Actions of homeomorphism groups of manifolds admitting a nontrivial finite free action
Lei Chen

Non-Hermitian Linear Response Theory
    Hui Zhai
    Received: date / Accepted: date
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In this paper, we study the action of _0(M), the identity component of the group of homeomorphisms of an n-dimensional manifold M with an _p-free action, on another manifold N of dimension n+k<2n. We prove that if M is not an _p-homology sphere, then N≅ M× K for a homology manifold K such that the action of _0(M) on M is standard and on K is trivial. In particular, for M=S^n a sphere, any nontrivial action is a generalization of the “coning-off" construction.



§ INTRODUCTION

For a manifold M, denote by (M) the group of homeomorphisms of M and by _0(M) the identity component of the group of homeomorphisms of M.  Let M,N be two connected, closed, topological manifolds, and let 
    ρ: _0(M)→(N)
 
be a group homomorphism. Recently the authors <cit.> showed that every orbit of the action ρ is a “submanifold" (continuous, injective image of a manifold) of N; in particular, every orbit is a continuous, injective image of a cover of _m(M) for some m, where _m(M) denotes the space of unordered m-tuples of distinct points on M. This orbit rigidity result means that any nontrivial action ρ is, in some sense, geometric. It reduces the study of _0(M) acting on N to the study of how to decompose N into a union of covers of _m(M) so that actions of _0(M) on pieces glue together. Do all possible decompositions geometric in some sense? We are puzzled by the following question: Given M, for which manifolds N, do there exist a nontrivial ρ? How many different actions can we find?

Some results in low dimensions
The followings have been proved in <cit.>.


  * if (M)>(N), then there is no nontrivial ρ; 

  * if (M)=(N), the existence of a nontrivial ρ is equivalent to the fact that an admissible cover (which is defined in <cit.>) of M can be embedded in N; 

  * if (M)+1=(N) and every admissible cover of M is compact, then ρ exists if and only if M=S^n. 

What will happen when (N) is bigger?

A zoo of examples


  * Product action: _0(M) acts on M^k× K (product of k copies of M with K) where the action is diagonal on M^k and trivial on K. 

  * Product action quotient by Bing's shrinking: Let p: K→ K' be a Bing's shrinking map <cit.>. Define M_p:=M× M× K/∼, where (m,m,k_1)∼ (m,m,k_2) if p(k_1)=p(k_2). The action of _0(M) on M× M× K/∼ is the diagonal on M× M and trivial on K. The fact that M_p is a topological manifold follows from the fact that p is a Bing's shrinking map.

  * The suspension action: _0(S^n) acts on S^n+l by lifting the standard action to the l-fold suspension Σ^l S^n=S^n+l.

  * Double suspension action: If H is an n-dimensional homology manifold, then the double suspension theorem <cit.> implies that there is a homeomorphism Σ^2H≅ S^n+2. This means that _0(H) acts on S^n+2 nontrivially with global fixed point set a circle. 

This is a showcase that the action can be quite weird, we can multiply and quotient under Bing's shrinking map. Therefore we ask the following general question.

Are there constructions other than modifying geometric actions? 

When the dimension n+k of N satisfies k+n<2n, we only have two types of orbit: either a global fixed point or a cover of M. The difficulty in this range is how to control the topology of the global fixed point set. In this paper, we make use of  the“shrinking property", which we discuss in Section 2 to control the local topology of the global fixed point set. We call τ∈_0(M) a free order p element if τ has order p and τ has no global fixed points on M. With the extra condition that there exists a free order p element in _0(M), we deduce the following.

Assume that admissible covers of M are trivial and there exists a free order p element in _0(M). Given a nontrivial homomorphism 
    ρ: _0(M)→(N),

then one of the following happens:


  * N≅ M× K for a -homology manifold K such that the action of _0(M) on M is standard and on K is trivial;

  * M is an _p-homology sphere and the global fixed point set is a union of finite (k-1)-dimensional _p-homology manifolds.





  * The above theorem applies for M=G× M' where M' is simply connected and G is a compact Lie group. 


  * We do not know if (2) of Theorem <ref> ever happens when M is not a ℤ-homology sphere. While if M is a -homology sphere, we know that such actions exist because of the double suspension theorem <cit.>.


  * We do not know if the above theorem depends on whether or not _0(M) contains a free order p element. We are not able to construct any non-product example when M is simply connected but not a sphere.


From the computation of this paper, the local homology of the global fixed set contains the cohomology of M with a degree shift. Spaces with big local homology seem hard to exist. Therefore, we have the following conjecture for general simply connected M. 

Let M be a simply connected manifold that is not homeomorphic to S^n. If _0(M) acts nontrivially on N of dimension n+k<2n, then the action has no global fixed points.


When M=S^n, we obtain the following rigidity theorem.

For (N)=n+k<2n, given a nontrivial homomorphism 
    ρ: _0(S^n)→(N),

there exists a -homology manifold K_1 with boundary K_0 such that 

    N≅  S^n × K_1/∼,

where (m,x)∼ (n,y) if x=y∈ K_0. The action of ρ on N is induced by the standard action on S^n and the trivial action on K_1.

The above construction is a higher dimensional generalization of the “coning-off" construction (or the suspension action) we discussed in <cit.>. When K_1=[0,1], the action ρ is the suspension action on S^n+1=Σ S^n. 


Proof idea of the paper The tool we use other than orbit classification theorem is the Borel-Moore homology. The key ingredient is the local Smith theory which is why our theorem is limited to manifolds M with finite actions. We also need to modify the “shrinking argument" that we use in <cit.>. 

Notation When we use -homology manifolds, we omit ; but for other coefficients, we keep the coefficient symbols. 

Acknowledgement We thank Shmuel Weinberger, Kathryn Mann and Benson Farb for helpful discussion. 




§ THE PROOF OF THEOREM <REF>

Let ρ: _0(M)→(N) be a homomorphism. Denote by 
    F=(ρ(_0(M)),

the global fixed point set of the action ρ. We have the following two ingredients of proof:


  * By the free order p element τ in _0(M), we know that F is a union of finitely many _p-homology manifolds (of various dimensions) by the local Smith theory;

  * A “shrinking property" near a global fixed that we obtained from ρ gives a way to compute the local homology of F.

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Admissible covers of M
Before we state the orbit classification theorem, we first discuss admissible covers of M.

A regular cover p: M'→ M is called an admissible cover if there exists a homomorphism
    L:_0(M)→(M'),
such that L(f) is a lift of f. 

For example, Simply-connected manifolds have trivial admissible covers; the universal cover of the circle S^1 is not admissible; however, the universal cover of S_g a genus g>1 surface is admissible; when M is connected Lie group, then admissible covers are trivial. More about which covers are admissible is discussed in <cit.>. 
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We now state the following orbit classification theorem and flat bundle structure theorem from <cit.>, which is the starting point of the whole argument.

Let M,N be two connected, closed manifolds and let ρ:_0(M)→(N) be a homomorphism when (N)<2(M). Then every orbit is either a global fixed point or a continuous image of an admissible cover of M. Let F denote the global fixed point set. There is a topological flat bundle (homeomorphic to M× K/π_1(M) for a topological space K with a π_1(M)-action)

    p: N-F→ M,

such that the action on N-F is a lift of the standard action on M.


Under the extra assumption that admissible covers of M are trivial, we have the following corollary.

With the assumptions as in Theorem <ref>, and that any admissible cover of M is trivial, we have that N-F≅ M× K such that _0(M) acts  standardly on M and trivially on K.





Denote by 

    ℰ: M× K→ N-F

the embedding obtained from the action ρ. 

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Combining Theorem <ref> with the existence of τ∈_0(M) such that τ is a free order p element, we know that the fixed point set of  ρ(τ) is also F since the action of ρ(τ) on N-F≅ M× K is free. With the help of ρ(τ), the topology of the fixed point set F is controlled by the local Smith theory.

We refer the reader to Bredon's book <cit.> for homology manifolds; see <cit.>. Morally, a homology manifold is a locally compact space whose local homology is the same as that of an actual manifold but it may not have the local Euclidean structure. We first state the following local Smith theory <cit.>.

Let ϕ∈(N) be an order p element. Then (ϕ) is a disjoint union of finitely many _p-homology manifolds (of possibly various dimensions).

By Theorem <ref>, we know that F=(ρ(τ)) is a union of finitely many _p-homology manifolds. To compute the local homology of F, we need the following “shrinking property". The “shrinking property" is also used in <cit.> to solve the extension problem of Ghys. 

For m∈ M, let {k_n∈ K} be a sequence such that ℰ(m,k_n)∈ N-F  converges to a global fixed point P of ρ. Then the sequence of compact spaces ℰ(M×{k_n}) converges to P; i.e., for any neighborhood U of P, there exist N such that for n>N, we have that ℰ(M×{k_n})⊂ U.


We prove the statement by contradiction. If not, then there exists a sequence {m_n∈ M} such that {ℰ(m_n,k_n)} has a subsequence that does not converge to P. Passing to a further subsequence, assume that m_n→ m', which is possible since M is compact. Let {f_n∈_0(M)} be a sequence of elements such that 

    f_n→ f and f_n(m)=m_n.
 
Since the action ρ is continuous (any homomorphism ρ is automatically continuous by <cit.>) and 

    ℰ(m,k_n)→ P,

we know that 

    ℰ(m_n,k_n)=ρ(f_n)(ℰ(m,k_n))→ρ(f)(P)=P.

This clearly contradicts the assumption that {ℰ(m_n,k_n)} does not converge to P.

We now start the investigation of the local homology of F.

We have the following two statements:


  * M is an _p-homology sphere (a topological manifold with the same _p-homology as a sphere).

  * F is a disjoint union of k-1-dimensional _p-homology manifolds (local homology, maybe not be a topological manifold).




Near a fixed point P∈ F, we take a decreasing sequence of open neighborhoods {U_m} such that U_m is homeomorphic to the Euclidean space and ⋂_m U_m={x}. For the following computation, we use Borel-Moore homology; see e.g., <cit.>.

Denote by A_m:=U_m∩ F. By <cit.>, we have the following equation about the stalk of the local homology of F:

    ℋ_*(F;_p)_x=lim_⟶ H_*(A_m;_p),

where the limit is taken over the induced map on homology of embeddings A_m+1↪ A_m.  If F is a homology manifold then ℋ_*(F;_p)_x=0 for *≠(F) at x.

We have the following universal coefficient theorem of Borel-Moore homology <cit.>,

    0→Ext(H^p+1_c(A_m;_p);_p)→ H_p(A_m;_p)→Hom(H^p_c(A_m;_p);_p)→ 0.



By Poincaré duality <cit.>, we have the following 

    H^p_c(A_m;_p)≅ H_k+n-p^c(U_m,U_m-A_m;_p) ≅ H_k+n-p-1^c(U_m)|_U_m- A_m(U_m- A_m;_p)

for p≠ k+n-1 (because H^c_*(U_m;_p)=0 for *≠ n+k). However, by Theorem <ref>, there is a projection of flat bundle

    p: U_m-A_m→ M,

and by Proposition <ref>, there is an embedding for any U_m:

    M U_m-A_m

such that p∘ e_m=id. 
Therefore we have the following embedding of homology group:

    e_m*: H_*^c(M;_p)↪ H_*^c(U_m-A_m;_p),

so that e_m+1* factors through e_m*. In summary, we have the following embedding (a combination of (<ref>) and (<ref>))

    H_k+n-p-1^c(M;_p)↪lim_⟵ H_k+n-p-1^c(U_m)|_U_m- A_m(U_m-A_m;_p)≅lim_⟵ H^p_c(A_m;_p).



Since F is a local _p-homology manifold, we know that H_*(F;_p)_x is nontrivial only when *=(F) at x. Since M is a closed n-dimensional manifold, we know that H_n(M;_p)≅_p. Therefore ℋ_k-1(F;_p)_x has an embedded copy of _p, which implies that each connected component of F is a (k-1)-dimensional _p-homology manifold. 

Therefore, we know also that H^*(M;_p) is nontrivial only when *=0 or n.

Theorem <ref> is an easy corollary of Lemma <ref> and the proof is left to the reader.




§ _0(S^N) ACTS ON N

In this section, we prove Theorem <ref>. Before that, we show that the space described in Theorem <ref> is still a homology manifold.

Let K_1 be a -homology manifold with boundary K_0. Then

    N= S^n × K_1/∼,

where (m,x)∼ (n,y) if x=y∈ K_0, is a homology manifold.


The key is that we can redefine N as a quotient of shrinking map. Let ϕ: S^n×∂ K_1→∂ D^n+1× K_0 be the identity map. Then

    N=S^n× K_1 ∪_ϕ D^n+1× K_0/∼,

where (x,t)∼ (y,t) for x,y ∈ D^n+1 and t∈ K_0. We know that S^n× K_1 ∪_ϕ D^n+1× K_0 is a homology manifold because it is a union of two homology manifolds glued along a common boundary. We know that N is a homology manifold because of Wilder's monotone mapping theorem <cit.> and that N is a quotient of a homology manifold by acyclic sets.



For (N)=k+n<2n, let 
    ρ:_0(S^n)→(N)

be a nontrivial homomorphism. 

Since (N)<2n, every orbit is either a global fixed point or a continuous image of S^n by Theorem <ref>. Therefore we have the following decomposition

    N≅ F∪ℰ(S^n× K)

where ℰ: S^n× K→ N-F is the embedding obtained from the trivial flat bundle structure of Theorem <ref>.
By <cit.>, we know that K is a homology manifold.
Firstly, we have the following description of F.

The fixed point set F is a finite union of closed k-1-dimensional 𝔽_2-homology manifold.


Let A=(/2)^n⊂ SO(n+1) be the subgroup consisting of diagonal matrices of entries ± 1 with determinant 1. Let A_0⊂ A be the subgroup of A consisting of matrices whose last entry 1. 

The action of A on S^n has no global fixed point. The action of A_0 on S^n has two fixed points (0,...,0,± 1)∈ S^n. Since the fixed point of ρ(A) on N is F, we know that F is a union of finitely many _2-homology manifolds.

Denote by C the global fixed point set of ρ(A_0). Since C is an _2-homology manifold and that C∩ N-F is a homology manifold of dimension k, we know that (C)=k as an _2-homology manifold. Since F divides C into two connected component. We know that F has dimension k-1.


We have the following stronger statement about F.

F is a k-1 dimensional homology manifold and F is the boundary of ℰ(m× K) for any m∈ M. 


Again, we use the subgroup A,A_0 as in the proof of Lemma <ref>. Again denote by C the global fixed point set of ρ(A_0).

Define B=diag(0,...,0,+1,-1)∈ SO(n+1), which is an involution in _0(S^n). The action ρ(B) restricts to an _2 action on C. We now use the Mayer–Vietoris sequence and local Smith theory to compute local homology of C at points of F.

Since the _2-dimension of F is k-1. A local version of <cit.> (See also <cit.>) gives us that for x∈ F,

    H_*(C/ρ(B);)_x=0

for any *. Denote  N:=(0,...,+1)∈ S^n and S:=(0,...,-1)∈ S^n.

By a local Mayer–Vietoris sequence, we have the following exact sequence

    ...→ H_*(F;)_x→ H_*(F∪ℰ({N}× K);)_x⊕ H_*(F∪ℰ({S}× K);)_x → H_*(C;)_x → ...

Therefore, we obtain that 
    H_*(F;)_x≅  H_*+1(C;)_x.
This implies that F is a k-1 dimensional homology manifold since C is a k dimensional homology manifold.

The fact that F is the boundary of ℰ({N}× K) follows from the fact that F cuts C into two pieces ℰ({N}× K) and ℰ({S}× K).

We now prove Theorem <ref>.

Denote by K the manifold with boundary ℰ({N}× K)∪ F. We have a continuous map:

    π: S^n×K→ N

by π(s,k)=ℰ(s,k) for k∈Int(K) and π(s,k)=k if k∈∂K=F.

The map π is continuous at points of S^n× K obviously and π is also continuous at points of S^n× F because of Proposition <ref> (the “shrinking property"). Therefore N is as described as in Theorem <ref>.






    	
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