Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

On isovariant rigidity of CAT(0) manifolds, I
by Qayum Khan (University of Notre Dame, U.S.A.)
joint with J.F. Davis and F.X. Connolly

RIMS Symposium (01 Sep 2010):
Transformation Groups and Surgery Theory
Kyoto University, JAPANTopological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Topological Rigidity, I

How close together are algebraic topology and geometric topology?

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Topological Rigidity, I

How close together are algebraic topology and geometric topology?
Conjecture (Borel, 1953)
Let X be an aspherical, closed topological manifold.

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Topological Rigidity, I

How close together are algebraic topology and geometric topology?
Conjecture (Borel, 1953)
Let X be an aspherical, closed topological manifold.
Suppose M is a closed topological manifold. If f : M → X is a
homotopy equivalence, then f is homotopic to a homeomorphism.

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Topological Rigidity, I

How close together are algebraic topology and geometric topology?
Conjecture (Borel, 1953)
Let X be an aspherical, closed topological manifold.
Suppose M is a closed topological manifold. If f : M → X is a
homotopy equivalence, then f is homotopic to a homeomorphism.
We say that X is topologically rigid if this is true.

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Topological Rigidity, II

Here is an important, seminal example.
Theorem
For all n ≥ 0, the n-torus T n is topologically rigid.

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Topological Rigidity, II

Here is an important, seminal example.
Theorem
For all n ≥ 0, the n-torus T n is topologically rigid.
The case n ≤ 2 is true by the classification of curves and surfaces.
The case n ≥ 5 was proven by Hsiang–Wall (1969).
The case n = 4 was proven by Freedman–Quinn (1990).
The case n = 3 was by Waldhausen (1968) and Perelman (2003).

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Topological Rigidity, III

That example is a member of a certain differential-geometric class.
Theorem (Farrell–Jones, 1993)
Let X be a closed smooth manifold of dimension ≥ 5.
Suppose X admits a Riemannian metricsuch that each sectional curvature is non-positive (≤ 0).
Then X is topologically rigid.

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Topological Rigidity, IV

That hypothesis can be weakened with geometric group theory.
Theorem (Bartels–Lück, 2009)
Let X be a closed topological manifold of dimension ≥ 5.
e admits a CAT(0) metricSuppose the universal cover Xe.
and the fundamental group π1 (X ) acts isometrically on X
Then X is topologically rigid.

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of the Torus, I
Recall the n-torus T n = Rn /Zn , which we discussed earlier.

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of the Torus, I
Recall the n-torus T n = Rn /Zn , which we discussed earlier.
Theorem (Connolly–Davis–Khan, 2010)
Consider all homeomorphisms σ : T n → T n such that:σ 2 = id on T n , andσ∗ = −id on H1 (T n ; Z).

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of the Torus, I
Recall the n-torus T n = Rn /Zn , which we discussed earlier.
Theorem (Connolly–Davis–Khan, 2010)
Consider all homeomorphisms σ : T n → T n such that:σ 2 = id on T n , andσ∗ = −id on H1 (T n ; Z).
1

If n ≡ 0, 1 (mod 4) or n = 3, then there is a unique conjugacy
class of such involutions σ, represented by σ0 : [x] 7→ [−x].

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of the Torus, I
Recall the n-torus T n = Rn /Zn , which we discussed earlier.
Theorem (Connolly–Davis–Khan, 2010)
Consider all homeomorphisms σ : T n → T n such that:σ 2 = id on T n , andσ∗ = −id on H1 (T n ; Z).
1

If n ≡ 0, 1 (mod 4) or n = 3, then there is a unique conjugacy
class of such involutions σ, represented by σ0 : [x] 7→ [−x].

2If n ≡ 2, 3 (mod 4) with n > 3, then there are infinitely many
conjugacy classes of such involutions σ, all locally linear.

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of the Torus, I
Recall the n-torus T n = Rn /Zn , which we discussed earlier.
Theorem (Connolly–Davis–Khan, 2010)
Consider all homeomorphisms σ : T n → T n such that:σ 2 = id on T n , andσ∗ = −id on H1 (T n ; Z).
1

If n ≡ 0, 1 (mod 4) or n = 3, then there is a unique conjugacy
class of such involutions σ, represented by σ0 : [x] 7→ [−x].

2If n ≡ 2, 3 (mod 4) with n > 3, then there are infinitely many
conjugacy classes of such involutions σ, all locally linear.

The proof follows from the next three technical theorems.

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of the Torus, II

Before moving on, consider the following surgery-theoretic concept.
Definition
Let X be a topological manifold equipped with a cocompact actionof a discrete group Γ. The equivariant structure set STOP (X , Γ)is the set of equivalence classes of all pairs (M, f ) such that:M ⊂ R∞ is a manifold equipped with a cocompact Γ-action,f : M → X is a Γ-equivariant homotopy equivalence.

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of the Torus, II

Before moving on, consider the following surgery-theoretic concept.
Definition
Let X be a topological manifold equipped with a cocompact actionof a discrete group Γ. The equivariant structure set STOP (X , Γ)is the set of equivalence classes of all pairs (M, f ) such that:M ⊂ R∞ is a manifold equipped with a cocompact Γ-action,f : M → X is a Γ-equivariant homotopy equivalence.
Here, the pair (M, f ) is equivalent to another such pair (M 0 , f 0 ) if
there exists a Γ-equivariant homeomorphism h : M → M 0 such thatf 0 ◦ h is equivariantly homotopic to f .

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of the Torus, III
Using Smith theory, we quantify those involutions to count them.
Theorem (Connolly–Davis–Khan, 2010)
The set of conjugacy classes of topological involutions
σ : T n → T n such that σ∗ = −id on H1 (T n )is in bijective correspondence with the equivariant structure set
STOP (T n , C2 ).

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of the Torus, III
Using Smith theory, we quantify those involutions to count them.
Theorem (Connolly–Davis–Khan, 2010)
The set of conjugacy classes of topological involutions
σ : T n → T n such that σ∗ = −id on H1 (T n )is in bijective correspondence with the equivariant structure set
STOP (T n , C2 ).
Here, the action of C2 on T n = S 1 × · · · × S 1 is generated by
σ0 : (z1 , . . . , zn ) 7→ (z 1 , . . . , z n ).

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of the Torus, IV
Consider the associated crystallographic group Γn acting on Rn :Γn := Zn o−1 C2 .

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of the Torus, IV
Consider the associated crystallographic group Γn acting on Rn :Γn := Zn o−1 C2 .

Theorem (Connolly–Davis–Khan, 2010)
Suppose n ≥ 4 and write ε := (−1)n . Then
STOP (T n , C2 ) ∼= STOP (Rn , Γn ) ∼=M

UNiln+ε (D).

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of the Torus, IV
Consider the associated crystallographic group Γn acting on Rn :Γn := Zn o−1 C2 .

Theorem (Connolly–Davis–Khan, 2010)
Suppose n ≥ 4 and write ε := (−1)n . Then
STOP (T n , C2 ) ∼= STOP (Rn , Γn ) ∼=M

UNiln+ε (D).

Here, the direct sum is indexed by conjugacy classes (D) of
maximal infinite dihedral subgroups D ∼= D∞ of Γn .

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of the Torus, V

The above abelian groups have already been calculated.
Theorem (Banagl–Connolly–Davis–Ranicki, 2004)
For each integer m, there is an isomorphism
0
if m ≡ 0
0if m ≡ 1
UNilm (D∞ ) ∼=⊕∞(Z/2Z)
if m ≡ 2
(Z/2Z ⊕ Z/4Z)⊕∞ if m ≡ 3(mod 4)(mod 4)(mod 4)(mod 4).

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of the Torus, V

The above abelian groups have already been calculated.
Theorem (Banagl–Connolly–Davis–Ranicki, 2004)
For each integer m, there is an isomorphism
0
if m ≡ 0
0if m ≡ 1
UNilm (D∞ ) ∼=⊕∞(Z/2Z)
if m ≡ 2
(Z/2Z ⊕ Z/4Z)⊕∞ if m ≡ 3(mod 4)(mod 4)(mod 4)(mod 4).

The last two answers are given by generalized Arf invariants.

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of CAT(0) Manifolds, I
The Borel Conjecture has an equivariant generalization.
Conjecture (Quinn, 1986, modified from original)
Let Γ be a discrete group with no elements of order two.
Let X and M be topological manifolds without boundary
equipped with cocompact, proper actions of Γ such that:

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of CAT(0) Manifolds, I
The Borel Conjecture has an equivariant generalization.
Conjecture (Quinn, 1986, modified from original)
Let Γ be a discrete group with no elements of order two.
Let X and M be topological manifolds without boundary
equipped with cocompact, proper actions of Γ such that:
X is a model for the classifying space E Γ = Efin Γ,Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of CAT(0) Manifolds, I
The Borel Conjecture has an equivariant generalization.
Conjecture (Quinn, 1986, modified from original)
Let Γ be a discrete group with no elements of order two.
Let X and M be topological manifolds without boundary
equipped with cocompact, proper actions of Γ such that:
X is a model for the classifying space E Γ = Efin Γ,each fixed set of X and of M is a locally flat submanifold,Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of CAT(0) Manifolds, I
The Borel Conjecture has an equivariant generalization.
Conjecture (Quinn, 1986, modified from original)
Let Γ be a discrete group with no elements of order two.
Let X and M be topological manifolds without boundary
equipped with cocompact, proper actions of Γ such that:
X is a model for the classifying space E Γ = Efin Γ,each fixed set of X and of M is a locally flat submanifold, andeach fixed set of X has codimension ≥ 3 in bigger fixed sets.

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of CAT(0) Manifolds, I
The Borel Conjecture has an equivariant generalization.
Conjecture (Quinn, 1986, modified from original)
Let Γ be a discrete group with no elements of order two.
Let X and M be topological manifolds without boundary
equipped with cocompact, proper actions of Γ such that:
X is a model for the classifying space E Γ = Efin Γ,each fixed set of X and of M is a locally flat submanifold, andeach fixed set of X has codimension ≥ 3 in bigger fixed sets.
Let f : M → X be a simple Γ-isovariant homotopy equivalence.
Then f is isovariantly homotopic to a homeomorphism.

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of CAT(0) Manifolds, I
The Borel Conjecture has an equivariant generalization.
Conjecture (Quinn, 1986, modified from original)
Let Γ be a discrete group with no elements of order two.
Let X and M be topological manifolds without boundary
equipped with cocompact, proper actions of Γ such that:
X is a model for the classifying space E Γ = Efin Γ,each fixed set of X and of M is a locally flat submanifold, andeach fixed set of X has codimension ≥ 3 in bigger fixed sets.
Let f : M → X be a simple Γ-isovariant homotopy equivalence.
Then f is isovariantly homotopic to a homeomorphism.
In this case, we say (X , Γ) is isovariantly topologically rigid.

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of CAT(0) Manifolds, II
Here, we mean “simple” in the equivariant sense.

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of CAT(0) Manifolds, II
Here, we mean “simple” in the equivariant sense.
Definition (Steinberger, Prassidis)
Let Γ be a discrete group. Let M and X be cocompact Γ-ANRs.
Suppose f : M → X is a Γ-equivariant homotopy equivalence.

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of CAT(0) Manifolds, II
Here, we mean “simple” in the equivariant sense.
Definition (Steinberger, Prassidis)
Let Γ be a discrete group. Let M and X be cocompact Γ-ANRs.
Suppose f : M → X is a Γ-equivariant homotopy equivalence. We
say that f is simple if the the torsion τ (f ) := f∗ (Cyl(f ), M)
represents the zero element in the abelian group Whtop (X , Γ).

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of CAT(0) Manifolds, II
Here, we mean “simple” in the equivariant sense.
Definition (Steinberger, Prassidis)
Let Γ be a discrete group. Let M and X be cocompact Γ-ANRs.
Suppose f : M → X is a Γ-equivariant homotopy equivalence. We
say that f is simple if the the torsion τ (f ) := f∗ (Cyl(f ), M)
represents the zero element in the abelian group Whtop (X , Γ).
Here, a strong Γ-deformation retraction pair (Y , X ) represents zero
if there exist Γ-cell-like pairs (Y 0 , Y ) and (Y 0 , X ).

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of CAT(0) Manifolds, II
Here, we mean “simple” in the equivariant sense.
Definition (Steinberger, Prassidis)
Let Γ be a discrete group. Let M and X be cocompact Γ-ANRs.
Suppose f : M → X is a Γ-equivariant homotopy equivalence. We
say that f is simple if the the torsion τ (f ) := f∗ (Cyl(f ), M)
represents the zero element in the abelian group Whtop (X , Γ).
Here, a strong Γ-deformation retraction pair (Y , X ) represents zero
if there exist Γ-cell-like pairs (Y 0 , Y ) and (Y 0 , X ).
RemarkIf we drop the “simple” hypothesis in the previous conjecture, thencounterexamples were constructed by Connolly–Koźniewski (1991).

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of CAT(0) Manifolds, III

Now, consider the following case with very little stratified data.

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of CAT(0) Manifolds, III

Now, consider the following case with very little stratified data.
Definition
Let X be a topological space equipped with an action of group Γ.

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of CAT(0) Manifolds, III

Now, consider the following case with very little stratified data.
Definition
Let X be a topological space equipped with an action of group Γ.
The action is pseudofree if the singular set Xsing is discrete:
Xsing := {x ∈ X | Γx 6= 1}.

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of CAT(0) Manifolds, IV

Modern tools allow now an initial attack on Equivariant Rigidity.

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of CAT(0) Manifolds, IV

Modern tools allow now an initial attack on Equivariant Rigidity.
Theorem (Connolly–Khan, 2010)
Let X be a topological manifold of dimension n ≥ 5 equipped witha CAT(0) metric. Let Γ be a discrete group of isometries of X .

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of CAT(0) Manifolds, IV

Modern tools allow now an initial attack on Equivariant Rigidity.
Theorem (Connolly–Khan, 2010)
Let X be a topological manifold of dimension n ≥ 5 equipped witha CAT(0) metric. Let Γ be a discrete group of isometries of X .
Suppose the action is cocompact, proper, and pseudofree.

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of CAT(0) Manifolds, IV

Modern tools allow now an initial attack on Equivariant Rigidity.
Theorem (Connolly–Khan, 2010)
Let X be a topological manifold of dimension n ≥ 5 equipped witha CAT(0) metric. Let Γ be a discrete group of isometries of X .
Suppose the action is cocompact, proper, and pseudofree. Then
M
STOP (X , Γ) ∼UNiln+1 (D, ε).
=Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of CAT(0) Manifolds, IV

Modern tools allow now an initial attack on Equivariant Rigidity.
Theorem (Connolly–Khan, 2010)
Let X be a topological manifold of dimension n ≥ 5 equipped witha CAT(0) metric. Let Γ be a discrete group of isometries of X .
Suppose the action is cocompact, proper, and pseudofree. Then
M
STOP (X , Γ) ∼UNiln+1 (D, ε).
=Here, the direct sum is indexed by conjugacy classes (D) of
maximal infinite dihedral subgroups D ∼= D∞ of Γ.

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of CAT(0) Manifolds, IV

Modern tools allow now an initial attack on Equivariant Rigidity.
Theorem (Connolly–Khan, 2010)
Let X be a topological manifold of dimension n ≥ 5 equipped witha CAT(0) metric. Let Γ be a discrete group of isometries of X .
Suppose the action is cocompact, proper, and pseudofree. Then
M
STOP (X , Γ) ∼UNiln+1 (D, ε).
=Here, the direct sum is indexed by conjugacy classes (D) of
maximal infinite dihedral subgroups D ∼= D∞ of Γ. Also, theorientation character ε : Γ → {±1} is given by g 7→ deg(g ).

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of CAT(0) Manifolds, V

Thus we affirm Quinn’s Conjecture in the following special case.

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of CAT(0) Manifolds, V

Thus we affirm Quinn’s Conjecture in the following special case.
Corollary (Connolly–Khan, 2010)
Let X be a topological manifold of dimension ≥ 5 equipped witha CAT(0) metric. Let Γ be a discrete group of isometries of X

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of CAT(0) Manifolds, V

Thus we affirm Quinn’s Conjecture in the following special case.
Corollary (Connolly–Khan, 2010)
Let X be a topological manifold of dimension ≥ 5 equipped witha CAT(0) metric. Let Γ be a discrete group of isometries of X
with no elements of order two.

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of CAT(0) Manifolds, V

Thus we affirm Quinn’s Conjecture in the following special case.
Corollary (Connolly–Khan, 2010)
Let X be a topological manifold of dimension ≥ 5 equipped witha CAT(0) metric. Let Γ be a discrete group of isometries of X
with no elements of order two. Suppose the action is cocompact,
proper, and pseudofree.

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of CAT(0) Manifolds, V

Thus we affirm Quinn’s Conjecture in the following special case.
Corollary (Connolly–Khan, 2010)
Let X be a topological manifold of dimension ≥ 5 equipped witha CAT(0) metric. Let Γ be a discrete group of isometries of X
with no elements of order two. Suppose the action is cocompact,
proper, and pseudofree. Then X is equivariantly topologically rigid.

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of CAT(0) Manifolds, V

Thus we affirm Quinn’s Conjecture in the following special case.
Corollary (Connolly–Khan, 2010)
Let X be a topological manifold of dimension ≥ 5 equipped witha CAT(0) metric. Let Γ be a discrete group of isometries of X
with no elements of order two. Suppose the action is cocompact,
proper, and pseudofree. Then X is equivariantly topologically rigid.
Consequently, we generalize the Bartels–Lück Theorem, which isthe case when Γ is torsion-free.

Topological Rigidity

Equivariant Rigidity of Torus

Equivariant Rigidity of CAT(0) Manifolds

Equivariant Rigidity of CAT(0) Manifolds, V

Thus we affirm Quinn’s Conjecture in the following special case.
Corollary (Connolly–Khan, 2010)
Let X be a topological manifold of dimension ≥ 5 equipped witha CAT(0) metric. Let Γ be a discrete group of isometries of X
with no elements of order two. Suppose the action is cocompact,
proper, and pseudofree. Then X is equivariantly topologically rigid.
Consequently, we generalize the Bartels–Lück Theorem, which isthe case when Γ is torsion-free. Our ideas further develop the
topological application of their recent solution to the Farrell–Jones
Conjecture. This is a powerful algebraic tool we discuss next time...