Non-product smooth actions on
Cartesian products of manifolds
Krzysztof Pawałowski (UAM Poznań Poland)

38th Symposium on Transformation Groups
November 18-20, 2011, Kobe, Japan

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

38th Symposium on Transformation Groups
November 18-20, 2011, Kobe, Japan

Non-product smooth actions on
Cartesian products of manifolds
Krzysztof Pawałowski (UAM Poznań Poland)
Joint work with Marek Kaluba and Wojciech Politarczyk
Graduate students of Adam Mickiewicz University

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Is an asymmetric manifold symmetric?

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Is an asymmetric manifold symmetric?
By an asymmetric smooth manifold we mean a smooth manifold
without a non-trivial

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Is an asymmetric manifold symmetric?
By an asymmetric smooth manifold we mean a smooth manifold
without a non-trivial smooth finite group action.

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Is an asymmetric manifold symmetric?
By an asymmetric smooth manifold we mean a smooth manifold
without a non-trivial smooth finite group action.

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Is an asymmetric manifold symmetric?
By an asymmetric smooth manifold we mean a smooth manifold
without a non-trivial smooth finite group action.
A popular phrase about smooth manifolds reads as follows.

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Is an asymmetric manifold symmetric?
By an asymmetric smooth manifold we mean a smooth manifold
without a non-trivial smooth finite group action.
A popular phrase about smooth manifolds reads as follows.
“If you choose a smooth manifold at random,

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Is an asymmetric manifold symmetric?
By an asymmetric smooth manifold we mean a smooth manifold
without a non-trivial smooth finite group action.
A popular phrase about smooth manifolds reads as follows.
“If you choose a smooth manifold at random, it is asymmetric.”

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Is an asymmetric manifold symmetric?
By an asymmetric smooth manifold we mean a smooth manifold
without a non-trivial smooth finite group action.
A popular phrase about smooth manifolds reads as follows.
“If you choose a smooth manifold at random, it is asymmetric.”
Now, we wish to show that this phrase can be complemented by
the following comment.

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Is an asymmetric manifold symmetric?
By an asymmetric smooth manifold we mean a smooth manifold
without a non-trivial smooth finite group action.
A popular phrase about smooth manifolds reads as follows.
“If you choose a smooth manifold at random, it is asymmetric.”
Now, we wish to show that this phrase can be complemented by
the following comment.
“If you choose an asymmetric smooth manifold,Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Is an asymmetric manifold symmetric?
By an asymmetric smooth manifold we mean a smooth manifold
without a non-trivial smooth finite group action.
A popular phrase about smooth manifolds reads as follows.
“If you choose a smooth manifold at random, it is asymmetric.”
Now, we wish to show that this phrase can be complemented by
the following comment.
“If you choose an asymmetric smooth manifold, it is symmetric
up to cartesian product operation with sphere factors.”

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Is an asymmetric manifold symmetric?
By an asymmetric smooth manifold we mean a smooth manifold
without a non-trivial smooth finite group action.
A popular phrase about smooth manifolds reads as follows.
“If you choose a smooth manifold at random, it is asymmetric.”
Now, we wish to show that this phrase can be complemented by
the following comment.
“If you choose an asymmetric smooth manifold, it is symmetric
up to cartesian product operation with sphere factors.”
Asymmetric smooth manifolds have been constructed, e.g., by A. Edmonds, Contemp.
Math. 36 (1985) 339–366, Kwasik and P. Vogel, Duke Math. J. 53 (1986) 759–764,
V. Puppe, Annals of Math. Studies 138 (1995) 283–302, and M. Kreck, J. Topology 2(2009) 249–261, Corrigendum: J. Topology 4 (2011) 254–255.
Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Theorems A and B

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Theorems A and B
We construct smooth group actions on Cartesian products ofsmooth manifolds,

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Theorems A and B
We construct smooth group actions on Cartesian products ofsmooth manifolds, such that the action is non-productKrzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Theorems A and B
We construct smooth group actions on Cartesian products ofsmooth manifolds, such that the action is non-product thought

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Theorems A and B
We construct smooth group actions on Cartesian products ofsmooth manifolds, such that the action is non-product thought
the product contains an asymmetric manifold as a factor.

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Theorems A and B
We construct smooth group actions on Cartesian products ofsmooth manifolds, such that the action is non-product thought
the product contains an asymmetric manifold as a factor.
Theorem A

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Theorems A and B
We construct smooth group actions on Cartesian products ofsmooth manifolds, such that the action is non-product thought
the product contains an asymmetric manifold as a factor.
Theorem A
Let G be a compact Lie group.

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Theorems A and B
We construct smooth group actions on Cartesian products ofsmooth manifolds, such that the action is non-product thought
the product contains an asymmetric manifold as a factor.
Theorem A
Let G be a compact Lie group. For any closed asymmetricsmooth m-manifold Mm with m ­ 3,

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Theorems A and B
We construct smooth group actions on Cartesian products ofsmooth manifolds, such that the action is non-product thought
the product contains an asymmetric manifold as a factor.
Theorem A
Let G be a compact Lie group. For any closed asymmetricsmooth m-manifold Mm with m ­ 3, there is a non-productsmooth action of G on a Cartesian product of the form

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Theorems A and B
We construct smooth group actions on Cartesian products ofsmooth manifolds, such that the action is non-product thought
the product contains an asymmetric manifold as a factor.
Theorem A
Let G be a compact Lie group. For any closed asymmetricsmooth m-manifold Mm with m ­ 3, there is a non-productsmooth action of G on a Cartesian product of the form
M = Mm × S n1 × · · · × S nk .

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Theorems A and B
We construct smooth group actions on Cartesian products ofsmooth manifolds, such that the action is non-product thought
the product contains an asymmetric manifold as a factor.
Theorem A
Let G be a compact Lie group. For any closed asymmetricsmooth m-manifold Mm with m ­ 3, there is a non-productsmooth action of G on a Cartesian product of the form
M = Mm × S n1 × · · · × S nk .
Every fixed point set connected component is homeomorphic
to the connected sum Mm #Σ m

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Theorems A and B
We construct smooth group actions on Cartesian products ofsmooth manifolds, such that the action is non-product thought
the product contains an asymmetric manifold as a factor.
Theorem A
Let G be a compact Lie group. For any closed asymmetricsmooth m-manifold Mm with m ­ 3, there is a non-productsmooth action of G on a Cartesian product of the form
M = Mm × S n1 × · · · × S nk .
Every fixed point set connected component is homeomorphic
to the connected sum Mm #Σ m for an appropriate homology
m-sphere Σ m .

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Theorems A and B
We construct smooth group actions on Cartesian products ofsmooth manifolds, such that the action is non-product thought
the product contains an asymmetric manifold as a factor.
Theorem A
Let G be a compact Lie group. For any closed asymmetricsmooth m-manifold Mm with m ­ 3, there is a non-productsmooth action of G on a Cartesian product of the form
M = Mm × S n1 × · · · × S nk .
Every fixed point set connected component is homeomorphic
to the connected sum Mm #Σ m for an appropriate homology
m-sphere Σ m . One may always assume that k = 1

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Theorems A and B
We construct smooth group actions on Cartesian products ofsmooth manifolds, such that the action is non-product thought
the product contains an asymmetric manifold as a factor.
Theorem A
Let G be a compact Lie group. For any closed asymmetricsmooth m-manifold Mm with m ­ 3, there is a non-productsmooth action of G on a Cartesian product of the form
M = Mm × S n1 × · · · × S nk .
Every fixed point set connected component is homeomorphic
to the connected sum Mm #Σ m for an appropriate homology
m-sphere Σ m . One may always assume that k = 1 and when
G = S 1,

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Theorems A and B
We construct smooth group actions on Cartesian products ofsmooth manifolds, such that the action is non-product thought
the product contains an asymmetric manifold as a factor.
Theorem A
Let G be a compact Lie group. For any closed asymmetricsmooth m-manifold Mm with m ­ 3, there is a non-productsmooth action of G on a Cartesian product of the form
M = Mm × S n1 × · · · × S nk .
Every fixed point set connected component is homeomorphic
to the connected sum Mm #Σ m for an appropriate homology
m-sphere Σ m . One may always assume that k = 1 and when
G = S 1 , that n1 = · · · = nk = 2 for some k ­ 1.

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Theorems A and B
We construct smooth group actions on Cartesian products ofsmooth manifolds, such that the action is non-product thought
the product contains an asymmetric manifold as a factor.
Theorem A
Let G be a compact Lie group. For any closed asymmetricsmooth m-manifold Mm with m ­ 3, there is a non-productsmooth action of G on a Cartesian product of the form
M = Mm × S n1 × · · · × S nk .
Every fixed point set connected component is homeomorphic
to the connected sum Mm #Σ m for an appropriate homology
m-sphere Σ m . One may always assume that k = 1 and when
G = S 1 , that n1 = · · · = nk = 2 for some k ­ 1.

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Theorems A and B

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Theorems A and B
We construct smooth group actions on Cartesian products of
symplectic manifolds,

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Theorems A and B
We construct smooth group actions on Cartesian products of
symplectic manifolds, such that the action is non-productKrzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Theorems A and B
We construct smooth group actions on Cartesian products of
symplectic manifolds, such that the action is non-product andnon-symplectic with respect to any symplectic structure.

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Theorems A and B
We construct smooth group actions on Cartesian products of
symplectic manifolds, such that the action is non-product andnon-symplectic with respect to any symplectic structure.
Theorem B

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Theorems A and B
We construct smooth group actions on Cartesian products of
symplectic manifolds, such that the action is non-product andnon-symplectic with respect to any symplectic structure.
Theorem B
For any compact Lie group G , there exists a smooth action of
G on a Cartesian product of symplectic manifolds of the form

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Theorems A and B
We construct smooth group actions on Cartesian products of
symplectic manifolds, such that the action is non-product andnon-symplectic with respect to any symplectic structure.
Theorem B
For any compact Lie group G , there exists a smooth action of
G on a Cartesian product of symplectic manifolds of the form
M = CP 2 × CP n1 × · · · × CP nk

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Theorems A and B
We construct smooth group actions on Cartesian products of
symplectic manifolds, such that the action is non-product andnon-symplectic with respect to any symplectic structure.
Theorem B
For any compact Lie group G , there exists a smooth action of
G on a Cartesian product of symplectic manifolds of the form
M = CP 2 × CP n1 × · · · × CP nk
such that each connected component of the fixed point set isdiffeomorphic to the connected sum CP 2 #Σ 4

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Theorems A and B
We construct smooth group actions on Cartesian products of
symplectic manifolds, such that the action is non-product andnon-symplectic with respect to any symplectic structure.
Theorem B
For any compact Lie group G , there exists a smooth action of
G on a Cartesian product of symplectic manifolds of the form
M = CP 2 × CP n1 × · · · × CP nk
such that each connected component of the fixed point set isdiffeomorphic to the connected sum CP 2 #Σ 4 for an arbitrary
homology 4-sphere Σ 4 .

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Theorems A and B
We construct smooth group actions on Cartesian products of
symplectic manifolds, such that the action is non-product andnon-symplectic with respect to any symplectic structure.
Theorem B
For any compact Lie group G , there exists a smooth action of
G on a Cartesian product of symplectic manifolds of the form
M = CP 2 × CP n1 × · · · × CP nk
such that each connected component of the fixed point set isdiffeomorphic to the connected sum CP 2 #Σ 4 for an arbitrary
homology 4-sphere Σ 4 . For an appropriate Σ 4 , the action of G
on M is non-productKrzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Theorems A and B
We construct smooth group actions on Cartesian products of
symplectic manifolds, such that the action is non-product andnon-symplectic with respect to any symplectic structure.
Theorem B
For any compact Lie group G , there exists a smooth action of
G on a Cartesian product of symplectic manifolds of the form
M = CP 2 × CP n1 × · · · × CP nk
such that each connected component of the fixed point set isdiffeomorphic to the connected sum CP 2 #Σ 4 for an arbitrary
homology 4-sphere Σ 4 . For an appropriate Σ 4 , the action of G
on M is non-product and non-symplectic with respect to any
possible symplectic structure on M.

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Theorems A and B
We construct smooth group actions on Cartesian products of
symplectic manifolds, such that the action is non-product andnon-symplectic with respect to any symplectic structure.
Theorem B
For any compact Lie group G , there exists a smooth action of
G on a Cartesian product of symplectic manifolds of the form
M = CP 2 × CP n1 × · · · × CP nk
such that each connected component of the fixed point set isdiffeomorphic to the connected sum CP 2 #Σ 4 for an arbitrary
homology 4-sphere Σ 4 . For an appropriate Σ 4 , the action of G
on M is non-product and non-symplectic with respect to any
possible symplectic structure on M. In the case where G = S 1 ,Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Theorems A and B
We construct smooth group actions on Cartesian products of
symplectic manifolds, such that the action is non-product andnon-symplectic with respect to any symplectic structure.
Theorem B
For any compact Lie group G , there exists a smooth action of
G on a Cartesian product of symplectic manifolds of the form
M = CP 2 × CP n1 × · · · × CP nk
such that each connected component of the fixed point set isdiffeomorphic to the connected sum CP 2 #Σ 4 for an arbitrary
homology 4-sphere Σ 4 . For an appropriate Σ 4 , the action of G
on M is non-product and non-symplectic with respect to any
possible symplectic structure on M. In the case where G = S 1 ,n1 = · · · = nk = 1 for some k ­ 1.
Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres
To prove Theorem A, consider the equivariant connected sum

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres
To prove Theorem A, consider the equivariant connected sum(Mm × S n1 × · · · × S nk ) # S m+n # S m+n # · · ·Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres
To prove Theorem A, consider the equivariant connected sum(Mm × S n1 × · · · × S nk ) # S m+n # S m+n # · · ·for n = n1 + · · · + nk ,Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres
To prove Theorem A, consider the equivariant connected sum(Mm × S n1 × · · · × S nk ) # S m+n # S m+n # · · ·for n = n1 + · · · + nk , where G acts trivially on X m

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres
To prove Theorem A, consider the equivariant connected sum(Mm × S n1 × · · · × S nk ) # S m+n # S m+n # · · ·for n = n1 + · · · + nk , where G acts trivially on X m and linearlyon S ni for i = 1, . . . , k.

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres
To prove Theorem A, consider the equivariant connected sum(Mm × S n1 × · · · × S nk ) # S m+n # S m+n # · · ·for n = n1 + · · · + nk , where G acts trivially on X m and linearlyon S ni for i = 1, . . . , k. Moreover, G acts smoothly on S m+n

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres
To prove Theorem A, consider the equivariant connected sum(Mm × S n1 × · · · × S nk ) # S m+n # S m+n # · · ·for n = n1 + · · · + nk , where G acts trivially on X m and linearlyon S ni for i = 1, . . . , k. Moreover, G acts smoothly on S m+n in
such a way that F (G , S m+n ) is homeomorphic to Σ m .

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres
To prove Theorem A, consider the equivariant connected sum(Mm × S n1 × · · · × S nk ) # S m+n # S m+n # · · ·for n = n1 + · · · + nk , where G acts trivially on X m and linearlyon S ni for i = 1, . . . , k. Moreover, G acts smoothly on S m+n in
such a way that F (G , S m+n ) is homeomorphic to Σ m .
The linear action of G on the product is determined

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres
To prove Theorem A, consider the equivariant connected sum(Mm × S n1 × · · · × S nk ) # S m+n # S m+n # · · ·for n = n1 + · · · + nk , where G acts trivially on X m and linearlyon S ni for i = 1, . . . , k. Moreover, G acts smoothly on S m+n in
such a way that F (G , S m+n ) is homeomorphic to Σ m .
The linear action of G on the product is determined by
Tx (S m+n ) = Rm ⊕ V n
at x ∈ Σ m ⊂ S m+n ,Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres
To prove Theorem A, consider the equivariant connected sum(Mm × S n1 × · · · × S nk ) # S m+n # S m+n # · · ·for n = n1 + · · · + nk , where G acts trivially on X m and linearlyon S ni for i = 1, . . . , k. Moreover, G acts smoothly on S m+n in
such a way that F (G , S m+n ) is homeomorphic to Σ m .
The linear action of G on the product is determined by
Tx (S m+n ) = Rm ⊕ V n
at x ∈ Σ m ⊂ S m+n , as follows.

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres
To prove Theorem A, consider the equivariant connected sum(Mm × S n1 × · · · × S nk ) # S m+n # S m+n # · · ·for n = n1 + · · · + nk , where G acts trivially on X m and linearlyon S ni for i = 1, . . . , k. Moreover, G acts smoothly on S m+n in
such a way that F (G , S m+n ) is homeomorphic to Σ m .
The linear action of G on the product is determined by
Tx (S m+n ) = Rm ⊕ V n
at x ∈ Σ m ⊂ S m+n , as follows. Consider any decompositionn
Vn ∼= V1n1 ⊕ · · · ⊕ Vk k .

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres
To prove Theorem A, consider the equivariant connected sum(Mm × S n1 × · · · × S nk ) # S m+n # S m+n # · · ·for n = n1 + · · · + nk , where G acts trivially on X m and linearlyon S ni for i = 1, . . . , k. Moreover, G acts smoothly on S m+n in
such a way that F (G , S m+n ) is homeomorphic to Σ m .
The linear action of G on the product is determined by
Tx (S m+n ) = Rm ⊕ V n
at x ∈ Σ m ⊂ S m+n , as follows. Consider any decompositionn
Vn ∼= V1n1 ⊕ · · · ⊕ Vk k .

For i = 1, . . . , k, let S ni = S(Vi ⊕ R),
Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres
To prove Theorem A, consider the equivariant connected sum(Mm × S n1 × · · · × S nk ) # S m+n # S m+n # · · ·for n = n1 + · · · + nk , where G acts trivially on X m and linearlyon S ni for i = 1, . . . , k. Moreover, G acts smoothly on S m+n in
such a way that F (G , S m+n ) is homeomorphic to Σ m .
The linear action of G on the product is determined by
Tx (S m+n ) = Rm ⊕ V n
at x ∈ Σ m ⊂ S m+n , as follows. Consider any decompositionn
Vn ∼= V1n1 ⊕ · · · ⊕ Vk k .

For i = 1, . . . , k, let S ni = S(Vi ⊕ R), where G acts trivially on R.
Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres
To prove Theorem A, consider the equivariant connected sum(Mm × S n1 × · · · × S nk ) # S m+n # S m+n # · · ·for n = n1 + · · · + nk , where G acts trivially on X m and linearlyon S ni for i = 1, . . . , k. Moreover, G acts smoothly on S m+n in
such a way that F (G , S m+n ) is homeomorphic to Σ m .
The linear action of G on the product is determined by
Tx (S m+n ) = Rm ⊕ V n
at x ∈ Σ m ⊂ S m+n , as follows. Consider any decompositionn
Vn ∼= V1n1 ⊕ · · · ⊕ Vk k .

For i = 1, . . . , k, let S ni = S(Vi ⊕ R), where G acts trivially on R.
Then G acts linearly on S ni with exactly two fixed points.
Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres
To prove Theorem B, consider the equivariant connected sum

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres
To prove Theorem B, consider the equivariant connected sum(CP 2 × CP n1 × · · · × CP nk ) # S 2n+4 # S 2n+4 # · · ·Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres
To prove Theorem B, consider the equivariant connected sum(CP 2 × CP n1 × · · · × CP nk ) # S 2n+4 # S 2n+4 # · · ·for n = n1 + · · · + nk ,Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres
To prove Theorem B, consider the equivariant connected sum(CP 2 × CP n1 × · · · × CP nk ) # S 2n+4 # S 2n+4 # · · ·for n = n1 + · · · + nk , where G acts trivially on CP 2

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres
To prove Theorem B, consider the equivariant connected sum(CP 2 × CP n1 × · · · × CP nk ) # S 2n+4 # S 2n+4 # · · ·for n = n1 + · · · + nk , where G acts trivially on CP 2 and linearlyon CP ni for i = 1, . . . , k.

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres
To prove Theorem B, consider the equivariant connected sum(CP 2 × CP n1 × · · · × CP nk ) # S 2n+4 # S 2n+4 # · · ·for n = n1 + · · · + nk , where G acts trivially on CP 2 and linearlyon CP ni for i = 1, . . . , k. Moreover, G acts smoothly on S 2n+4

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres
To prove Theorem B, consider the equivariant connected sum(CP 2 × CP n1 × · · · × CP nk ) # S 2n+4 # S 2n+4 # · · ·for n = n1 + · · · + nk , where G acts trivially on CP 2 and linearlyon CP ni for i = 1, . . . , k. Moreover, G acts smoothly on S 2n+4
in such a way that F (G , S 2n+4 ) is diffeomorphic to Σ 4 .

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres
To prove Theorem B, consider the equivariant connected sum(CP 2 × CP n1 × · · · × CP nk ) # S 2n+4 # S 2n+4 # · · ·for n = n1 + · · · + nk , where G acts trivially on CP 2 and linearlyon CP ni for i = 1, . . . , k. Moreover, G acts smoothly on S 2n+4
in such a way that F (G , S 2n+4 ) is diffeomorphic to Σ 4 .
The linear action of G on the product is determined

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres
To prove Theorem B, consider the equivariant connected sum(CP 2 × CP n1 × · · · × CP nk ) # S 2n+4 # S 2n+4 # · · ·for n = n1 + · · · + nk , where G acts trivially on CP 2 and linearlyon CP ni for i = 1, . . . , k. Moreover, G acts smoothly on S 2n+4
in such a way that F (G , S 2n+4 ) is diffeomorphic to Σ 4 .
The linear action of G on the product is determined by
Tx (S 2n+4 ) = V 2n ⊕ R4
at x ∈ Σ 4 ⊂ S 2n+4 ,Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres
To prove Theorem B, consider the equivariant connected sum(CP 2 × CP n1 × · · · × CP nk ) # S 2n+4 # S 2n+4 # · · ·for n = n1 + · · · + nk , where G acts trivially on CP 2 and linearlyon CP ni for i = 1, . . . , k. Moreover, G acts smoothly on S 2n+4
in such a way that F (G , S 2n+4 ) is diffeomorphic to Σ 4 .
The linear action of G on the product is determined by
Tx (S 2n+4 ) = V 2n ⊕ R4
at x ∈ Σ 4 ⊂ S 2n+4 , where V 2n is the realification of a complex
G -module W n .

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres
To prove Theorem B, consider the equivariant connected sum(CP 2 × CP n1 × · · · × CP nk ) # S 2n+4 # S 2n+4 # · · ·for n = n1 + · · · + nk , where G acts trivially on CP 2 and linearlyon CP ni for i = 1, . . . , k. Moreover, G acts smoothly on S 2n+4
in such a way that F (G , S 2n+4 ) is diffeomorphic to Σ 4 .
The linear action of G on the product is determined by
Tx (S 2n+4 ) = V 2n ⊕ R4
at x ∈ Σ 4 ⊂ S 2n+4 , where V 2n is the realification of a complex
G -module W n . Letn
Wn ∼= W1n1 ⊕ · · · ⊕ Wk k

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres
To prove Theorem B, consider the equivariant connected sum(CP 2 × CP n1 × · · · × CP nk ) # S 2n+4 # S 2n+4 # · · ·for n = n1 + · · · + nk , where G acts trivially on CP 2 and linearlyon CP ni for i = 1, . . . , k. Moreover, G acts smoothly on S 2n+4
in such a way that F (G , S 2n+4 ) is diffeomorphic to Σ 4 .
The linear action of G on the product is determined by
Tx (S 2n+4 ) = V 2n ⊕ R4
at x ∈ Σ 4 ⊂ S 2n+4 , where V 2n is the realification of a complex
G -module W n . Letn
Wn ∼= W1n1 ⊕ · · · ⊕ Wk k

be the decomposition of W n into irreducible summands Wini .
Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

For i = 1, . . . , k,

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

For i = 1, . . . , k, set
S 2ni +1 = S(Wi ⊕ C)

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

For i = 1, . . . , k, set
S 2ni +1 = S(Wi ⊕ C) and CP ni = S(Wi ⊕ C)/S 1 ,Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

For i = 1, . . . , k, set
S 2ni +1 = S(Wi ⊕ C) and CP ni = S(Wi ⊕ C)/S 1 ,where G acts trivially on C.

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

For i = 1, . . . , k, set
S 2ni +1 = S(Wi ⊕ C) and CP ni = S(Wi ⊕ C)/S 1 ,where G acts trivially on C.
If dim Wi > 1, G acts on CP ni with exactly one fixed point.

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

For i = 1, . . . , k, set
S 2ni +1 = S(Wi ⊕ C) and CP ni = S(Wi ⊕ C)/S 1 ,where G acts trivially on C.
If dim Wi > 1, G acts on CP ni with exactly one fixed point.
If dim Wi = 1, G acts on CP 1 = S 2 with exactly two fixed points.

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

For i = 1, . . . , k, set
S 2ni +1 = S(Wi ⊕ C) and CP ni = S(Wi ⊕ C)/S 1 ,where G acts trivially on C.
If dim Wi > 1, G acts on CP ni with exactly one fixed point.
If dim Wi = 1, G acts on CP 1 = S 2 with exactly two fixed points.
At any fixed point x ∈ CP ni ,Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

For i = 1, . . . , k, set
S 2ni +1 = S(Wi ⊕ C) and CP ni = S(Wi ⊕ C)/S 1 ,where G acts trivially on C.
If dim Wi > 1, G acts on CP ni with exactly one fixed point.
If dim Wi = 1, G acts on CP 1 = S 2 with exactly two fixed points.
At any fixed point x ∈ CP ni , the tangent G -module Tx (CP ni ) is
the realification of Wi

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Absorbing group actions from spheres

For i = 1, . . . , k, set
S 2ni +1 = S(Wi ⊕ C) and CP ni = S(Wi ⊕ C)/S 1 ,where G acts trivially on C.
If dim Wi > 1, G acts on CP ni with exactly one fixed point.
If dim Wi = 1, G acts on CP 1 = S 2 with exactly two fixed points.
At any fixed point x ∈ CP ni , the tangent G -module Tx (CP ni ) is
the realification of Wi and thus at any fixed point, the tangent
G -module to the product is isomorphic to V 2n ⊕ R4 .

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Equivariant thickening

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Equivariant thickening
Topology 28 (1989) 273–289

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Equivariant thickening
Topology 28 (1989) 273–289

Theorem (K. Pawałowski)

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Equivariant thickening
Topology 28 (1989) 273–289

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

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Equivariant thickening
Topology 28 (1989) 273–289

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

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Equivariant thickening
Topology 28 (1989) 273–289

Theorem (K. Pawałowski)
Let G be a compact Lie group. Let F be a compact smoothmanifold. Let X be a finite contractible G -CW complex with
XG = F.

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Equivariant thickening
Topology 28 (1989) 273–289

Theorem (K. Pawałowski)
Let G be a compact Lie group. Let F be a compact smoothmanifold. Let X be a finite contractible G -CW complex with
X G = F . Let ξ be a G -vector bundle over X such that
ξ|F ∼= τF ⊕ ε ⊕ νKrzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Equivariant thickening
Topology 28 (1989) 273–289

Theorem (K. Pawałowski)
Let G be a compact Lie group. Let F be a compact smoothmanifold. Let X be a finite contractible G -CW complex with
X G = F . Let ξ be a G -vector bundle over X such that
ξ|F ∼= τF ⊕ ε ⊕ ν
for a product vector bundle ε over F with the trivial action of G

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Equivariant thickening
Topology 28 (1989) 273–289

Theorem (K. Pawałowski)
Let G be a compact Lie group. Let F be a compact smoothmanifold. Let X be a finite contractible G -CW complex with
X G = F . Let ξ be a G -vector bundle over X such that
ξ|F ∼= τF ⊕ ε ⊕ ν
for a product vector bundle ε over F with the trivial action of G
and a G -vector bundle ν over F with dim ν G = 0.

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Equivariant thickening
Topology 28 (1989) 273–289

Theorem (K. Pawałowski)
Let G be a compact Lie group. Let F be a compact smoothmanifold. Let X be a finite contractible G -CW complex with
X G = F . Let ξ be a G -vector bundle over X such that
ξ|F ∼= τF ⊕ ε ⊕ ν
for a product vector bundle ε over F with the trivial action of G
and a G -vector bundle ν over F with dim ν G = 0. Then there isa smooth action of G on a disk D

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Equivariant thickening
Topology 28 (1989) 273–289

Theorem (K. Pawałowski)
Let G be a compact Lie group. Let F be a compact smoothmanifold. Let X be a finite contractible G -CW complex with
X G = F . Let ξ be a G -vector bundle over X such that
ξ|F ∼= τF ⊕ ε ⊕ ν
for a product vector bundle ε over F with the trivial action of G
and a G -vector bundle ν over F with dim ν G = 0. Then there isa smooth action of G on a disk D such that the fixed point set
is diffeomorphic to F and νF ⊂D ∼= ν as G -vector bundles.

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Equivariant thickening
Topology 28 (1989) 273–289

Theorem (K. Pawałowski)
Let G be a compact Lie group. Let F be a compact smoothmanifold. Let X be a finite contractible G -CW complex with
X G = F . Let ξ be a G -vector bundle over X such that
ξ|F ∼= τF ⊕ ε ⊕ ν
for a product vector bundle ε over F with the trivial action of G
and a G -vector bundle ν over F with dim ν G = 0. Then there isa smooth action of G on a disk D such that the fixed point set
is diffeomorphic to F and νF ⊂D ∼= ν as G -vector bundles.
We may always assume that at a chosen point x ∈ F ,Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Equivariant thickening
Topology 28 (1989) 273–289

Theorem (K. Pawałowski)
Let G be a compact Lie group. Let F be a compact smoothmanifold. Let X be a finite contractible G -CW complex with
X G = F . Let ξ be a G -vector bundle over X such that
ξ|F ∼= τF ⊕ ε ⊕ ν
for a product vector bundle ε over F with the trivial action of G
and a G -vector bundle ν over F with dim ν G = 0. Then there isa smooth action of G on a disk D such that the fixed point set
is diffeomorphic to F and νF ⊂D ∼= ν as G -vector bundles.
We may always assume that at a chosen point x ∈ F , the fiber ofν over x
Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Equivariant thickening
Topology 28 (1989) 273–289

Theorem (K. Pawałowski)
Let G be a compact Lie group. Let F be a compact smoothmanifold. Let X be a finite contractible G -CW complex with
X G = F . Let ξ be a G -vector bundle over X such that
ξ|F ∼= τF ⊕ ε ⊕ ν
for a product vector bundle ε over F with the trivial action of G
and a G -vector bundle ν over F with dim ν G = 0. Then there isa smooth action of G on a disk D such that the fixed point set
is diffeomorphic to F and νF ⊂D ∼= ν as G -vector bundles.
We may always assume that at a chosen point x ∈ F , the fiber ofν over x is the realification of a complex G -module.
Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology disks

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology disks
By a homology m-disk we mean a compact Z-acyclic smoothmanifold of dimension m ­ 0.

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology disks
By a homology m-disk we mean a compact Z-acyclic smoothmanifold of dimension m ­ 0.
TheoremKrzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology disks
By a homology m-disk we mean a compact Z-acyclic smoothmanifold of dimension m ­ 0.
Theorem
Let G be a compact Lie group.

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology disks
By a homology m-disk we mean a compact Z-acyclic smoothmanifold of dimension m ­ 0.
Theorem
Let G be a compact Lie group. Let ∆m be a homology m-disk.

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology disks
By a homology m-disk we mean a compact Z-acyclic smoothmanifold of dimension m ­ 0.
Theorem
Let G be a compact Lie group. Let ∆m be a homology m-disk.
Then there is a smooth action of G on a disk D m+n

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology disks
By a homology m-disk we mean a compact Z-acyclic smoothmanifold of dimension m ­ 0.
Theorem
Let G be a compact Lie group. Let ∆m be a homology m-disk.
Then there is a smooth action of G on a disk D m+n such thatthe fixed point set F (G , D m+n ) is diffeomorphic to ∆m .

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology disks
By a homology m-disk we mean a compact Z-acyclic smoothmanifold of dimension m ­ 0.
Theorem
Let G be a compact Lie group. Let ∆m be a homology m-disk.
Then there is a smooth action of G on a disk D m+n such thatthe fixed point set F (G , D m+n ) is diffeomorphic to ∆m .
Proof.

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology disks
By a homology m-disk we mean a compact Z-acyclic smoothmanifold of dimension m ­ 0.
Theorem
Let G be a compact Lie group. Let ∆m be a homology m-disk.
Then there is a smooth action of G on a disk D m+n such thatthe fixed point set F (G , D m+n ) is diffeomorphic to ∆m .
Proof. There exists a finite contractible G -CW complex X suchthat X G = ∆m .

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology disks
By a homology m-disk we mean a compact Z-acyclic smoothmanifold of dimension m ­ 0.
Theorem
Let G be a compact Lie group. Let ∆m be a homology m-disk.
Then there is a smooth action of G on a disk D m+n such thatthe fixed point set F (G , D m+n ) is diffeomorphic to ∆m .
Proof. There exists a finite contractible G -CW complex X suchthat X G = ∆m . In fact, if:
G is connected, set X = G ∗ ∆m with the joint action of G .
G /G0 is of prime power order,

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology disks
By a homology m-disk we mean a compact Z-acyclic smoothmanifold of dimension m ­ 0.
Theorem
Let G be a compact Lie group. Let ∆m be a homology m-disk.
Then there is a smooth action of G on a disk D m+n such thatthe fixed point set F (G , D m+n ) is diffeomorphic to ∆m .
Proof. There exists a finite contractible G -CW complex X suchthat X G = ∆m . In fact, if:
G is connected, set X = G ∗ ∆m with the joint action of G .
G /G0 is of prime power order, such an X exists by the workof L. Jones (1971).
G /G0 is not of prime power order,

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology disks
By a homology m-disk we mean a compact Z-acyclic smoothmanifold of dimension m ­ 0.
Theorem
Let G be a compact Lie group. Let ∆m be a homology m-disk.
Then there is a smooth action of G on a disk D m+n such thatthe fixed point set F (G , D m+n ) is diffeomorphic to ∆m .
Proof. There exists a finite contractible G -CW complex X suchthat X G = ∆m . In fact, if:
G is connected, set X = G ∗ ∆m with the joint action of G .
G /G0 is of prime power order, such an X exists by the workof L. Jones (1971).
G /G0 is not of prime power order, such an X exists by thework of B. Oliver (1975).
Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology disks

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology disksAs ∆m is Z-acyclic,

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology disksAs ∆m is Z-acyclic, ∆m is stably parallelizable.

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology disksAs ∆m is Z-acyclic, ∆m is stably parallelizable.
By making use of the product bundle X × (Rm ⊕ V n ) over X

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology disksAs ∆m is Z-acyclic, ∆m is stably parallelizable.
By making use of the product bundle X × (Rm ⊕ V n ) over X for
an appropriate n-dimensional real G -module V n ,Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology disksAs ∆m is Z-acyclic, ∆m is stably parallelizable.
By making use of the product bundle X × (Rm ⊕ V n ) over X for
an appropriate n-dimensional real G -module V n , one can thicken
up X to the standard disk D m+n with a smooth action of G

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology disksAs ∆m is Z-acyclic, ∆m is stably parallelizable.
By making use of the product bundle X × (Rm ⊕ V n ) over X for
an appropriate n-dimensional real G -module V n , one can thicken
up X to the standard disk D m+n with a smooth action of G suchthat F (G , D m+n ) is diffeomorphic to ∆m .
Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology disksAs ∆m is Z-acyclic, ∆m is stably parallelizable.
By making use of the product bundle X × (Rm ⊕ V n ) over X for
an appropriate n-dimensional real G -module V n , one can thicken
up X to the standard disk D m+n with a smooth action of G suchthat F (G , D m+n ) is diffeomorphic to ∆m .
We may always assume that n is even and at a chosen pointx ∈ ∆m ,Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology disksAs ∆m is Z-acyclic, ∆m is stably parallelizable.
By making use of the product bundle X × (Rm ⊕ V n ) over X for
an appropriate n-dimensional real G -module V n , one can thicken
up X to the standard disk D m+n with a smooth action of G suchthat F (G , D m+n ) is diffeomorphic to ∆m .
We may always assume that n is even and at a chosen pointx ∈ ∆m , the normal G -module is the realification of a complex
G -module.

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology spheres

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology spheres
By a homology m-sphere we mean a closed smooth manifold Σ mof dimension m ­ 0, with the homology H∗ (Σ m ; Z) = H∗ (S m ; Z).

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology spheres
By a homology m-sphere we mean a closed smooth manifold Σ mof dimension m ­ 0, with the homology H∗ (Σ m ; Z) = H∗ (S m ; Z).
Trans. Amer. Math. Soc. 144 (1969) 67–72

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology spheres
By a homology m-sphere we mean a closed smooth manifold Σ mof dimension m ­ 0, with the homology H∗ (Σ m ; Z) = H∗ (S m ; Z).
Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology spheres
By a homology m-sphere we mean a closed smooth manifold Σ mof dimension m ­ 0, with the homology H∗ (Σ m ; Z) = H∗ (S m ; Z).
Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)
Any homology 4-sphere bounds a contractible 5-disk.

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology spheres
By a homology m-sphere we mean a closed smooth manifold Σ mof dimension m ­ 0, with the homology H∗ (Σ m ; Z) = H∗ (S m ; Z).
Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)
Any homology 4-sphere bounds a contractible 5-disk.
For any homology m-sphere with m ­ 5, there is a unique
homotopy m-sphere

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology spheres
By a homology m-sphere we mean a closed smooth manifold Σ mof dimension m ­ 0, with the homology H∗ (Σ m ; Z) = H∗ (S m ; Z).
Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)
Any homology 4-sphere bounds a contractible 5-disk.
For any homology m-sphere with m ­ 5, there is a unique
homotopy m-sphere such that the connected sum of thetwo spheres bounds a homotopy (m + 1)-disk.

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology spheres
By a homology m-sphere we mean a closed smooth manifold Σ mof dimension m ­ 0, with the homology H∗ (Σ m ; Z) = H∗ (S m ; Z).
Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)
Any homology 4-sphere bounds a contractible 5-disk.
For any homology m-sphere with m ­ 5, there is a unique
homotopy m-sphere such that the connected sum of thetwo spheres bounds a homotopy (m + 1)-disk.
Math. Research Letters 7 (2000) 757–766

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology spheres
By a homology m-sphere we mean a closed smooth manifold Σ mof dimension m ­ 0, with the homology H∗ (Σ m ; Z) = H∗ (S m ; Z).
Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)
Any homology 4-sphere bounds a contractible 5-disk.
For any homology m-sphere with m ­ 5, there is a unique
homotopy m-sphere such that the connected sum of thetwo spheres bounds a homotopy (m + 1)-disk.
Math. Research Letters 7 (2000) 757–766

Theorem (Y. Fukumoto, M. Furuta)

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology spheres
By a homology m-sphere we mean a closed smooth manifold Σ mof dimension m ­ 0, with the homology H∗ (Σ m ; Z) = H∗ (S m ; Z).
Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)
Any homology 4-sphere bounds a contractible 5-disk.
For any homology m-sphere with m ­ 5, there is a unique
homotopy m-sphere such that the connected sum of thetwo spheres bounds a homotopy (m + 1)-disk.
Math. Research Letters 7 (2000) 757–766

Theorem (Y. Fukumoto, M. Furuta)
There exist homology 3-spheres which bound homology
4-disks,
Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology spheres
By a homology m-sphere we mean a closed smooth manifold Σ mof dimension m ­ 0, with the homology H∗ (Σ m ; Z) = H∗ (S m ; Z).
Trans. Amer. Math. Soc. 144 (1969) 67–72

Theorem (M. Kervaire)
Any homology 4-sphere bounds a contractible 5-disk.
For any homology m-sphere with m ­ 5, there is a unique
homotopy m-sphere such that the connected sum of thetwo spheres bounds a homotopy (m + 1)-disk.
Math. Research Letters 7 (2000) 757–766

Theorem (Y. Fukumoto, M. Furuta)
There exist homology 3-spheres which bound homology
4-disks, and some of the homology 4-disks are contractible.
Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology spheres

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology spheres
Let G be a compact Lie group.

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology spheres
Let G be a compact Lie group.
Corollary

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology spheres
Let G be a compact Lie group.
Corollary

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology spheres
Let G be a compact Lie group.
Corollary
There are families of homology 3-spheres such that for eachof the homology 3-sphere Σ 3 ,Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology spheres
Let G be a compact Lie group.
Corollary
There are families of homology 3-spheres such that for eachof the homology 3-sphere Σ 3 , there exists a smooth actionof G on a sphere S n+3 such that F is diffeomorphic to Σ 3 .

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology spheres
Let G be a compact Lie group.
Corollary
There are families of homology 3-spheres such that for eachof the homology 3-sphere Σ 3 , there exists a smooth actionof G on a sphere S n+3 such that F is diffeomorphic to Σ 3 .
For any homology 4-spheres Σ 4 ,Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology spheres
Let G be a compact Lie group.
Corollary
There are families of homology 3-spheres such that for eachof the homology 3-sphere Σ 3 , there exists a smooth actionof G on a sphere S n+3 such that F is diffeomorphic to Σ 3 .
For any homology 4-spheres Σ 4 , there exists a smooth actionof G on a sphere S n+4 such that F is diffeomorphic to Σ 4 .

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology spheres
Let G be a compact Lie group.
Corollary
There are families of homology 3-spheres such that for eachof the homology 3-sphere Σ 3 , there exists a smooth actionof G on a sphere S n+3 such that F is diffeomorphic to Σ 3 .
For any homology 4-spheres Σ 4 , there exists a smooth actionof G on a sphere S n+4 such that F is diffeomorphic to Σ 4 .
For any homology m-spheres Σ m with m ­ 5,

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology spheres
Let G be a compact Lie group.
Corollary
There are families of homology 3-spheres such that for eachof the homology 3-sphere Σ 3 , there exists a smooth actionof G on a sphere S n+3 such that F is diffeomorphic to Σ 3 .
For any homology 4-spheres Σ 4 , there exists a smooth actionof G on a sphere S n+4 such that F is diffeomorphic to Σ 4 .
For any homology m-spheres Σ m with m ­ 5, there exists
a smooth action of G on a sphere S m+n such that F is
homeomorphic to Σ m .

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology spheres
Let G be a compact Lie group.
Corollary
There are families of homology 3-spheres such that for eachof the homology 3-sphere Σ 3 , there exists a smooth actionof G on a sphere S n+3 such that F is diffeomorphic to Σ 3 .
For any homology 4-spheres Σ 4 , there exists a smooth actionof G on a sphere S n+4 such that F is diffeomorphic to Σ 4 .
For any homology m-spheres Σ m with m ­ 5, there exists
a smooth action of G on a sphere S m+n such that F is
homeomorphic to Σ m .
We may always assume that n is even and at a chosen pointx ∈ Σm,Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Homology spheres
Let G be a compact Lie group.
Corollary
There are families of homology 3-spheres such that for eachof the homology 3-sphere Σ 3 , there exists a smooth actionof G on a sphere S n+3 such that F is diffeomorphic to Σ 3 .
For any homology 4-spheres Σ 4 , there exists a smooth actionof G on a sphere S n+4 such that F is diffeomorphic to Σ 4 .
For any homology m-spheres Σ m with m ­ 5, there exists
a smooth action of G on a sphere S m+n such that F is
homeomorphic to Σ m .
We may always assume that n is even and at a chosen pointx ∈ Σ m , the normal G -module is the realification of a complex
G -module.
Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

For a closed 4-manifold X ,Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

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 Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

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 Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

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 Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

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 Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

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 Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

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 Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

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 Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums
To appear in J. Sympl. Geom.

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums
To appear in J. Sympl. Geom.

Proposition (M. Kaluba, W. Politarczyk)

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums
To appear in J. Sympl. Geom.

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

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums
To appear in J. Sympl. Geom.

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

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums
To appear in J. Sympl. Geom.

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 Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums
To appear in J. Sympl. Geom.

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 Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums
To appear in J. Sympl. Geom.

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 Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums
To appear in J. Sympl. Geom.

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 Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums
To appear in J. Sympl. Geom.

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 Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums
To appear in J. Sympl. Geom.

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 Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums
To appear in J. Sympl. Geom.

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 Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums
To appear in J. Sympl. Geom.

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 Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums
To appear in J. Sympl. Geom.

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 Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums
To appear in J. Sympl. Geom.

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
the result of Taubes (1994)

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums
To appear in J. Sympl. Geom.

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
the result of Taubes (1994) that E is not symplectic

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums
To appear in J. Sympl. Geom.

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
the result of Taubes (1994) that E is not symplectic and thus,
X #M is not symplectic too.
Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

Osaka J. Math. 28 (1991) 243–253

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

Osaka J. Math. 28 (1991) 243–253

Lemma (Y. Sato)

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

Osaka J. Math. 28 (1991) 243–253

Lemma (Y. Sato)
There exists a homology 4-sphere Σ 4 with π1 (Σ 4 ) ∼= SL(2, 5).

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

Osaka J. Math. 28 (1991) 243–253

Lemma (Y. Sato)
There exists a homology 4-sphere Σ 4 with π1 (Σ 4 ) ∼= SL(2, 5).
By the result of Kaluba and Politarczyk (to appear in J. S. Geom.)
applied for X = CP 2

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

Osaka J. Math. 28 (1991) 243–253

Lemma (Y. Sato)
There exists a homology 4-sphere Σ 4 with π1 (Σ 4 ) ∼= SL(2, 5).
By the result of Kaluba and Politarczyk (to appear in J. S. Geom.)
applied for X = CP 2 and M = Σ 4 , Sato’s homology 4-sphere,

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

Non-symplectic connected sums

Osaka J. Math. 28 (1991) 243–253

Lemma (Y. Sato)
There exists a homology 4-sphere Σ 4 with π1 (Σ 4 ) ∼= SL(2, 5).
By the result of Kaluba and Politarczyk (to appear in J. S. Geom.)
applied for X = CP 2 and M = Σ 4 , Sato’s homology 4-sphere,
the connected sum CP 2 #Σ 4 is not a symplectic manifold.


Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds

ARIGATO GOSAIMASTA!

Krzysztof Pawałowski (UAM Poznań Poland)

Non-product smooth actions on Cartesian products of manifolds