrices with positive determinant;i.e., the classifying map f : M → BGLn (R) of E lifts to a map
fe: M → BGL+n (R)with respect to the map BGL+n (R) → BGLn (R).
An orientation on two of the three vector bundles E1 , E2 , andE1 ⊕ E2 over M determines an orientation on the third.
By definition, a smooth manifold M is oriented if the tangent
bundle T (M) is oriented, as a real vector bundle over M.
Since any complex vector bundle E over M is oriented, and the
product vector bundle M × Rk over M is oriented for any k ­ 1,
any stable almost complex manifold M is oriented.
Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Orientation of stable almost complex manifolds
A way of saying that a real vector bundle E over a manifold M
is oriented is to claim that the structure group of the principal
bundle associated to E is reduced from the group GLn (R) to
the subgroup GL+n (R) of matrices with positive determinant;i.e., the classifying map f : M → BGLn (R) of E lifts to a map
fe: M → BGL+n (R)with respect to the map BGL+n (R) → BGLn (R).
An orientation on two of the three vector bundles E1 , E2 , andE1 ⊕ E2 over M determines an orientation on the third.
By definition, a smooth manifold M is oriented if the tangent
bundle T (M) is oriented, as a real vector bundle over M.
Since any complex vector bundle E over M is oriented, and the
product vector bundle M × Rk over M is oriented for any k ­ 1,
any stable almost complex manifold M is oriented.
Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Classifying maps and (stable) almost complex manifolds

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Classifying maps and (stable) almost complex manifolds
An oriented manifold M is almost complex

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Classifying maps and (stable) almost complex manifolds
An oriented manifold M is almost complex if and only if theclassifying map fM : M → BGL+
2n (R) of T (M)Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Classifying maps and (stable) almost complex manifolds
An oriented manifold M is almost complex if and only if theclassifying map fM : M → BGL+
2n (R) of T (M) lifts to a map
ff
M : M → BGLn (C)Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Classifying maps and (stable) almost complex manifolds
An oriented manifold M is almost complex if and only if theclassifying map fM : M → BGL+
2n (R) of T (M) lifts to a map
ff
M : M → BGLn (C)with respect to the forgetful map BGLn (C) → BGL+
2n (R).

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Classifying maps and (stable) almost complex manifolds
An oriented manifold M is almost complex if and only if theclassifying map fM : M → BGL+
2n (R) of T (M) lifts to a map
ff
M : M → BGLn (C)with respect to the forgetful map BGLn (C) → BGL+
2n (R).
An oriented manifold M is stable almost complex

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Classifying maps and (stable) almost complex manifolds
An oriented manifold M is almost complex if and only if theclassifying map fM : M → BGL+
2n (R) of T (M) lifts to a map
ff
M : M → BGLn (C)with respect to the forgetful map BGLn (C) → BGL+
2n (R).
An oriented manifold M is stable almost complex if and only if theclassifying map fM : M → BGL+
2n (R) of T (M)Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Classifying maps and (stable) almost complex manifolds
An oriented manifold M is almost complex if and only if theclassifying map fM : M → BGL+
2n (R) of T (M) lifts to a map
ff
M : M → BGLn (C)with respect to the forgetful map BGLn (C) → BGL+
2n (R).
An oriented manifold M is stable almost complex if and only if theclassifying map fM : M → BGL+
2n (R) of T (M) composed with
BGL+
2n (R) → BGL(R) = lim BGLn (R)nKrzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Classifying maps and (stable) almost complex manifolds
An oriented manifold M is almost complex if and only if theclassifying map fM : M → BGL+
2n (R) of T (M) lifts to a map
ff
M : M → BGLn (C)with respect to the forgetful map BGLn (C) → BGL+
2n (R).
An oriented manifold M is stable almost complex if and only if theclassifying map fM : M → BGL+
2n (R) of T (M) composed with
BGL+
2n (R) → BGL(R) = lim BGLn (R)nlifts to a map
ff
M : M → BGL(C) = lim BGLn (C)nKrzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Classifying maps and (stable) almost complex manifolds
An oriented manifold M is almost complex if and only if theclassifying map fM : M → BGL+
2n (R) of T (M) lifts to a map
ff
M : M → BGLn (C)with respect to the forgetful map BGLn (C) → BGL+
2n (R).
An oriented manifold M is stable almost complex if and only if theclassifying map fM : M → BGL+
2n (R) of T (M) composed with
BGL+
2n (R) → BGL(R) = lim BGLn (R)nlifts to a map
ff
M : M → BGL(C) = lim BGLn (C)nwith respect to the forgetful map BGL(C) → BGL(R).
Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Classifying maps and (stable) almost complex manifolds
An oriented manifold M is almost complex if and only if theclassifying map fM : M → BGL+
2n (R) of T (M) lifts to a map
ff
M : M → BGLn (C)with respect to the forgetful map BGLn (C) → BGL+
2n (R).
An oriented manifold M is stable almost complex if and only if theclassifying map fM : M → BGL+
2n (R) of T (M) composed with
BGL+
2n (R) → BGL(R) = lim BGLn (R)nlifts to a map
ff
M : M → BGL(C) = lim BGLn (C)nwith respect to the forgetful map BGL(C) → BGL(R).
Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Symplectic and homotopically symplectic manifolds

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Symplectic and homotopically symplectic manifolds
By a closed symplectic manifold (M 2n , ω)Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Symplectic and homotopically symplectic manifolds
By a closed symplectic manifold (M 2n , ω) we mean a closed
smooth manifold M 2n

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Symplectic and homotopically symplectic manifolds
By a closed symplectic manifold (M 2n , ω) we mean a closed
smooth manifold M 2n with a smooth 2-form ω

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Symplectic and homotopically symplectic manifolds
By a closed symplectic manifold (M 2n , ω) we mean a closed
smooth manifold M 2n with a smooth 2-form ω which is bothclosed, i.e., the exterior derivative dω = 0.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Symplectic and homotopically symplectic manifolds
By a closed symplectic manifold (M 2n , ω) we mean a closed
smooth manifold M 2n with a smooth 2-form ω which is bothclosed, i.e., the exterior derivative dω = 0.
2n (M 2n ).
non-degenerate, i.e., ω n 6= 0 in HdR

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Symplectic and homotopically symplectic manifolds
By a closed symplectic manifold (M 2n , ω) we mean a closed
smooth manifold M 2n with a smooth 2-form ω which is bothclosed, i.e., the exterior derivative dω = 0.
2n (M 2n ).
non-degenerate, i.e., ω n 6= 0 in HdRCorollary
Any closed symplectic manifold (M 2n , ω) admits an almost
complex structure.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Symplectic and homotopically symplectic manifolds
By a closed symplectic manifold (M 2n , ω) we mean a closed
smooth manifold M 2n with a smooth 2-form ω which is bothclosed, i.e., the exterior derivative dω = 0.
2n (M 2n ).
non-degenerate, i.e., ω n 6= 0 in HdRCorollary
Any closed symplectic manifold (M 2n , ω) admits an almost
complex structure. The structure is determined by the 2-form ω.
By a closed homotopically symplectic manifold (M 2n , ω)Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Symplectic and homotopically symplectic manifolds
By a closed symplectic manifold (M 2n , ω) we mean a closed
smooth manifold M 2n with a smooth 2-form ω which is bothclosed, i.e., the exterior derivative dω = 0.
2n (M 2n ).
non-degenerate, i.e., ω n 6= 0 in HdRCorollary
Any closed symplectic manifold (M 2n , ω) admits an almost
complex structure. The structure is determined by the 2-form ω.
By a closed homotopically symplectic manifold (M 2n , ω) we meana closed almost complex manifold M 2n

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Symplectic and homotopically symplectic manifolds
By a closed symplectic manifold (M 2n , ω) we mean a closed
smooth manifold M 2n with a smooth 2-form ω which is bothclosed, i.e., the exterior derivative dω = 0.
2n (M 2n ).
non-degenerate, i.e., ω n 6= 0 in HdRCorollary
Any closed symplectic manifold (M 2n , ω) admits an almost
complex structure. The structure is determined by the 2-form ω.
By a closed homotopically symplectic manifold (M 2n , ω) we meana closed almost complex manifold M 2n with a cohomology classc ∈ H 2 (M 2n ; R) such that c n 6= 0.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Symplectic and homotopically symplectic manifolds
By a closed symplectic manifold (M 2n , ω) we mean a closed
smooth manifold M 2n with a smooth 2-form ω which is bothclosed, i.e., the exterior derivative dω = 0.
2n (M 2n ).
non-degenerate, i.e., ω n 6= 0 in HdRCorollary
Any closed symplectic manifold (M 2n , ω) admits an almost
complex structure. The structure is determined by the 2-form ω.
By a closed homotopically symplectic manifold (M 2n , ω) we meana closed almost complex manifold M 2n with a cohomology classc ∈ H 2 (M 2n ; R) such that c n 6= 0.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Four classes of manifolds

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Four classes of manifolds

We have defined the following four classes of manifolds:

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Four classes of manifolds

We have defined the following four classes of manifolds:{symplectic mfds} ⊂ {homotopically symplectic mfds} ⊂Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Four classes of manifolds

We have defined the following four classes of manifolds:{symplectic mfds} ⊂ {homotopically symplectic mfds} ⊂{almost complex mfds} ⊂ {stable almost complex mfds}.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Four classes of manifolds

We have defined the following four classes of manifolds:{symplectic mfds} ⊂ {homotopically symplectic mfds} ⊂{almost complex mfds} ⊂ {stable almost complex mfds}.

We shall give examples of manifolds showing that all inclusions
above are proper.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Four classes of manifolds

We have defined the following four classes of manifolds:{symplectic mfds} ⊂ {homotopically symplectic mfds} ⊂{almost complex mfds} ⊂ {stable almost complex mfds}.

We shall give examples of manifolds showing that all inclusions
above are proper. Every stable almost complex manifold is alsooriented.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Four classes of manifolds

We have defined the following four classes of manifolds:{symplectic mfds} ⊂ {homotopically symplectic mfds} ⊂{almost complex mfds} ⊂ {stable almost complex mfds}.

We shall give examples of manifolds showing that all inclusions
above are proper. Every stable almost complex manifold is alsooriented. We shall see that there are oriented smooth manifolds
which are not stable almost complex.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Four classes of manifolds

We have defined the following four classes of manifolds:{symplectic mfds} ⊂ {homotopically symplectic mfds} ⊂{almost complex mfds} ⊂ {stable almost complex mfds}.

We shall give examples of manifolds showing that all inclusions
above are proper. Every stable almost complex manifold is alsooriented. We shall see that there are oriented smooth manifolds
which are not stable almost complex.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Oriented non-stable almost complex manifolds

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Oriented non-stable almost complex manifolds
The following result allows us to give examples of smooth oriented
connected manifolds which are not stable almost complex.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Oriented non-stable almost complex manifolds
The following result allows us to give examples of smooth oriented
connected manifolds which are not stable almost complex.
Math. Nachr. 192 (1998) 159–172.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Oriented non-stable almost complex manifolds
The following result allows us to give examples of smooth oriented
connected manifolds which are not stable almost complex.
Math. Nachr. 192 (1998) 159–172.

Corollary (A. Dessai)

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Oriented non-stable almost complex manifolds
The following result allows us to give examples of smooth oriented
connected manifolds which are not stable almost complex.
Math. Nachr. 192 (1998) 159–172.

Corollary (A. Dessai)
Let M be a closed smooth 10-manifold with H1 (M; Z) = 0

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Oriented non-stable almost complex manifolds
The following result allows us to give examples of smooth oriented
connected manifolds which are not stable almost complex.
Math. Nachr. 192 (1998) 159–172.

Corollary (A. Dessai)
Let M be a closed smooth 10-manifold with H1 (M; Z) = 0 andsuch that Hi (M; Z) has no 2-torsion for i = 1, 2.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Oriented non-stable almost complex manifolds
The following result allows us to give examples of smooth oriented
connected manifolds which are not stable almost complex.
Math. Nachr. 192 (1998) 159–172.

Corollary (A. Dessai)
Let M be a closed smooth 10-manifold with H1 (M; Z) = 0 andsuch that Hi (M; Z) has no 2-torsion for i = 1, 2. Assume alsothat H 2 (M; Z) is generated by an element x such that x 2 6≡ 0(mod 2).

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Oriented non-stable almost complex manifolds
The following result allows us to give examples of smooth oriented
connected manifolds which are not stable almost complex.
Math. Nachr. 192 (1998) 159–172.

Corollary (A. Dessai)
Let M be a closed smooth 10-manifold with H1 (M; Z) = 0 andsuch that Hi (M; Z) has no 2-torsion for i = 1, 2. Assume alsothat H 2 (M; Z) is generated by an element x such that x 2 6≡ 0(mod 2). Then M admits a stable almost complex structure if
and only if

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Oriented non-stable almost complex manifolds
The following result allows us to give examples of smooth oriented
connected manifolds which are not stable almost complex.
Math. Nachr. 192 (1998) 159–172.

Corollary (A. Dessai)
Let M be a closed smooth 10-manifold with H1 (M; Z) = 0 andsuch that Hi (M; Z) has no 2-torsion for i = 1, 2. Assume alsothat H 2 (M; Z) is generated by an element x such that x 2 6≡ 0(mod 2). Then M admits a stable almost complex structure if
and only if w2 (M) · w4 (M)2 = 0.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Oriented non-stable almost complex manifolds
The following result allows us to give examples of smooth oriented
connected manifolds which are not stable almost complex.
Math. Nachr. 192 (1998) 159–172.

Corollary (A. Dessai)
Let M be a closed smooth 10-manifold with H1 (M; Z) = 0 andsuch that Hi (M; Z) has no 2-torsion for i = 1, 2. Assume alsothat H 2 (M; Z) is generated by an element x such that x 2 6≡ 0(mod 2). Then M admits a stable almost complex structure if
and only if w2 (M) · w4 (M)2 = 0.
In particular, the existence of a stable almost complex structure on
the manifold M above

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Oriented non-stable almost complex manifolds
The following result allows us to give examples of smooth oriented
connected manifolds which are not stable almost complex.
Math. Nachr. 192 (1998) 159–172.

Corollary (A. Dessai)
Let M be a closed smooth 10-manifold with H1 (M; Z) = 0 andsuch that Hi (M; Z) has no 2-torsion for i = 1, 2. Assume alsothat H 2 (M; Z) is generated by an element x such that x 2 6≡ 0(mod 2). Then M admits a stable almost complex structure if
and only if w2 (M) · w4 (M)2 = 0.
In particular, the existence of a stable almost complex structure on
the manifold M above depends only on the homotopy type of M.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Oriented non-stable almost complex manifolds
The following result allows us to give examples of smooth oriented
connected manifolds which are not stable almost complex.
Math. Nachr. 192 (1998) 159–172.

Corollary (A. Dessai)
Let M be a closed smooth 10-manifold with H1 (M; Z) = 0 andsuch that Hi (M; Z) has no 2-torsion for i = 1, 2. Assume alsothat H 2 (M; Z) is generated by an element x such that x 2 6≡ 0(mod 2). Then M admits a stable almost complex structure if
and only if w2 (M) · w4 (M)2 = 0.
In particular, the existence of a stable almost complex structure on
the manifold M above depends only on the homotopy type of M.
Once we choose, e.g., a simply connected manifold M

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Oriented non-stable almost complex manifolds
The following result allows us to give examples of smooth oriented
connected manifolds which are not stable almost complex.
Math. Nachr. 192 (1998) 159–172.

Corollary (A. Dessai)
Let M be a closed smooth 10-manifold with H1 (M; Z) = 0 andsuch that Hi (M; Z) has no 2-torsion for i = 1, 2. Assume alsothat H 2 (M; Z) is generated by an element x such that x 2 6≡ 0(mod 2). Then M admits a stable almost complex structure if
and only if w2 (M) · w4 (M)2 = 0.
In particular, the existence of a stable almost complex structure on
the manifold M above depends only on the homotopy type of M.
Once we choose, e.g., a simply connected manifold M satisfying
the hypotheses above,
Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Oriented non-stable almost complex manifolds
The following result allows us to give examples of smooth oriented
connected manifolds which are not stable almost complex.
Math. Nachr. 192 (1998) 159–172.

Corollary (A. Dessai)
Let M be a closed smooth 10-manifold with H1 (M; Z) = 0 andsuch that Hi (M; Z) has no 2-torsion for i = 1, 2. Assume alsothat H 2 (M; Z) is generated by an element x such that x 2 6≡ 0(mod 2). Then M admits a stable almost complex structure if
and only if w2 (M) · w4 (M)2 = 0.
In particular, the existence of a stable almost complex structure on
the manifold M above depends only on the homotopy type of M.
Once we choose, e.g., a simply connected manifold M satisfying
the hypotheses above, and such that w2 (M) · w4 (M)2 6= 0,
Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Oriented non-stable almost complex manifolds
The following result allows us to give examples of smooth oriented
connected manifolds which are not stable almost complex.
Math. Nachr. 192 (1998) 159–172.

Corollary (A. Dessai)
Let M be a closed smooth 10-manifold with H1 (M; Z) = 0 andsuch that Hi (M; Z) has no 2-torsion for i = 1, 2. Assume alsothat H 2 (M; Z) is generated by an element x such that x 2 6≡ 0(mod 2). Then M admits a stable almost complex structure if
and only if w2 (M) · w4 (M)2 = 0.
In particular, the existence of a stable almost complex structure on
the manifold M above depends only on the homotopy type of M.
Once we choose, e.g., a simply connected manifold M satisfying
the hypotheses above, and such that w2 (M) · w4 (M)2 6= 0, we get
the required example of a manifold M.
Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Even dimensional spheres and their products

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Even dimensional spheres and their products
Amer. J. Math. 75 (1953) 409–448.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Even dimensional spheres and their products
Amer. J. Math. 75 (1953) 409–448.

Theorem (A. Borel, J.-P. Serre)

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Even dimensional spheres and their products
Amer. J. Math. 75 (1953) 409–448.

Theorem (A. Borel, J.-P. Serre)
The spheres S 2 and S 6 are the only spheres

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Even dimensional spheres and their products
Amer. J. Math. 75 (1953) 409–448.

Theorem (A. Borel, J.-P. Serre)
The spheres S 2 and S 6 are the only spheres that admit almost
complex structures.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Even dimensional spheres and their products
Amer. J. Math. 75 (1953) 409–448.

Theorem (A. Borel, J.-P. Serre)
The spheres S 2 and S 6 are the only spheres that admit almost
complex structures.
As H 2 (S 6 ) = 0, S 6 is not a homotopically symplectic manifold

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Even dimensional spheres and their products
Amer. J. Math. 75 (1953) 409–448.

Theorem (A. Borel, J.-P. Serre)
The spheres S 2 and S 6 are the only spheres that admit almost
complex structures.
As H 2 (S 6 ) = 0, S 6 is not a homotopically symplectic manifold
while S 2 = CP 1 is clearly

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Even dimensional spheres and their products
Amer. J. Math. 75 (1953) 409–448.

Theorem (A. Borel, J.-P. Serre)
The spheres S 2 and S 6 are the only spheres that admit almost
complex structures.
As H 2 (S 6 ) = 0, S 6 is not a homotopically symplectic manifold
while S 2 = CP 1 is clearly a symplectic manifold.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Even dimensional spheres and their products
Amer. J. Math. 75 (1953) 409–448.

Theorem (A. Borel, J.-P. Serre)
The spheres S 2 and S 6 are the only spheres that admit almost
complex structures.
As H 2 (S 6 ) = 0, S 6 is not a homotopically symplectic manifold
while S 2 = CP 1 is clearly a symplectic manifold.
Topology Appl. 36 (1990) 39–42.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Even dimensional spheres and their products
Amer. J. Math. 75 (1953) 409–448.

Theorem (A. Borel, J.-P. Serre)
The spheres S 2 and S 6 are the only spheres that admit almost
complex structures.
As H 2 (S 6 ) = 0, S 6 is not a homotopically symplectic manifold
while S 2 = CP 1 is clearly a symplectic manifold.
Topology Appl. 36 (1990) 39–42.

Theorem (B. Datta, S. Subramanian)

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Even dimensional spheres and their products
Amer. J. Math. 75 (1953) 409–448.

Theorem (A. Borel, J.-P. Serre)
The spheres S 2 and S 6 are the only spheres that admit almost
complex structures.
As H 2 (S 6 ) = 0, S 6 is not a homotopically symplectic manifold
while S 2 = CP 1 is clearly a symplectic manifold.
Topology Appl. 36 (1990) 39–42.

Theorem (B. Datta, S. Subramanian)S 2 × S 4 , S 2 × S 6 , and S 6 × S 6

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Even dimensional spheres and their products
Amer. J. Math. 75 (1953) 409–448.

Theorem (A. Borel, J.-P. Serre)
The spheres S 2 and S 6 are the only spheres that admit almost
complex structures.
As H 2 (S 6 ) = 0, S 6 is not a homotopically symplectic manifold
while S 2 = CP 1 is clearly a symplectic manifold.
Topology Appl. 36 (1990) 39–42.

Theorem (B. Datta, S. Subramanian)S 2 × S 4 , S 2 × S 6 , and S 6 × S 6 are the only products of evendimensional spheres

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Even dimensional spheres and their products
Amer. J. Math. 75 (1953) 409–448.

Theorem (A. Borel, J.-P. Serre)
The spheres S 2 and S 6 are the only spheres that admit almost
complex structures.
As H 2 (S 6 ) = 0, S 6 is not a homotopically symplectic manifold
while S 2 = CP 1 is clearly a symplectic manifold.
Topology Appl. 36 (1990) 39–42.

Theorem (B. Datta, S. Subramanian)S 2 × S 4 , S 2 × S 6 , and S 6 × S 6 are the only products of evendimensional spheres that admit almost complex structures.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Even dimensional spheres and their products
Amer. J. Math. 75 (1953) 409–448.

Theorem (A. Borel, J.-P. Serre)
The spheres S 2 and S 6 are the only spheres that admit almost
complex structures.
As H 2 (S 6 ) = 0, S 6 is not a homotopically symplectic manifold
while S 2 = CP 1 is clearly a symplectic manifold.
Topology Appl. 36 (1990) 39–42.

Theorem (B. Datta, S. Subramanian)S 2 × S 4 , S 2 × S 6 , and S 6 × S 6 are the only products of evendimensional spheres that admit almost complex structures.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Odd dimensional spheres and their products

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Odd dimensional spheres and their products
By using the principal bundle
S 1 × S 1 → S 2m+1 × S 2n+1 → CP m × CP n ,Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Odd dimensional spheres and their products
By using the principal bundle
S 1 × S 1 → S 2m+1 × S 2n+1 → CP m × CP n ,we can prove that S 2m+1 × S 2n+1 is a complex manifold

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Odd dimensional spheres and their products
By using the principal bundle
S 1 × S 1 → S 2m+1 × S 2n+1 → CP m × CP n ,we can prove that S 2m+1 × S 2n+1 is a complex manifold (H. Hopf,
Interscience Publishers, Inc., New York, 1948, pp. 167–185).

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Odd dimensional spheres and their products
By using the principal bundle
S 1 × S 1 → S 2m+1 × S 2n+1 → CP m × CP n ,we can prove that S 2m+1 × S 2n+1 is a complex manifold (H. Hopf,
Interscience Publishers, Inc., New York, 1948, pp. 167–185).
The Cartesian product S 2m+1 × S 2n+1 of two spheres of odd
dimension is a complex manifold

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Odd dimensional spheres and their products
By using the principal bundle
S 1 × S 1 → S 2m+1 × S 2n+1 → CP m × CP n ,we can prove that S 2m+1 × S 2n+1 is a complex manifold (H. Hopf,
Interscience Publishers, Inc., New York, 1948, pp. 167–185).
The Cartesian product S 2m+1 × S 2n+1 of two spheres of odd
dimension is a complex manifold and therefore, the manifold
M = S 2m+1 × S 2n+1 is almost complex.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Odd dimensional spheres and their products
By using the principal bundle
S 1 × S 1 → S 2m+1 × S 2n+1 → CP m × CP n ,we can prove that S 2m+1 × S 2n+1 is a complex manifold (H. Hopf,
Interscience Publishers, Inc., New York, 1948, pp. 167–185).
The Cartesian product S 2m+1 × S 2n+1 of two spheres of odd
dimension is a complex manifold and therefore, the manifold
M = S 2m+1 × S 2n+1 is almost complex. Also, if m, n ­ 1,
H 2 (M) = 0

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Odd dimensional spheres and their products
By using the principal bundle
S 1 × S 1 → S 2m+1 × S 2n+1 → CP m × CP n ,we can prove that S 2m+1 × S 2n+1 is a complex manifold (H. Hopf,
Interscience Publishers, Inc., New York, 1948, pp. 167–185).
The Cartesian product S 2m+1 × S 2n+1 of two spheres of odd
dimension is a complex manifold and therefore, the manifold
M = S 2m+1 × S 2n+1 is almost complex. Also, if m, n ­ 1,
H 2 (M) = 0 and hence, M is not homotopically symplectic.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Odd dimensional spheres and their products
By using the principal bundle
S 1 × S 1 → S 2m+1 × S 2n+1 → CP m × CP n ,we can prove that S 2m+1 × S 2n+1 is a complex manifold (H. Hopf,
Interscience Publishers, Inc., New York, 1948, pp. 167–185).
The Cartesian product S 2m+1 × S 2n+1 of two spheres of odd
dimension is a complex manifold and therefore, the manifold
M = S 2m+1 × S 2n+1 is almost complex. Also, if m, n ­ 1,
H 2 (M) = 0 and hence, M is not homotopically symplectic.
Odd dimensional spheres are not almost complex by the dimension
reason.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Odd dimensional spheres and their products
By using the principal bundle
S 1 × S 1 → S 2m+1 × S 2n+1 → CP m × CP n ,we can prove that S 2m+1 × S 2n+1 is a complex manifold (H. Hopf,
Interscience Publishers, Inc., New York, 1948, pp. 167–185).
The Cartesian product S 2m+1 × S 2n+1 of two spheres of odd
dimension is a complex manifold and therefore, the manifold
M = S 2m+1 × S 2n+1 is almost complex. Also, if m, n ­ 1,
H 2 (M) = 0 and hence, M is not homotopically symplectic.
Odd dimensional spheres are not almost complex by the dimension
reason. Clearly, all spheres (even and odd dimensional) are stable
almost complex because they are stably parallelizable.
Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Odd dimensional spheres and their products
By using the principal bundle
S 1 × S 1 → S 2m+1 × S 2n+1 → CP m × CP n ,we can prove that S 2m+1 × S 2n+1 is a complex manifold (H. Hopf,
Interscience Publishers, Inc., New York, 1948, pp. 167–185).
The Cartesian product S 2m+1 × S 2n+1 of two spheres of odd
dimension is a complex manifold and therefore, the manifold
M = S 2m+1 × S 2n+1 is almost complex. Also, if m, n ­ 1,
H 2 (M) = 0 and hence, M is not homotopically symplectic.
Odd dimensional spheres are not almost complex by the dimension
reason. Clearly, all spheres (even and odd dimensional) are stable
almost complex because they are stably parallelizable.
Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Examples of stable parallelizable manifolds
The manifolds S 6 and S 2m+1 × S 2n+1 are almost complex

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Examples of stable parallelizable manifolds
The manifolds S 6 and S 2m+1 × S 2n+1 are almost complex but
they are not homotopically symplectic.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Examples of stable parallelizable manifolds
The manifolds S 6 and S 2m+1 × S 2n+1 are almost complex but
they are not homotopically symplectic.
Therefore,{homotopically symplectic mfds} ⊂ {almost complex mfds}

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Examples of stable parallelizable manifolds
The manifolds S 6 and S 2m+1 × S 2n+1 are almost complex but
they are not homotopically symplectic.
Therefore,{homotopically symplectic mfds} ⊂ {almost complex mfds}
is a proper inclusion.
The manifolds S 4 and S 8 , S 10 , . . . are stable almost complex

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Examples of stable parallelizable manifolds
The manifolds S 6 and S 2m+1 × S 2n+1 are almost complex but
they are not homotopically symplectic.
Therefore,{homotopically symplectic mfds} ⊂ {almost complex mfds}
is a proper inclusion.
The manifolds S 4 and S 8 , S 10 , . . . are stable almost complex
but they are not almost complex.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Examples of stable parallelizable manifolds
The manifolds S 6 and S 2m+1 × S 2n+1 are almost complex but
they are not homotopically symplectic.
Therefore,{homotopically symplectic mfds} ⊂ {almost complex mfds}
is a proper inclusion.
The manifolds S 4 and S 8 , S 10 , . . . are stable almost complex
but they are not almost complex.
Therefore,{almost complex mfds} ⊂ {stable almost complex mfds}

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Examples of stable parallelizable manifolds
The manifolds S 6 and S 2m+1 × S 2n+1 are almost complex but
they are not homotopically symplectic.
Therefore,{homotopically symplectic mfds} ⊂ {almost complex mfds}
is a proper inclusion.
The manifolds S 4 and S 8 , S 10 , . . . are stable almost complex
but they are not almost complex.
Therefore,{almost complex mfds} ⊂ {stable almost complex mfds}
is a proper inclusion.
Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex 4-manifolds

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex 4-manifolds
Actualités Sci. Ind. 1183 (1952)

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex 4-manifolds
Actualités Sci. Ind. 1183 (1952)

Theorem (W.-T. Wu)

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex 4-manifolds
Actualités Sci. Ind. 1183 (1952)

Theorem (W.-T. Wu)
A closed 4-manifolds M is almost complex if and only if

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex 4-manifolds
Actualités Sci. Ind. 1183 (1952)

Theorem (W.-T. Wu)
A closed 4-manifolds M is almost complex if and only if there
exists a class c ∈ H 2 (M, Z)

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex 4-manifolds
Actualités Sci. Ind. 1183 (1952)

Theorem (W.-T. Wu)
A closed 4-manifolds M is almost complex if and only if there
exists a class c ∈ H 2 (M, Z) whose reduction mod 2 is w2 (M),Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex 4-manifolds
Actualités Sci. Ind. 1183 (1952)

Theorem (W.-T. Wu)
A closed 4-manifolds M is almost complex if and only if there
exists a class c ∈ H 2 (M, Z) whose reduction mod 2 is w2 (M),the second Stiefel-Whitney class of M,

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex 4-manifolds
Actualités Sci. Ind. 1183 (1952)

Theorem (W.-T. Wu)
A closed 4-manifolds M is almost complex if and only if there
exists a class c ∈ H 2 (M, Z) whose reduction mod 2 is w2 (M),the second Stiefel-Whitney class of M, i.e., c ≡ w2 (M) (mod 2),andc 2 = 2χ(M) + 3σ(M)Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex 4-manifolds
Actualités Sci. Ind. 1183 (1952)

Theorem (W.-T. Wu)
A closed 4-manifolds M is almost complex if and only if there
exists a class c ∈ H 2 (M, Z) whose reduction mod 2 is w2 (M),the second Stiefel-Whitney class of M, i.e., c ≡ w2 (M) (mod 2),andc 2 = 2χ(M) + 3σ(M)for the Euler characteristic χ(M) and the signature σ(M) of M.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex 4-manifolds
Actualités Sci. Ind. 1183 (1952)

Theorem (W.-T. Wu)
A closed 4-manifolds M is almost complex if and only if there
exists a class c ∈ H 2 (M, Z) whose reduction mod 2 is w2 (M),the second Stiefel-Whitney class of M, i.e., c ≡ w2 (M) (mod 2),andc 2 = 2χ(M) + 3σ(M)for the Euler characteristic χ(M) and the signature σ(M) of M.

The connected sum (S 2 × S 2 )#(S 2 × S 2 ) is stable almost
complexKrzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex 4-manifolds
Actualités Sci. Ind. 1183 (1952)

Theorem (W.-T. Wu)
A closed 4-manifolds M is almost complex if and only if there
exists a class c ∈ H 2 (M, Z) whose reduction mod 2 is w2 (M),the second Stiefel-Whitney class of M, i.e., c ≡ w2 (M) (mod 2),andc 2 = 2χ(M) + 3σ(M)for the Euler characteristic χ(M) and the signature σ(M) of M.

The connected sum (S 2 × S 2 )#(S 2 × S 2 ) is stable almost
complex but it is not almost complex.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex 4-manifolds
Actualités Sci. Ind. 1183 (1952)

Theorem (W.-T. Wu)
A closed 4-manifolds M is almost complex if and only if there
exists a class c ∈ H 2 (M, Z) whose reduction mod 2 is w2 (M),the second Stiefel-Whitney class of M, i.e., c ≡ w2 (M) (mod 2),andc 2 = 2χ(M) + 3σ(M)for the Euler characteristic χ(M) and the signature σ(M) of M.

The connected sum (S 2 × S 2 )#(S 2 × S 2 ) is stable almost
complex but it is not almost complex.
The connected sum #k CP 2 is always stable almost complex

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex 4-manifolds
Actualités Sci. Ind. 1183 (1952)

Theorem (W.-T. Wu)
A closed 4-manifolds M is almost complex if and only if there
exists a class c ∈ H 2 (M, Z) whose reduction mod 2 is w2 (M),the second Stiefel-Whitney class of M, i.e., c ≡ w2 (M) (mod 2),andc 2 = 2χ(M) + 3σ(M)for the Euler characteristic χ(M) and the signature σ(M) of M.

The connected sum (S 2 × S 2 )#(S 2 × S 2 ) is stable almost
complex but it is not almost complex.
The connected sum #k CP 2 is always stable almost complex
and #k CP 2 is almost complex if and only if k is odd.
In particular, CP 2 #CP 2 is not almost complex.
Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex connected sum

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex connected sum
Osaka J. Math. 28 (1991) 243–253

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex connected sum
Osaka J. Math. 28 (1991) 243–253

Lemma (Y. Sato)

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex connected sum
Osaka J. Math. 28 (1991) 243–253

Lemma (Y. Sato)
There exists a homology 4-sphere Σ 4 with π1 (Σ 4 ) ∼= SL2 (F5 ).

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex connected sum
Osaka J. Math. 28 (1991) 243–253

Lemma (Y. Sato)
There exists a homology 4-sphere Σ 4 with π1 (Σ 4 ) ∼= SL2 (F5 ).
As we have noted, CP 2 #CP 2 is not almost complex.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex connected sum
Osaka J. Math. 28 (1991) 243–253

Lemma (Y. Sato)
There exists a homology 4-sphere Σ 4 with π1 (Σ 4 ) ∼= SL2 (F5 ).
As we have noted, CP 2 #CP 2 is not almost complex. However,the connected sum CP 2 #Σ 4 is an almost complex manifold

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex connected sum
Osaka J. Math. 28 (1991) 243–253

Lemma (Y. Sato)
There exists a homology 4-sphere Σ 4 with π1 (Σ 4 ) ∼= SL2 (F5 ).
As we have noted, CP 2 #CP 2 is not almost complex. However,the connected sum CP 2 #Σ 4 is an almost complex manifold
because the cohomology rings H ∗ (CP 2 #Σ 4 ) and H ∗ (CP 2 ) are
isomorphic

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex connected sum
Osaka J. Math. 28 (1991) 243–253

Lemma (Y. Sato)
There exists a homology 4-sphere Σ 4 with π1 (Σ 4 ) ∼= SL2 (F5 ).
As we have noted, CP 2 #CP 2 is not almost complex. However,the connected sum CP 2 #Σ 4 is an almost complex manifold
because the cohomology rings H ∗ (CP 2 #Σ 4 ) and H ∗ (CP 2 ) are
isomorphic and to apply Wu’s theorem to confirm that CP 2 #Σ 4
is almost complex,

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex connected sum
Osaka J. Math. 28 (1991) 243–253

Lemma (Y. Sato)
There exists a homology 4-sphere Σ 4 with π1 (Σ 4 ) ∼= SL2 (F5 ).
As we have noted, CP 2 #CP 2 is not almost complex. However,the connected sum CP 2 #Σ 4 is an almost complex manifold
because the cohomology rings H ∗ (CP 2 #Σ 4 ) and H ∗ (CP 2 ) are
isomorphic and to apply Wu’s theorem to confirm that CP 2 #Σ 4
is almost complex, one may take as c ∈ H ∗ (CP 2 #Σ 4 )Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex connected sum
Osaka J. Math. 28 (1991) 243–253

Lemma (Y. Sato)
There exists a homology 4-sphere Σ 4 with π1 (Σ 4 ) ∼= SL2 (F5 ).
As we have noted, CP 2 #CP 2 is not almost complex. However,the connected sum CP 2 #Σ 4 is an almost complex manifold
because the cohomology rings H ∗ (CP 2 #Σ 4 ) and H ∗ (CP 2 ) are
isomorphic and to apply Wu’s theorem to confirm that CP 2 #Σ 4
is almost complex, one may take as c ∈ H ∗ (CP 2 #Σ 4 ) the classcorresponding to the first Chern class c1 (CP 2 ).
Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Almost complex connected sum
Osaka J. Math. 28 (1991) 243–253

Lemma (Y. Sato)
There exists a homology 4-sphere Σ 4 with π1 (Σ 4 ) ∼= SL2 (F5 ).
As we have noted, CP 2 #CP 2 is not almost complex. However,the connected sum CP 2 #Σ 4 is an almost complex manifold
because the cohomology rings H ∗ (CP 2 #Σ 4 ) and H ∗ (CP 2 ) are
isomorphic and to apply Wu’s theorem to confirm that CP 2 #Σ 4
is almost complex, one may take as c ∈ H ∗ (CP 2 #Σ 4 ) the classcorresponding to the first Chern class c1 (CP 2 ).
Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Homotopically symplectic connected sum

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Homotopically symplectic connected sum
The connected sum CP 2 #Σ 4 is homotopically symplectic

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Homotopically symplectic connected sum
The connected sum CP 2 #Σ 4 is homotopically symplectic
because it is an almost complex manifold such thatKrzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Homotopically symplectic connected sum
The connected sum CP 2 #Σ 4 is homotopically symplectic
because it is an almost complex manifold such thatH ∗ (CP 2 #Σ 4 ; R) ∼= H ∗ (CP 2 ; R)

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Homotopically symplectic connected sum
The connected sum CP 2 #Σ 4 is homotopically symplectic
because it is an almost complex manifold such thatH ∗ (CP 2 #Σ 4 ; R) ∼= H ∗ (CP 2 ; R)and there exists an element c ∈ H ∗ (CP 2 #Σ 4 ; R) with c 2 6= 0,

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Homotopically symplectic connected sum
The connected sum CP 2 #Σ 4 is homotopically symplectic
because it is an almost complex manifold such thatH ∗ (CP 2 #Σ 4 ; R) ∼= H ∗ (CP 2 ; R)and there exists an element c ∈ H ∗ (CP 2 #Σ 4 ; R) with c 2 6= 0,
namely the element c corresponding to c1 (CP 2 ).

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Homotopically symplectic connected sum
The connected sum CP 2 #Σ 4 is homotopically symplectic
because it is an almost complex manifold such thatH ∗ (CP 2 #Σ 4 ; R) ∼= H ∗ (CP 2 ; R)and there exists an element c ∈ H ∗ (CP 2 #Σ 4 ; R) with c 2 6= 0,
namely the element c corresponding to c1 (CP 2 ).
We shall see that CP 2 #Σ 4 is not a symplectic manifold.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Homotopically symplectic connected sum
The connected sum CP 2 #Σ 4 is homotopically symplectic
because it is an almost complex manifold such thatH ∗ (CP 2 #Σ 4 ; R) ∼= H ∗ (CP 2 ; R)and there exists an element c ∈ H ∗ (CP 2 #Σ 4 ; R) with c 2 6= 0,
namely the element c corresponding to c1 (CP 2 ).
We shall see that CP 2 #Σ 4 is not a symplectic manifold.
Therefore,{symplectic mfds} ⊂ {homotopically symplectic mfds}

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Homotopically symplectic connected sum
The connected sum CP 2 #Σ 4 is homotopically symplectic
because it is an almost complex manifold such thatH ∗ (CP 2 #Σ 4 ; R) ∼= H ∗ (CP 2 ; R)and there exists an element c ∈ H ∗ (CP 2 #Σ 4 ; R) with c 2 6= 0,
namely the element c corresponding to c1 (CP 2 ).
We shall see that CP 2 #Σ 4 is not a symplectic manifold.
Therefore,{symplectic mfds} ⊂ {homotopically symplectic mfds}
is a proper inclusion.
Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Homotopically symplectic connected sum
The connected sum CP 2 #Σ 4 is homotopically symplectic
because it is an almost complex manifold such thatH ∗ (CP 2 #Σ 4 ; R) ∼= H ∗ (CP 2 ; R)and there exists an element c ∈ H ∗ (CP 2 #Σ 4 ; R) with c 2 6= 0,
namely the element c corresponding to c1 (CP 2 ).
We shall see that CP 2 #Σ 4 is not a symplectic manifold.
Therefore,{symplectic mfds} ⊂ {homotopically symplectic mfds}
is a proper inclusion.
Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spacesTaubes Theorem

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spacesTaubes Theorem
For a closed 4-manifold X ,Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spacesTaubes Theorem
For a closed 4-manifold X , let b2+ (X ) be the number of positive
entries in a diagonalization of the intersection form of X over Q

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spacesTaubes Theorem
For a closed 4-manifold X , let b2+ (X ) be the number of positive
entries in a diagonalization of the intersection form of X over Q
H 2 (X , Q) × H 2 (X , Q) → Q(a, b) 7→ ha ∪ b, [X ]i.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spacesTaubes Theorem
For a closed 4-manifold X , let b2+ (X ) be the number of positive
entries in a diagonalization of the intersection form of X over Q
H 2 (X , Q) × H 2 (X , Q) → Q(a, b) 7→ ha ∪ b, [X ]i.
Math. Research Letters 1 (1994) 809–822

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spacesTaubes Theorem
For a closed 4-manifold X , let b2+ (X ) be the number of positive
entries in a diagonalization of the intersection form of X over Q
H 2 (X , Q) × H 2 (X , Q) → Q(a, b) 7→ ha ∪ b, [X ]i.
Math. Research Letters 1 (1994) 809–822

Theorem (M. Taubes)

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spacesTaubes Theorem
For a closed 4-manifold X , let b2+ (X ) be the number of positive
entries in a diagonalization of the intersection form of X over Q
H 2 (X , Q) × H 2 (X , Q) → Q(a, b) 7→ ha ∪ b, [X ]i.
Math. Research Letters 1 (1994) 809–822

Theorem (M. Taubes)
Let X and Y be two closed oriented smooth 4-manifolds

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spacesTaubes Theorem
For a closed 4-manifold X , let b2+ (X ) be the number of positive
entries in a diagonalization of the intersection form of X over Q
H 2 (X , Q) × H 2 (X , Q) → Q(a, b) 7→ ha ∪ b, [X ]i.
Math. Research Letters 1 (1994) 809–822

Theorem (M. Taubes)
Let X and Y be two closed oriented smooth 4-manifolds suchthat b2+ (X ) > 0 and b2+ (Y ) > 0.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spacesTaubes Theorem
For a closed 4-manifold X , let b2+ (X ) be the number of positive
entries in a diagonalization of the intersection form of X over Q
H 2 (X , Q) × H 2 (X , Q) → Q(a, b) 7→ ha ∪ b, [X ]i.
Math. Research Letters 1 (1994) 809–822

Theorem (M. Taubes)
Let X and Y be two closed oriented smooth 4-manifolds suchthat b2+ (X ) > 0 and b2+ (Y ) > 0. Then the connected sum X #Y
is not a symplectic manifold.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spacesTaubes Theorem
For a closed 4-manifold X , let b2+ (X ) be the number of positive
entries in a diagonalization of the intersection form of X over Q
H 2 (X , Q) × H 2 (X , Q) → Q(a, b) 7→ ha ∪ b, [X ]i.
Math. Research Letters 1 (1994) 809–822

Theorem (M. Taubes)
Let X and Y be two closed oriented smooth 4-manifolds suchthat b2+ (X ) > 0 and b2+ (Y ) > 0. Then the connected sum X #Y
is not a symplectic manifold.
In particular, for k ­ 2,

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spacesTaubes Theorem
For a closed 4-manifold X , let b2+ (X ) be the number of positive
entries in a diagonalization of the intersection form of X over Q
H 2 (X , Q) × H 2 (X , Q) → Q(a, b) 7→ ha ∪ b, [X ]i.
Math. Research Letters 1 (1994) 809–822

Theorem (M. Taubes)
Let X and Y be two closed oriented smooth 4-manifolds suchthat b2+ (X ) > 0 and b2+ (Y ) > 0. Then the connected sum X #Y
is not a symplectic manifold.
In particular, for k ­ 2, the connected sum of k copies of CP 2 ,Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spacesTaubes Theorem
For a closed 4-manifold X , let b2+ (X ) be the number of positive
entries in a diagonalization of the intersection form of X over Q
H 2 (X , Q) × H 2 (X , Q) → Q(a, b) 7→ ha ∪ b, [X ]i.
Math. Research Letters 1 (1994) 809–822

Theorem (M. Taubes)
Let X and Y be two closed oriented smooth 4-manifolds suchthat b2+ (X ) > 0 and b2+ (Y ) > 0. Then the connected sum X #Y
is not a symplectic manifold.
In particular, for k ­ 2, the connected sum of k copies of CP 2 ,#k CP 2 = CP 2 # · · · #CP 2

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spacesTaubes Theorem
For a closed 4-manifold X , let b2+ (X ) be the number of positive
entries in a diagonalization of the intersection form of X over Q
H 2 (X , Q) × H 2 (X , Q) → Q(a, b) 7→ ha ∪ b, [X ]i.
Math. Research Letters 1 (1994) 809–822

Theorem (M. Taubes)
Let X and Y be two closed oriented smooth 4-manifolds suchthat b2+ (X ) > 0 and b2+ (Y ) > 0. Then the connected sum X #Y
is not a symplectic manifold.
In particular, for k ­ 2, the connected sum of k copies of CP 2 ,#k CP 2 = CP 2 # · · · #CP 2
is not a symplectic manifold.
Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spacesTaubes Theorem
For a closed 4-manifold X , let b2+ (X ) be the number of positive
entries in a diagonalization of the intersection form of X over Q
H 2 (X , Q) × H 2 (X , Q) → Q(a, b) 7→ ha ∪ b, [X ]i.
Math. Research Letters 1 (1994) 809–822

Theorem (M. Taubes)
Let X and Y be two closed oriented smooth 4-manifolds suchthat b2+ (X ) > 0 and b2+ (Y ) > 0. Then the connected sum X #Y
is not a symplectic manifold.
In particular, for k ­ 2, the connected sum of k copies of CP 2 ,#k CP 2 = CP 2 # · · · #CP 2
is not a symplectic manifold.
Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spacesNon-symplectic connected sum

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spacesNon-symplectic connected sum
J. Sympl. Geom. Vol. 10 (2012) 17–26.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spacesNon-symplectic connected sum
J. Sympl. Geom. Vol. 10 (2012) 17–26.

Proposition (M. Kaluba, W. Politarczyk)

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spacesNon-symplectic connected sum
J. Sympl. Geom. Vol. 10 (2012) 17–26.

Proposition (M. Kaluba, W. Politarczyk)
Let X and M be two closed oriented smooth 4-manifolds

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spacesNon-symplectic connected sum
J. Sympl. Geom. Vol. 10 (2012) 17–26.

Proposition (M. Kaluba, W. Politarczyk)
Let X and M be two closed oriented smooth 4-manifolds suchthat b2+ (X ) > 0

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spacesNon-symplectic connected sum
J. Sympl. Geom. Vol. 10 (2012) 17–26.

Proposition (M. Kaluba, W. Politarczyk)
Let X and M be two closed oriented smooth 4-manifolds suchthat b2+ (X ) > 0 and π1 (M) has a subgroup of finite index k > 1.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spacesNon-symplectic connected sum
J. Sympl. Geom. Vol. 10 (2012) 17–26.

Proposition (M. Kaluba, W. Politarczyk)
Let X and M be two closed oriented smooth 4-manifolds suchthat b2+ (X ) > 0 and π1 (M) has a subgroup of finite index k > 1.
Then the connected sum X #M is not a symplectic manifold.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spacesNon-symplectic connected sum
J. Sympl. Geom. Vol. 10 (2012) 17–26.

Proposition (M. Kaluba, W. Politarczyk)
Let X and M be two closed oriented smooth 4-manifolds suchthat b2+ (X ) > 0 and π1 (M) has a subgroup of finite index k > 1.
Then the connected sum X #M is not a symplectic manifold.
As π1 (M) has a subgroup of finite index k > 1,

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spacesNon-symplectic connected sum
J. Sympl. Geom. Vol. 10 (2012) 17–26.

Proposition (M. Kaluba, W. Politarczyk)
Let X and M be two closed oriented smooth 4-manifolds suchthat b2+ (X ) > 0 and π1 (M) has a subgroup of finite index k > 1.
Then the connected sum X #M is not a symplectic manifold.
As π1 (M) has a subgroup of finite index k > 1, there exists
f → M.
a k-sheeted covering M

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spacesNon-symplectic connected sum
J. Sympl. Geom. Vol. 10 (2012) 17–26.

Proposition (M. Kaluba, W. Politarczyk)
Let X and M be two closed oriented smooth 4-manifolds suchthat b2+ (X ) > 0 and π1 (M) has a subgroup of finite index k > 1.
Then the connected sum X #M is not a symplectic manifold.
As π1 (M) has a subgroup of finite index k > 1, there exists
f → M. Then the k-sheeted coveringa k-sheeted covering M
E → X #M has the form

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spacesNon-symplectic connected sum
J. Sympl. Geom. Vol. 10 (2012) 17–26.

Proposition (M. Kaluba, W. Politarczyk)
Let X and M be two closed oriented smooth 4-manifolds suchthat b2+ (X ) > 0 and π1 (M) has a subgroup of finite index k > 1.
Then the connected sum X #M is not a symplectic manifold.
As π1 (M) has a subgroup of finite index k > 1, there exists
f → M. Then the k-sheeted coveringa k-sheeted covering M
E → X #M has the formf
E = kX #M

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spacesNon-symplectic connected sum
J. Sympl. Geom. Vol. 10 (2012) 17–26.

Proposition (M. Kaluba, W. Politarczyk)
Let X and M be two closed oriented smooth 4-manifolds suchthat b2+ (X ) > 0 and π1 (M) has a subgroup of finite index k > 1.
Then the connected sum X #M is not a symplectic manifold.
As π1 (M) has a subgroup of finite index k > 1, there exists
f → M. Then the k-sheeted coveringa k-sheeted covering M
E → X #M has the form

f = X # (k − 1)X #M
f .
E = kX #M

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spacesNon-symplectic connected sum
J. Sympl. Geom. Vol. 10 (2012) 17–26.

Proposition (M. Kaluba, W. Politarczyk)
Let X and M be two closed oriented smooth 4-manifolds suchthat b2+ (X ) > 0 and π1 (M) has a subgroup of finite index k > 1.
Then the connected sum X #M is not a symplectic manifold.
As π1 (M) has a subgroup of finite index k > 1, there exists
f → M. Then the k-sheeted coveringa k-sheeted covering M
E → X #M has the form

f = X # (k − 1)X #M
f .
E = kX #M
f
Set Y = (k − 1)X #M.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spacesNon-symplectic connected sum
J. Sympl. Geom. Vol. 10 (2012) 17–26.

Proposition (M. Kaluba, W. Politarczyk)
Let X and M be two closed oriented smooth 4-manifolds suchthat b2+ (X ) > 0 and π1 (M) has a subgroup of finite index k > 1.
Then the connected sum X #M is not a symplectic manifold.
As π1 (M) has a subgroup of finite index k > 1, there exists
f → M. Then the k-sheeted coveringa k-sheeted covering M
E → X #M has the form

f = X # (k − 1)X #M
f .
E = kX #M
f As b + (Y ) ­ b + (X ) > 0,
Set Y = (k − 1)X #M.
2
2

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spacesNon-symplectic connected sum
J. Sympl. Geom. Vol. 10 (2012) 17–26.

Proposition (M. Kaluba, W. Politarczyk)
Let X and M be two closed oriented smooth 4-manifolds suchthat b2+ (X ) > 0 and π1 (M) has a subgroup of finite index k > 1.
Then the connected sum X #M is not a symplectic manifold.
As π1 (M) has a subgroup of finite index k > 1, there exists
f → M. Then the k-sheeted coveringa k-sheeted covering M
E → X #M has the form

f = X # (k − 1)X #M
f .
E = kX #M
f As b + (Y ) ­ b + (X ) > 0, it follows by
Set Y = (k − 1)X #M.
2
2
Taubes Theorem (Math. Research Letters 1 (1994) 809–822)

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spacesNon-symplectic connected sum
J. Sympl. Geom. Vol. 10 (2012) 17–26.

Proposition (M. Kaluba, W. Politarczyk)
Let X and M be two closed oriented smooth 4-manifolds suchthat b2+ (X ) > 0 and π1 (M) has a subgroup of finite index k > 1.
Then the connected sum X #M is not a symplectic manifold.
As π1 (M) has a subgroup of finite index k > 1, there exists
f → M. Then the k-sheeted coveringa k-sheeted covering M
E → X #M has the form

f = X # (k − 1)X #M
f .
E = kX #M
f As b + (Y ) ­ b + (X ) > 0, it follows by
Set Y = (k − 1)X #M.
2
2
Taubes Theorem (Math. Research Letters 1 (1994) 809–822)
that E is not symplectic
Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spacesNon-symplectic connected sum
J. Sympl. Geom. Vol. 10 (2012) 17–26.

Proposition (M. Kaluba, W. Politarczyk)
Let X and M be two closed oriented smooth 4-manifolds suchthat b2+ (X ) > 0 and π1 (M) has a subgroup of finite index k > 1.
Then the connected sum X #M is not a symplectic manifold.
As π1 (M) has a subgroup of finite index k > 1, there exists
f → M. Then the k-sheeted coveringa k-sheeted covering M
E → X #M has the form

f = X # (k − 1)X #M
f .
E = kX #M
f As b + (Y ) ­ b + (X ) > 0, it follows by
Set Y = (k − 1)X #M.
2
2
Taubes Theorem (Math. Research Letters 1 (1994) 809–822)
that E is not symplectic and so, X #M is not symplectic too.
Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Kaluba-Politarczyk Theorem

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Kaluba-Politarczyk Theorem
J. Sympl. Geom. Vol. 10 (2012) 17–26.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Kaluba-Politarczyk Theorem
J. Sympl. Geom. Vol. 10 (2012) 17–26.

Theorem (M. Kaluba, W. Politarczyk)

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Kaluba-Politarczyk Theorem
J. Sympl. Geom. Vol. 10 (2012) 17–26.

Theorem (M. Kaluba, W. Politarczyk)
For any compact Lie group G ,Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Kaluba-Politarczyk Theorem
J. Sympl. Geom. Vol. 10 (2012) 17–26.

Theorem (M. Kaluba, W. Politarczyk)
For any compact Lie group G , there exists a smooth action of G
on a complex projective space CP n

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Kaluba-Politarczyk Theorem
J. Sympl. Geom. Vol. 10 (2012) 17–26.

Theorem (M. Kaluba, W. Politarczyk)
For any compact Lie group G , there exists a smooth action of G
on a complex projective space CP n such that the fixed point set
contains a connected component

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Kaluba-Politarczyk Theorem
J. Sympl. Geom. Vol. 10 (2012) 17–26.

Theorem (M. Kaluba, W. Politarczyk)
For any compact Lie group G , there exists a smooth action of G
on a complex projective space CP n such that the fixed point set
contains a connected component diffeomorphic to CP 2 #Σ 4 .

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Kaluba-Politarczyk Theorem
J. Sympl. Geom. Vol. 10 (2012) 17–26.

Theorem (M. Kaluba, W. Politarczyk)
For any compact Lie group G , there exists a smooth action of G
on a complex projective space CP n such that the fixed point set
contains a connected component diffeomorphic to CP 2 #Σ 4 .
Construct a smooth action of G on a disk D 2n+5

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Kaluba-Politarczyk Theorem
J. Sympl. Geom. Vol. 10 (2012) 17–26.

Theorem (M. Kaluba, W. Politarczyk)
For any compact Lie group G , there exists a smooth action of G
on a complex projective space CP n such that the fixed point set
contains a connected component diffeomorphic to CP 2 #Σ 4 .
Construct a smooth action of G on a disk D 2n+5 withF (G  D 2n+5 ) ∼= ∆5 for a contractible compact smooth5
manifold ∆ of dimension 5,

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Kaluba-Politarczyk Theorem
J. Sympl. Geom. Vol. 10 (2012) 17–26.

Theorem (M. Kaluba, W. Politarczyk)
For any compact Lie group G , there exists a smooth action of G
on a complex projective space CP n such that the fixed point set
contains a connected component diffeomorphic to CP 2 #Σ 4 .
Construct a smooth action of G on a disk D 2n+5 withF (G  D 2n+5 ) ∼= ∆5 for a contractible compact smooth5
manifold ∆ of dimension 5, such that ∂∆5 = Σ 4 .

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Kaluba-Politarczyk Theorem
J. Sympl. Geom. Vol. 10 (2012) 17–26.

Theorem (M. Kaluba, W. Politarczyk)
For any compact Lie group G , there exists a smooth action of G
on a complex projective space CP n such that the fixed point set
contains a connected component diffeomorphic to CP 2 #Σ 4 .
Construct a smooth action of G on a disk D 2n+5 withF (G  D 2n+5 ) ∼= ∆5 for a contractible compact smooth5
manifold ∆ of dimension 5, such that ∂∆5 = Σ 4 .
The restricted action of G on ∂D 2n+5 = S 2n+4

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Kaluba-Politarczyk Theorem
J. Sympl. Geom. Vol. 10 (2012) 17–26.

Theorem (M. Kaluba, W. Politarczyk)
For any compact Lie group G , there exists a smooth action of G
on a complex projective space CP n such that the fixed point set
contains a connected component diffeomorphic to CP 2 #Σ 4 .
Construct a smooth action of G on a disk D 2n+5 withF (G  D 2n+5 ) ∼= ∆5 for a contractible compact smooth5
manifold ∆ of dimension 5, such that ∂∆5 = Σ 4 .
The restricted action of G on ∂D 2n+5 = S 2n+4 is a smoothaction of G on S 2n+4

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Kaluba-Politarczyk Theorem
J. Sympl. Geom. Vol. 10 (2012) 17–26.

Theorem (M. Kaluba, W. Politarczyk)
For any compact Lie group G , there exists a smooth action of G
on a complex projective space CP n such that the fixed point set
contains a connected component diffeomorphic to CP 2 #Σ 4 .
Construct a smooth action of G on a disk D 2n+5 withF (G  D 2n+5 ) ∼= ∆5 for a contractible compact smooth5
manifold ∆ of dimension 5, such that ∂∆5 = Σ 4 .
The restricted action of G on ∂D 2n+5 = S 2n+4 is a smoothaction of G on S 2n+4 such that F (G  S 2n+4 ) ∼= Σ4.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Kaluba-Politarczyk Theorem
J. Sympl. Geom. Vol. 10 (2012) 17–26.

Theorem (M. Kaluba, W. Politarczyk)
For any compact Lie group G , there exists a smooth action of G
on a complex projective space CP n such that the fixed point set
contains a connected component diffeomorphic to CP 2 #Σ 4 .
Construct a smooth action of G on a disk D 2n+5 withF (G  D 2n+5 ) ∼= ∆5 for a contractible compact smooth5
manifold ∆ of dimension 5, such that ∂∆5 = Σ 4 .
The restricted action of G on ∂D 2n+5 = S 2n+4 is a smoothaction of G on S 2n+4 such that F (G  S 2n+4 ) ∼
= Σ4.
n+2
2n+4
∼
The G -equivariant connected sum CP= CP n+2
c #x S

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Kaluba-Politarczyk Theorem
J. Sympl. Geom. Vol. 10 (2012) 17–26.

Theorem (M. Kaluba, W. Politarczyk)
For any compact Lie group G , there exists a smooth action of G
on a complex projective space CP n such that the fixed point set
contains a connected component diffeomorphic to CP 2 #Σ 4 .
Construct a smooth action of G on a disk D 2n+5 withF (G  D 2n+5 ) ∼= ∆5 for a contractible compact smooth5
manifold ∆ of dimension 5, such that ∂∆5 = Σ 4 .
The restricted action of G on ∂D 2n+5 = S 2n+4 is a smoothaction of G on S 2n+4 such that F (G  S 2n+4 ) ∼
= Σ4.
n+2
2n+4
∼
The G -equivariant connected sum CP= CP n+2
c #x Sat two G -fixed point c ∈ CP 2 ⊂ CP n+2 and x ∈ Σ 4 ⊂ S 2n+4

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Kaluba-Politarczyk Theorem
J. Sympl. Geom. Vol. 10 (2012) 17–26.

Theorem (M. Kaluba, W. Politarczyk)
For any compact Lie group G , there exists a smooth action of G
on a complex projective space CP n such that the fixed point set
contains a connected component diffeomorphic to CP 2 #Σ 4 .
Construct a smooth action of G on a disk D 2n+5 withF (G  D 2n+5 ) ∼= ∆5 for a contractible compact smooth5
manifold ∆ of dimension 5, such that ∂∆5 = Σ 4 .
The restricted action of G on ∂D 2n+5 = S 2n+4 is a smoothaction of G on S 2n+4 such that F (G  S 2n+4 ) ∼
= Σ4.
n+2
2n+4
∼
The G -equivariant connected sum CP= CP n+2
c #x Sat two G -fixed point c ∈ CP 2 ⊂ CP n+2 and x ∈ Σ 4 ⊂ S 2n+4
yields a smooth action of G on CP n+2

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Kaluba-Politarczyk Theorem
J. Sympl. Geom. Vol. 10 (2012) 17–26.

Theorem (M. Kaluba, W. Politarczyk)
For any compact Lie group G , there exists a smooth action of G
on a complex projective space CP n such that the fixed point set
contains a connected component diffeomorphic to CP 2 #Σ 4 .
Construct a smooth action of G on a disk D 2n+5 withF (G  D 2n+5 ) ∼= ∆5 for a contractible compact smooth5
manifold ∆ of dimension 5, such that ∂∆5 = Σ 4 .
The restricted action of G on ∂D 2n+5 = S 2n+4 is a smoothaction of G on S 2n+4 such that F (G  S 2n+4 ) ∼
= Σ4.
n+2
2n+4
∼
The G -equivariant connected sum CP= CP n+2
c #x Sat two G -fixed point c ∈ CP 2 ⊂ CP n+2 and x ∈ Σ 4 ⊂ S 2n+4
yields a smooth action of G on CP n+2 such that
F (G  CP n+2 ) = CP 2 #Σ 4 .
Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Proof of Kaluba–Politarczyk Theorem

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Proof of Kaluba–Politarczyk Theorem
Trans. Amer. Math. Soc. 144 (1969) 67–72

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Proof of Kaluba–Politarczyk Theorem
Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Proof of Kaluba–Politarczyk Theorem
Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)
Any homology 4-sphere is diffeomorphic to the boundary ofa compact contractible smooth 5-manifold.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Proof of Kaluba–Politarczyk Theorem
Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)
Any homology 4-sphere is diffeomorphic to the boundary ofa compact contractible smooth 5-manifold.
In particular, Sato’s homology 4-sphere Σ 4 bounds a compact
contractible smooth 5-manifold ∆5 .

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Proof of Kaluba–Politarczyk Theorem
Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)
Any homology 4-sphere is diffeomorphic to the boundary ofa compact contractible smooth 5-manifold.
In particular, Sato’s homology 4-sphere Σ 4 bounds a compact
contractible smooth 5-manifold ∆5 . As π1 (∂∆5 ) ∼= SL2 (F5 ),Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Proof of Kaluba–Politarczyk Theorem
Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)
Any homology 4-sphere is diffeomorphic to the boundary ofa compact contractible smooth 5-manifold.
In particular, Sato’s homology 4-sphere Σ 4 bounds a compact
contractible smooth 5-manifold ∆5 . As π1 (∂∆5 ) ∼= SL2 (F5 ),∆5 is not homeomorphic to D 5 .

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Proof of Kaluba–Politarczyk Theorem
Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)
Any homology 4-sphere is diffeomorphic to the boundary ofa compact contractible smooth 5-manifold.
In particular, Sato’s homology 4-sphere Σ 4 bounds a compact
contractible smooth 5-manifold ∆5 . As π1 (∂∆5 ) ∼= SL2 (F5 ),∆5 is not homeomorphic to D 5 .
Let G be a compact Lie group.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Proof of Kaluba–Politarczyk Theorem
Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)
Any homology 4-sphere is diffeomorphic to the boundary ofa compact contractible smooth 5-manifold.
In particular, Sato’s homology 4-sphere Σ 4 bounds a compact
contractible smooth 5-manifold ∆5 . As π1 (∂∆5 ) ∼= SL2 (F5 ),∆5 is not homeomorphic to D 5 .
Let G be a compact Lie group. As ∆5 is contractible,

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Proof of Kaluba–Politarczyk Theorem
Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)
Any homology 4-sphere is diffeomorphic to the boundary ofa compact contractible smooth 5-manifold.
In particular, Sato’s homology 4-sphere Σ 4 bounds a compact
contractible smooth 5-manifold ∆5 . As π1 (∂∆5 ) ∼= SL2 (F5 ),∆5 is not homeomorphic to D 5 .
Let G be a compact Lie group. As ∆5 is contractible, so is the join
X = G ∗ ∆5 ,Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Proof of Kaluba–Politarczyk Theorem
Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)
Any homology 4-sphere is diffeomorphic to the boundary ofa compact contractible smooth 5-manifold.
In particular, Sato’s homology 4-sphere Σ 4 bounds a compact
contractible smooth 5-manifold ∆5 . As π1 (∂∆5 ) ∼= SL2 (F5 ),∆5 is not homeomorphic to D 5 .
Let G be a compact Lie group. As ∆5 is contractible, so is the join
X = G ∗ ∆5 , which admits the structure of a finite G -CW complex
such that F (G  X ) = ∆5 .

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Proof of Kaluba–Politarczyk Theorem
Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)
Any homology 4-sphere is diffeomorphic to the boundary ofa compact contractible smooth 5-manifold.
In particular, Sato’s homology 4-sphere Σ 4 bounds a compact
contractible smooth 5-manifold ∆5 . As π1 (∂∆5 ) ∼= SL2 (F5 ),∆5 is not homeomorphic to D 5 .
Let G be a compact Lie group. As ∆5 is contractible, so is the join
X = G ∗ ∆5 , which admits the structure of a finite G -CW complex
such that F (G  X ) = ∆5 . By using the product G -vector bundle
X × V n over X

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Proof of Kaluba–Politarczyk Theorem
Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)
Any homology 4-sphere is diffeomorphic to the boundary ofa compact contractible smooth 5-manifold.
In particular, Sato’s homology 4-sphere Σ 4 bounds a compact
contractible smooth 5-manifold ∆5 . As π1 (∂∆5 ) ∼= SL2 (F5 ),∆5 is not homeomorphic to D 5 .
Let G be a compact Lie group. As ∆5 is contractible, so is the join
X = G ∗ ∆5 , which admits the structure of a finite G -CW complex
such that F (G  X ) = ∆5 . By using the product G -vector bundle
X × V n over X for an appropriate complex G -module V n ,Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Proof of Kaluba–Politarczyk Theorem
Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)
Any homology 4-sphere is diffeomorphic to the boundary ofa compact contractible smooth 5-manifold.
In particular, Sato’s homology 4-sphere Σ 4 bounds a compact
contractible smooth 5-manifold ∆5 . As π1 (∂∆5 ) ∼= SL2 (F5 ),∆5 is not homeomorphic to D 5 .
Let G be a compact Lie group. As ∆5 is contractible, so is the join
X = G ∗ ∆5 , which admits the structure of a finite G -CW complex
such that F (G  X ) = ∆5 . By using the product G -vector bundle
X × V n over X for an appropriate complex G -module V n , we canthicken up X into a disk D 2n+5

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Proof of Kaluba–Politarczyk Theorem
Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)
Any homology 4-sphere is diffeomorphic to the boundary ofa compact contractible smooth 5-manifold.
In particular, Sato’s homology 4-sphere Σ 4 bounds a compact
contractible smooth 5-manifold ∆5 . As π1 (∂∆5 ) ∼= SL2 (F5 ),∆5 is not homeomorphic to D 5 .
Let G be a compact Lie group. As ∆5 is contractible, so is the join
X = G ∗ ∆5 , which admits the structure of a finite G -CW complex
such that F (G  X ) = ∆5 . By using the product G -vector bundle
X × V n over X for an appropriate complex G -module V n , we canthicken up X into a disk D 2n+5 equipped with a smooth action ofG

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Proof of Kaluba–Politarczyk Theorem
Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)
Any homology 4-sphere is diffeomorphic to the boundary ofa compact contractible smooth 5-manifold.
In particular, Sato’s homology 4-sphere Σ 4 bounds a compact
contractible smooth 5-manifold ∆5 . As π1 (∂∆5 ) ∼= SL2 (F5 ),∆5 is not homeomorphic to D 5 .
Let G be a compact Lie group. As ∆5 is contractible, so is the join
X = G ∗ ∆5 , which admits the structure of a finite G -CW complex
such that F (G  X ) = ∆5 . By using the product G -vector bundle
X × V n over X for an appropriate complex G -module V n , we canthicken up X into a disk D 2n+5 equipped with a smooth action of
G such that F (G  D 2n+5 ) = ∆5

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Proof of Kaluba–Politarczyk Theorem
Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)
Any homology 4-sphere is diffeomorphic to the boundary ofa compact contractible smooth 5-manifold.
In particular, Sato’s homology 4-sphere Σ 4 bounds a compact
contractible smooth 5-manifold ∆5 . As π1 (∂∆5 ) ∼= SL2 (F5 ),∆5 is not homeomorphic to D 5 .
Let G be a compact Lie group. As ∆5 is contractible, so is the join
X = G ∗ ∆5 , which admits the structure of a finite G -CW complex
such that F (G  X ) = ∆5 . By using the product G -vector bundle
X × V n over X for an appropriate complex G -module V n , we canthicken up X into a disk D 2n+5 equipped with a smooth action of
G such that F (G  D 2n+5 ) = ∆5 and at any x ∈ F (G  D 2n+5 ),Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Proof of Kaluba–Politarczyk Theorem
Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)
Any homology 4-sphere is diffeomorphic to the boundary ofa compact contractible smooth 5-manifold.
In particular, Sato’s homology 4-sphere Σ 4 bounds a compact
contractible smooth 5-manifold ∆5 . As π1 (∂∆5 ) ∼= SL2 (F5 ),∆5 is not homeomorphic to D 5 .
Let G be a compact Lie group. As ∆5 is contractible, so is the join
X = G ∗ ∆5 , which admits the structure of a finite G -CW complex
such that F (G  X ) = ∆5 . By using the product G -vector bundle
X × V n over X for an appropriate complex G -module V n , we canthicken up X into a disk D 2n+5 equipped with a smooth action of
G such that F (G  D 2n+5 ) = ∆5 and at any x ∈ F (G  D 2n+5 ),the normal G -module is the realification of V n .
Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Proof of Kaluba–Politarczyk Theorem

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Proof of Kaluba–Politarczyk Theorem
Therefore, we obtain a smooth action of G on the sphere S 2n+4

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Proof of Kaluba–Politarczyk Theorem
Therefore, we obtain a smooth action of G on the sphere S 2n+4
such that F (G  S 2n+4 ) is diffeomorphic to Σ 4

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Proof of Kaluba–Politarczyk Theorem
Therefore, we obtain a smooth action of G on the sphere S 2n+4
such that F (G  S 2n+4 ) is diffeomorphic to Σ 4 and at any pointx ∈ F (G  S 2n+4 ),Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Proof of Kaluba–Politarczyk Theorem
Therefore, we obtain a smooth action of G on the sphere S 2n+4
such that F (G  S 2n+4 ) is diffeomorphic to Σ 4 and at any pointx ∈ F (G  S 2n+4 ), the normal G -module is the realification of V n .

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Proof of Kaluba–Politarczyk Theorem
Therefore, we obtain a smooth action of G on the sphere S 2n+4
such that F (G  S 2n+4 ) is diffeomorphic to Σ 4 and at any pointx ∈ F (G  S 2n+4 ), the normal G -module is the realification of V n .

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Proof of Kaluba–Politarczyk Theorem
Therefore, we obtain a smooth action of G on the sphere S 2n+4
such that F (G  S 2n+4 ) is diffeomorphic to Σ 4 and at any pointx ∈ F (G  S 2n+4 ), the normal G -module is the realification of V n .
Let S 2n+5 = S(V n ⊕ C3 ) be equipped with the linear action of G ,Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Proof of Kaluba–Politarczyk Theorem
Therefore, we obtain a smooth action of G on the sphere S 2n+4
such that F (G  S 2n+4 ) is diffeomorphic to Σ 4 and at any pointx ∈ F (G  S 2n+4 ), the normal G -module is the realification of V n .
Let S 2n+5 = S(V n ⊕ C3 ) be equipped with the linear action of G ,where G acts trivially on C3 .

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Proof of Kaluba–Politarczyk Theorem
Therefore, we obtain a smooth action of G on the sphere S 2n+4
such that F (G  S 2n+4 ) is diffeomorphic to Σ 4 and at any pointx ∈ F (G  S 2n+4 ), the normal G -module is the realification of V n .
Let S 2n+5 = S(V n ⊕ C3 ) be equipped with the linear action of G ,where G acts trivially on C3 . Then CP n+2 = S(V n ⊕ C3 )/S 1

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Proof of Kaluba–Politarczyk Theorem
Therefore, we obtain a smooth action of G on the sphere S 2n+4
such that F (G  S 2n+4 ) is diffeomorphic to Σ 4 and at any pointx ∈ F (G  S 2n+4 ), the normal G -module is the realification of V n .
Let S 2n+5 = S(V n ⊕ C3 ) be equipped with the linear action of G ,where G acts trivially on C3 . Then CP n+2 = S(V n ⊕ C3 )/S 1 hasa smooth action of G

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Proof of Kaluba–Politarczyk Theorem
Therefore, we obtain a smooth action of G on the sphere S 2n+4
such that F (G  S 2n+4 ) is diffeomorphic to Σ 4 and at any pointx ∈ F (G  S 2n+4 ), the normal G -module is the realification of V n .
Let S 2n+5 = S(V n ⊕ C3 ) be equipped with the linear action of G ,where G acts trivially on C3 . Then CP n+2 = S(V n ⊕ C3 )/S 1 hasa smooth action of G such that F (G  CP n+2 ) ⊃ CP 2 as someconnected component

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Proof of Kaluba–Politarczyk Theorem
Therefore, we obtain a smooth action of G on the sphere S 2n+4
such that F (G  S 2n+4 ) is diffeomorphic to Σ 4 and at any pointx ∈ F (G  S 2n+4 ), the normal G -module is the realification of V n .
Let S 2n+5 = S(V n ⊕ C3 ) be equipped with the linear action of G ,where G acts trivially on C3 . Then CP n+2 = S(V n ⊕ C3 )/S 1 hasa smooth action of G such that F (G  CP n+2 ) ⊃ CP 2 as someconnected component and at any point c ∈ CP 2 ,Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Proof of Kaluba–Politarczyk Theorem
Therefore, we obtain a smooth action of G on the sphere S 2n+4
such that F (G  S 2n+4 ) is diffeomorphic to Σ 4 and at any pointx ∈ F (G  S 2n+4 ), the normal G -module is the realification of V n .
Let S 2n+5 = S(V n ⊕ C3 ) be equipped with the linear action of G ,where G acts trivially on C3 . Then CP n+2 = S(V n ⊕ C3 )/S 1 hasa smooth action of G such that F (G  CP n+2 ) ⊃ CP 2 as someconnected component and at any point c ∈ CP 2 , the normal
G -module is the realification of V n .

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Proof of Kaluba–Politarczyk Theorem
Therefore, we obtain a smooth action of G on the sphere S 2n+4
such that F (G  S 2n+4 ) is diffeomorphic to Σ 4 and at any pointx ∈ F (G  S 2n+4 ), the normal G -module is the realification of V n .
Let S 2n+5 = S(V n ⊕ C3 ) be equipped with the linear action of G ,where G acts trivially on C3 . Then CP n+2 = S(V n ⊕ C3 )/S 1 hasa smooth action of G such that F (G  CP n+2 ) ⊃ CP 2 as someconnected component and at any point c ∈ CP 2 , the normal
G -module is the realification of V n .
By forming the G -equivariant connected sum at x and c,
CP n+2 c #x S 2n+4 ∼= CP n+2

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Proof of Kaluba–Politarczyk Theorem
Therefore, we obtain a smooth action of G on the sphere S 2n+4
such that F (G  S 2n+4 ) is diffeomorphic to Σ 4 and at any pointx ∈ F (G  S 2n+4 ), the normal G -module is the realification of V n .
Let S 2n+5 = S(V n ⊕ C3 ) be equipped with the linear action of G ,where G acts trivially on C3 . Then CP n+2 = S(V n ⊕ C3 )/S 1 hasa smooth action of G such that F (G  CP n+2 ) ⊃ CP 2 as someconnected component and at any point c ∈ CP 2 , the normal
G -module is the realification of V n .
By forming the G -equivariant connected sum at x and c,
CP n+2 c #x S 2n+4 ∼= CP n+2
we get a smooth action of G on CP n+2 such that F (G  CP n+2 )Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Proof of Kaluba–Politarczyk Theorem
Therefore, we obtain a smooth action of G on the sphere S 2n+4
such that F (G  S 2n+4 ) is diffeomorphic to Σ 4 and at any pointx ∈ F (G  S 2n+4 ), the normal G -module is the realification of V n .
Let S 2n+5 = S(V n ⊕ C3 ) be equipped with the linear action of G ,where G acts trivially on C3 . Then CP n+2 = S(V n ⊕ C3 )/S 1 hasa smooth action of G such that F (G  CP n+2 ) ⊃ CP 2 as someconnected component and at any point c ∈ CP 2 , the normal
G -module is the realification of V n .
By forming the G -equivariant connected sum at x and c,
CP n+2 c #x S 2n+4 ∼= CP n+2
we get a smooth action of G on CP n+2 such that F (G  CP n+2 )contains a connected component diffeomorphic to CP 2 #Σ 4 .

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Main Theorem

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Main Theorem
Main Theorem

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Main Theorem
Main Theorem
Let G be a compact Lie group such that G0 is nonabelian

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Main Theorem
Main Theorem
Let G be a compact Lie group such that G0 is nonabelian or
G /G0 is not of prime power order.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Main Theorem
Main Theorem
Let G be a compact Lie group such that G0 is nonabelian or
G /G0 is not of prime power order. There exists a smooth actionof G on some complex projective space CP d+n

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Main Theorem
Main Theorem
Let G be a compact Lie group such that G0 is nonabelian or
G /G0 is not of prime power order. There exists a smooth actionof G on some complex projective space CP d+n such that the
fixed point set F (G  CP d+n )Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Main Theorem
Main Theorem
Let G be a compact Lie group such that G0 is nonabelian or
G /G0 is not of prime power order. There exists a smooth actionof G on some complex projective space CP d+n such that the
fixed point set F (G  CP d+n ) contains connected components
M of dimension 2d which are:Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Main Theorem
Main Theorem
Let G be a compact Lie group such that G0 is nonabelian or
G /G0 is not of prime power order. There exists a smooth actionof G on some complex projective space CP d+n such that the
fixed point set F (G  CP d+n ) contains connected components
M of dimension 2d which are:stable almost complex and not almost complex,

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Main Theorem
Main Theorem
Let G be a compact Lie group such that G0 is nonabelian or
G /G0 is not of prime power order. There exists a smooth actionof G on some complex projective space CP d+n such that the
fixed point set F (G  CP d+n ) contains connected components
M of dimension 2d which are:stable almost complex and not almost complex,e.g., S 4 , S 8 , S 10 , S 12 , . . . ,Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Main Theorem
Main Theorem
Let G be a compact Lie group such that G0 is nonabelian or
G /G0 is not of prime power order. There exists a smooth actionof G on some complex projective space CP d+n such that the
fixed point set F (G  CP d+n ) contains connected components
M of dimension 2d which are:stable almost complex and not almost complex,e.g., S 4 , S 8 , S 10 , S 12 , . . . ,almost complex and not homotopically symplectic,

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Main Theorem
Main Theorem
Let G be a compact Lie group such that G0 is nonabelian or
G /G0 is not of prime power order. There exists a smooth actionof G on some complex projective space CP d+n such that the
fixed point set F (G  CP d+n ) contains connected components
M of dimension 2d which are:stable almost complex and not almost complex,e.g., S 4 , S 8 , S 10 , S 12 , . . . ,almost complex and not homotopically symplectic,
e.g., S 6 , S 2 × S 4 , S 2 × S 6 , S 6 × S 6 , and S 2m+1 × S 2n+1 ,homotopically symplectic and not symplectic,

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Main Theorem
Main Theorem
Let G be a compact Lie group such that G0 is nonabelian or
G /G0 is not of prime power order. There exists a smooth actionof G on some complex projective space CP d+n such that the
fixed point set F (G  CP d+n ) contains connected components
M of dimension 2d which are:stable almost complex and not almost complex,e.g., S 4 , S 8 , S 10 , S 12 , . . . ,almost complex and not homotopically symplectic,
e.g., S 6 , S 2 × S 4 , S 2 × S 6 , S 6 × S 6 , and S 2m+1 × S 2n+1 ,homotopically symplectic and not symplectic,
e.g., CP 2 #Σ 4 , where Σ 4 is Sato’s homology sphere.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Main Theorem
Main Theorem
Let G be a compact Lie group such that G0 is nonabelian or
G /G0 is not of prime power order. There exists a smooth actionof G on some complex projective space CP d+n such that the
fixed point set F (G  CP d+n ) contains connected components
M of dimension 2d which are:stable almost complex and not almost complex,e.g., S 4 , S 8 , S 10 , S 12 , . . . ,almost complex and not homotopically symplectic,
e.g., S 6 , S 2 × S 4 , S 2 × S 6 , S 6 × S 6 , and S 2m+1 × S 2n+1 ,homotopically symplectic and not symplectic,
e.g., CP 2 #Σ 4 , where Σ 4 is Sato’s homology sphere.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The idea of the proof of Main Theorem

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The idea of the proof of Main Theorem
The case where a connected component M of the fixed point set

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The idea of the proof of Main Theorem
The case where a connected component M of the fixed point set
F (G  CP d+n ) is homotopically symplectic and not symplectic

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The idea of the proof of Main Theorem
The case where a connected component M of the fixed point set
F (G  CP d+n ) is homotopically symplectic and not symplectic iscovered by the Kaluba-Politarczyk Theorem.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The idea of the proof of Main Theorem
The case where a connected component M of the fixed point set
F (G  CP d+n ) is homotopically symplectic and not symplectic iscovered by the Kaluba-Politarczyk Theorem. In the remaining twocases of M,

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The idea of the proof of Main Theorem
The case where a connected component M of the fixed point set
F (G  CP d+n ) is homotopically symplectic and not symplectic iscovered by the Kaluba-Politarczyk Theorem. In the remaining twocases of M, we argue as follows.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The idea of the proof of Main Theorem
The case where a connected component M of the fixed point set
F (G  CP d+n ) is homotopically symplectic and not symplectic iscovered by the Kaluba-Politarczyk Theorem. In the remaining twocases of M, we argue as follows.
First, we construct a smooth action of G on S 2d+2n such thatKrzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The idea of the proof of Main Theorem
The case where a connected component M of the fixed point set
F (G  CP d+n ) is homotopically symplectic and not symplectic iscovered by the Kaluba-Politarczyk Theorem. In the remaining twocases of M, we argue as follows.
First, we construct a smooth action of G on S 2d+2n such that
F (G  S 2d+2n ) contains two connected components:

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The idea of the proof of Main Theorem
The case where a connected component M of the fixed point set
F (G  CP d+n ) is homotopically symplectic and not symplectic iscovered by the Kaluba-Politarczyk Theorem. In the remaining twocases of M, we argue as follows.
First, we construct a smooth action of G on S 2d+2n such that
F (G  S 2d+2n ) contains two connected components: one is
the sphere S 2d

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The idea of the proof of Main Theorem
The case where a connected component M of the fixed point set
F (G  CP d+n ) is homotopically symplectic and not symplectic iscovered by the Kaluba-Politarczyk Theorem. In the remaining twocases of M, we argue as follows.
First, we construct a smooth action of G on S 2d+2n such that
F (G  S 2d+2n ) contains two connected components: one is
the sphere S 2d and the other one is the manifold M.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The idea of the proof of Main Theorem
The case where a connected component M of the fixed point set
F (G  CP d+n ) is homotopically symplectic and not symplectic iscovered by the Kaluba-Politarczyk Theorem. In the remaining twocases of M, we argue as follows.
First, we construct a smooth action of G on S 2d+2n such that
F (G  S 2d+2n ) contains two connected components: one is
the sphere S 2d and the other one is the manifold M.
Moreover, at any point x ∈ S 2d ,Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The idea of the proof of Main Theorem
The case where a connected component M of the fixed point set
F (G  CP d+n ) is homotopically symplectic and not symplectic iscovered by the Kaluba-Politarczyk Theorem. In the remaining twocases of M, we argue as follows.
First, we construct a smooth action of G on S 2d+2n such that
F (G  S 2d+2n ) contains two connected components: one is
the sphere S 2d and the other one is the manifold M.
Moreover, at any point x ∈ S 2d , the normal G -module is

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The idea of the proof of Main Theorem
The case where a connected component M of the fixed point set
F (G  CP d+n ) is homotopically symplectic and not symplectic iscovered by the Kaluba-Politarczyk Theorem. In the remaining twocases of M, we argue as follows.
First, we construct a smooth action of G on S 2d+2n such that
F (G  S 2d+2n ) contains two connected components: one is
the sphere S 2d and the other one is the manifold M.
Moreover, at any point x ∈ S 2d , the normal G -module is
the realification of a complex n-dimensional G -module V n .

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The idea of the proof of Main Theorem
The case where a connected component M of the fixed point set
F (G  CP d+n ) is homotopically symplectic and not symplectic iscovered by the Kaluba-Politarczyk Theorem. In the remaining twocases of M, we argue as follows.
First, we construct a smooth action of G on S 2d+2n such that
F (G  S 2d+2n ) contains two connected components: one is
the sphere S 2d and the other one is the manifold M.
Moreover, at any point x ∈ S 2d , the normal G -module is
the realification of a complex n-dimensional G -module V n .
Then, we consider the equivariant projectivization

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The idea of the proof of Main Theorem
The case where a connected component M of the fixed point set
F (G  CP d+n ) is homotopically symplectic and not symplectic iscovered by the Kaluba-Politarczyk Theorem. In the remaining twocases of M, we argue as follows.
First, we construct a smooth action of G on S 2d+2n such that
F (G  S 2d+2n ) contains two connected components: one is
the sphere S 2d and the other one is the manifold M.
Moreover, at any point x ∈ S 2d , the normal G -module is
the realification of a complex n-dimensional G -module V n .
Then, we consider the equivariant projectivization
CP d+n = P(Cd ⊕ C1 ⊕ V n ) = S(Cd ⊕ C1 ⊕ V n )/S 1 .

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The idea of the proof of Main Theorem
The case where a connected component M of the fixed point set
F (G  CP d+n ) is homotopically symplectic and not symplectic iscovered by the Kaluba-Politarczyk Theorem. In the remaining twocases of M, we argue as follows.
First, we construct a smooth action of G on S 2d+2n such that
F (G  S 2d+2n ) contains two connected components: one is
the sphere S 2d and the other one is the manifold M.
Moreover, at any point x ∈ S 2d , the normal G -module is
the realification of a complex n-dimensional G -module V n .
Then, we consider the equivariant projectivization
CP d+n = P(Cd ⊕ C1 ⊕ V n ) = S(Cd ⊕ C1 ⊕ V n )/S 1 .
Hence, F (G  CP d+n ) ⊃ CP d as a connected component

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The idea of the proof of Main Theorem
The case where a connected component M of the fixed point set
F (G  CP d+n ) is homotopically symplectic and not symplectic iscovered by the Kaluba-Politarczyk Theorem. In the remaining twocases of M, we argue as follows.
First, we construct a smooth action of G on S 2d+2n such that
F (G  S 2d+2n ) contains two connected components: one is
the sphere S 2d and the other one is the manifold M.
Moreover, at any point x ∈ S 2d , the normal G -module is
the realification of a complex n-dimensional G -module V n .
Then, we consider the equivariant projectivization
CP d+n = P(Cd ⊕ C1 ⊕ V n ) = S(Cd ⊕ C1 ⊕ V n )/S 1 .
Hence, F (G  CP d+n ) ⊃ CP d as a connected component
and V n occurs as the normal G -module at any c ∈ CP d .

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The idea of the proof of Main Theorem
The case where a connected component M of the fixed point set
F (G  CP d+n ) is homotopically symplectic and not symplectic iscovered by the Kaluba-Politarczyk Theorem. In the remaining twocases of M, we argue as follows.
First, we construct a smooth action of G on S 2d+2n such that
F (G  S 2d+2n ) contains two connected components: one is
the sphere S 2d and the other one is the manifold M.
Moreover, at any point x ∈ S 2d , the normal G -module is
the realification of a complex n-dimensional G -module V n .
Then, we consider the equivariant projectivization
CP d+n = P(Cd ⊕ C1 ⊕ V n ) = S(Cd ⊕ C1 ⊕ V n )/S 1 .
Hence, F (G  CP d+n ) ⊃ CP d as a connected component
and V n occurs as the normal G -module at any c ∈ CP d .
Now, we form CP d+n c #x S 2d+2n ∼= CP d+n

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The idea of the proof of Main Theorem
The case where a connected component M of the fixed point set
F (G  CP d+n ) is homotopically symplectic and not symplectic iscovered by the Kaluba-Politarczyk Theorem. In the remaining twocases of M, we argue as follows.
First, we construct a smooth action of G on S 2d+2n such that
F (G  S 2d+2n ) contains two connected components: one is
the sphere S 2d and the other one is the manifold M.
Moreover, at any point x ∈ S 2d , the normal G -module is
the realification of a complex n-dimensional G -module V n .
Then, we consider the equivariant projectivization
CP d+n = P(Cd ⊕ C1 ⊕ V n ) = S(Cd ⊕ C1 ⊕ V n )/S 1 .
Hence, F (G  CP d+n ) ⊃ CP d as a connected component
and V n occurs as the normal G -module at any c ∈ CP d .
Now, we form CP d+n c #x S 2d+2n ∼= CP d+n at two pointsc ∈ CP d ⊂ CP d+n and x ∈ S 2d ⊂ S 2d+2n
Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The idea of the proof of Main Theorem
The case where a connected component M of the fixed point set
F (G  CP d+n ) is homotopically symplectic and not symplectic iscovered by the Kaluba-Politarczyk Theorem. In the remaining twocases of M, we argue as follows.
First, we construct a smooth action of G on S 2d+2n such that
F (G  S 2d+2n ) contains two connected components: one is
the sphere S 2d and the other one is the manifold M.
Moreover, at any point x ∈ S 2d , the normal G -module is
the realification of a complex n-dimensional G -module V n .
Then, we consider the equivariant projectivization
CP d+n = P(Cd ⊕ C1 ⊕ V n ) = S(Cd ⊕ C1 ⊕ V n )/S 1 .
Hence, F (G  CP d+n ) ⊃ CP d as a connected component
and V n occurs as the normal G -module at any c ∈ CP d .
Now, we form CP d+n c #x S 2d+2n ∼= CP d+n at two pointsc ∈ CP d ⊂ CP d+n and x ∈ S 2d ⊂ S 2d+2n to obtain that
F (G  CP d+n ) ⊃ M as a connected component.
Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Doubling of actions
In order to obtain the required smooth action of G on S 2d+2n ,Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Doubling of actions
In order to obtain the required smooth action of G on S 2d+2n ,first we construct a smooth action of G on D 2d+2n

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Doubling of actions
In order to obtain the required smooth action of G on S 2d+2n ,first we construct a smooth action of G on D 2d+2n such that
F (G  D 2d+2n ) ⊃ D 2d t M

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Doubling of actions
In order to obtain the required smooth action of G on S 2d+2n ,first we construct a smooth action of G on D 2d+2n such that
F (G  D 2d+2n ) ⊃ D 2d t M with ∂M = ∅.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Doubling of actions
In order to obtain the required smooth action of G on S 2d+2n ,first we construct a smooth action of G on D 2d+2n such that
F (G  D 2d+2n ) ⊃ D 2d t M with ∂M = ∅.
By forming the G -equivariant double

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Doubling of actions
In order to obtain the required smooth action of G on S 2d+2n ,first we construct a smooth action of G on D 2d+2n such that
F (G  D 2d+2n ) ⊃ D 2d t M with ∂M = ∅.
By forming the G -equivariant double∂(D 2d+2n × D 1 ) = S 2d+2n ,Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Doubling of actions
In order to obtain the required smooth action of G on S 2d+2n ,first we construct a smooth action of G on D 2d+2n such that
F (G  D 2d+2n ) ⊃ D 2d t M with ∂M = ∅.
By forming the G -equivariant double∂(D 2d+2n × D 1 ) = S 2d+2n ,we obtain a smooth action of G on S 2d+2n

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Doubling of actions
In order to obtain the required smooth action of G on S 2d+2n ,first we construct a smooth action of G on D 2d+2n such that
F (G  D 2d+2n ) ⊃ D 2d t M with ∂M = ∅.
By forming the G -equivariant double∂(D 2d+2n × D 1 ) = S 2d+2n ,we obtain a smooth action of G on S 2d+2n such that
F (G  S 2d+2n ) ⊃ S 2d t M t M.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Doubling of actions
In order to obtain the required smooth action of G on S 2d+2n ,first we construct a smooth action of G on D 2d+2n such that
F (G  D 2d+2n ) ⊃ D 2d t M with ∂M = ∅.
By forming the G -equivariant double∂(D 2d+2n × D 1 ) = S 2d+2n ,we obtain a smooth action of G on S 2d+2n such that
F (G  S 2d+2n ) ⊃ S 2d t M t M.

As far as the action of G on D 2n+2d is concerned,
Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Doubling of actions
In order to obtain the required smooth action of G on S 2d+2n ,first we construct a smooth action of G on D 2d+2n such that
F (G  D 2d+2n ) ⊃ D 2d t M with ∂M = ∅.
By forming the G -equivariant double∂(D 2d+2n × D 1 ) = S 2d+2n ,we obtain a smooth action of G on S 2d+2n such that
F (G  S 2d+2n ) ⊃ S 2d t M t M.

As far as the action of G on D 2n+2d is concerned, it is obtainedby the equivariant thickening technique.
Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Doubling of actions
In order to obtain the required smooth action of G on S 2d+2n ,first we construct a smooth action of G on D 2d+2n such that
F (G  D 2d+2n ) ⊃ D 2d t M with ∂M = ∅.
By forming the G -equivariant double∂(D 2d+2n × D 1 ) = S 2d+2n ,we obtain a smooth action of G on S 2d+2n such that
F (G  S 2d+2n ) ⊃ S 2d t M t M.

As far as the action of G on D 2n+2d is concerned, it is obtainedby the equivariant thickening technique.
Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The equivariant thickening techniqueKrzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The equivariant thickening technique

Topology 28 (1989) 273–289

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The equivariant thickening technique

Topology 28 (1989) 273–289

Theorem (K. Pawałowski)

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The equivariant thickening technique

Topology 28 (1989) 273–289

Theorem (K. Pawałowski)
Let G be a compact Lie group.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The equivariant thickening technique

Topology 28 (1989) 273–289

Theorem (K. Pawałowski)
Let G be a compact Lie group. Let M be a compact smoothmanifold.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The equivariant thickening technique

Topology 28 (1989) 273–289

Theorem (K. Pawałowski)
Let G be a compact Lie group. Let M be a compact smoothmanifold. Let X be a finite contractible G -CW complex withF (G  X ) = D 2d t M.

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The equivariant thickening technique

Topology 28 (1989) 273–289

Theorem (K. Pawałowski)
Let G be a compact Lie group. Let M be a compact smoothmanifold. Let X be a finite contractible G -CW complex withF (G  X ) = D 2d t M. Let E be a G -vector bundle over X
such that F (G  E |D 2d tM )Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The equivariant thickening technique

Topology 28 (1989) 273–289

Theorem (K. Pawałowski)
Let G be a compact Lie group. Let M be a compact smoothmanifold. Let X be a finite contractible G -CW complex withF (G  X ) = D 2d t M. Let E be a G -vector bundle over X
such that F (G  E |D 2d tM ) is stably isomorphic to T (D 2d t M).

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The equivariant thickening technique

Topology 28 (1989) 273–289

Theorem (K. Pawałowski)
Let G be a compact Lie group. Let M be a compact smoothmanifold. Let X be a finite contractible G -CW complex withF (G  X ) = D 2d t M. Let E be a G -vector bundle over X
such that F (G  E |D 2d tM ) is stably isomorphic to T (D 2d t M).
Then there exists a smooth action

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The equivariant thickening technique

Topology 28 (1989) 273–289

Theorem (K. Pawałowski)
Let G be a compact Lie group. Let M be a compact smoothmanifold. Let X be a finite contractible G -CW complex withF (G  X ) = D 2d t M. Let E be a G -vector bundle over X
such that F (G  E |D 2d tM ) is stably isomorphic to T (D 2d t M).
Then there exists a smooth action of G on a disk D 2d+2n withF (G  D 2d+2n ) ∼= D 2d t M

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The equivariant thickening technique

Topology 28 (1989) 273–289

Theorem (K. Pawałowski)
Let G be a compact Lie group. Let M be a compact smoothmanifold. Let X be a finite contractible G -CW complex withF (G  X ) = D 2d t M. Let E be a G -vector bundle over X
such that F (G  E |D 2d tM ) is stably isomorphic to T (D 2d t M).
Then there exists a smooth action of G on a disk D 2d+2n withF (G  D 2d+2n ) ∼= D 2d t M
and at any point x ∈ D 2d ,Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The equivariant thickening technique

Topology 28 (1989) 273–289

Theorem (K. Pawałowski)
Let G be a compact Lie group. Let M be a compact smoothmanifold. Let X be a finite contractible G -CW complex withF (G  X ) = D 2d t M. Let E be a G -vector bundle over X
such that F (G  E |D 2d tM ) is stably isomorphic to T (D 2d t M).
Then there exists a smooth action of G on a disk D 2d+2n withF (G  D 2d+2n ) ∼= D 2d t M
and at any point x ∈ D 2d , the normal G -module is isomorphic

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The equivariant thickening technique

Topology 28 (1989) 273–289

Theorem (K. Pawałowski)
Let G be a compact Lie group. Let M be a compact smoothmanifold. Let X be a finite contractible G -CW complex withF (G  X ) = D 2d t M. Let E be a G -vector bundle over X
such that F (G  E |D 2d tM ) is stably isomorphic to T (D 2d t M).
Then there exists a smooth action of G on a disk D 2d+2n withF (G  D 2d+2n ) ∼= D 2d t M
and at any point x ∈ D 2d , the normal G -module is isomorphic
to the realification of a complex n-dimensional G -module V n .

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The equivariant thickening technique

Topology 28 (1989) 273–289

Theorem (K. Pawałowski)
Let G be a compact Lie group. Let M be a compact smoothmanifold. Let X be a finite contractible G -CW complex withF (G  X ) = D 2d t M. Let E be a G -vector bundle over X
such that F (G  E |D 2d tM ) is stably isomorphic to T (D 2d t M).
Then there exists a smooth action of G on a disk D 2d+2n withF (G  D 2d+2n ) ∼= D 2d t M
and at any point x ∈ D 2d , the normal G -module is isomorphic
to the realification of a complex n-dimensional G -module V n .
Also, there exists a G -homotopy equivalence f : D 2d+2n → X

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The equivariant thickening technique

Topology 28 (1989) 273–289

Theorem (K. Pawałowski)
Let G be a compact Lie group. Let M be a compact smoothmanifold. Let X be a finite contractible G -CW complex withF (G  X ) = D 2d t M. Let E be a G -vector bundle over X
such that F (G  E |D 2d tM ) is stably isomorphic to T (D 2d t M).
Then there exists a smooth action of G on a disk D 2d+2n withF (G  D 2d+2n ) ∼= D 2d t M
and at any point x ∈ D 2d , the normal G -module is isomorphic
to the realification of a complex n-dimensional G -module V n .
Also, there exists a G -homotopy equivalence f : D 2d+2n → Xsuch that f ∗ (E ) is stably G -isomorphic to T (D 2d+2n ).

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

The equivariant thickening technique

Topology 28 (1989) 273–289

Theorem (K. Pawałowski)
Let G be a compact Lie group. Let M be a compact smoothmanifold. Let X be a finite contractible G -CW complex withF (G  X ) = D 2d t M. Let E be a G -vector bundle over X
such that F (G  E |D 2d tM ) is stably isomorphic to T (D 2d t M).
Then there exists a smooth action of G on a disk D 2d+2n withF (G  D 2d+2n ) ∼= D 2d t M
and at any point x ∈ D 2d , the normal G -module is isomorphic
to the realification of a complex n-dimensional G -module V n .
Also, there exists a G -homotopy equivalence f : D 2d+2n → Xsuch that f ∗ (E ) is stably G -isomorphic to T (D 2d+2n ).

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces

Dōmo arigatō gozaimasu!

Krzysztof M. Pawałowski Adam Mickiewicz University · Poznań · Poland
Smooth actions on complex projective spaces