# null-bordant vs null-homologous sub-manifolds of $\infty$-d spaces/CW complexes

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Preliminaries: Let $$\Sigma$$ be a closed manifold, $$X$$ be a CW complex and $$f:\Sigma \to X$$ be a map. We say that the pair $$(\Sigma,f)$$ is null-homologous (over $$\mathbb{Z}_2$$) if $$f_*[\Sigma] = 0 \in H_*(X;\mathbb{Z}_2)$$, and we say that $$(\Sigma,f)$$ is null-bordant if there exists a manifold with boundary $$Y$$ and a map $$g:Y \to X$$ such that $$\partial Y = \Sigma$$ and $$g|_\Sigma = f$$.

The (non-oriented) bordism group $$\Omega_*(X)$$ of $$X$$ is the graded group generated by bordism classes $$[\Sigma,f]$$ of maps $$f:\Sigma \to X$$, with addition given by disjoint union. Note that there is a natural map $$\Omega_*(X) \to H_*(X;\mathbb{Z}_2)$$ given by $$[\Sigma,f] \mapsto f_*[\Sigma]$$ where $$[\Sigma]$$ is the $$\mathbb{Z}_2$$-fundamental class of $$\Sigma$$.

The Stieffel-Whitney numbers $$sw_{\alpha,I}[\Sigma,f] \in \mathbb{Z}_2$$ of a bordism class $$[\Sigma,f] \in \Omega_n(X)$$ are numerical invariants associated to $$[\Sigma,f]$$. Given a choice of a cohomology class $$\alpha \in H^k(X;\mathbb{Z}_2)$$ and a finite sequence of numbers $$I = (i_j)_1^N$$ with $$i_j \in \{0,\dots,n\}$$ such that $$|\alpha| + i_1 + \dots + i_N = n$$, the associated Stieffel-Whitney number is given by the following formula: $$sw_{\alpha,I}[\Sigma,f] = \langle w_{i_1}(\Sigma)\dots w_{i_k}(\Sigma) f^*\alpha,[\Sigma]\rangle$$ Here $$w_i(\Sigma) \in H^i(\Sigma;\mathbb{Z}_2)$$ denotes the $$i$$-th Stieffel-Whitney class of $$\Sigma$$.

Main Part: I am interested in the following result, which is Proposition 17.2 in [1]. It generalizes Thom's characterization of (non-oriented) null-bordant manifolds.

Theorem 1: Suppose that $$X$$ is a finite CW complex. Then a class $$[\Sigma,f]$$ is 0 if and only the Stieffel-Whitney numbers $$sw_{\alpha,I}[\Sigma,f]$$ are zero for all $$\alpha$$ and $$I$$.

This theorem yields lots of useful corollaries relating the property of being null-bordant to the property of being null-homologous. For instance, we have the following result.

Corollary 2: Let $$X$$ be a finite CW complex and let $$[\Sigma,f] \in \Omega_n(X)$$ be a bordism class such that $$w(\Sigma) = 1 \in H^*(\Sigma;\mathbb{Z}_2)$$. Here $$w(X)$$ is the total Stieffel-Whitney class of $$\Sigma$$. Then $$[\Sigma,f] = 0$$ if and only if $$f_*[\Sigma] = 0$$.

My question is about extending these results to infinite CW complexes.

Question: Can the finiteness hypothesis on $$X$$ in Theorem 1 and/or Corollary 2 be weakened to admit (some) non-finite CW complexes?

Remark On Proving Corollary 2: One possible way to try to prove Corollary 2 for infinite CW complexes is the following.

Suppose that $$X$$ and $$[\Sigma,f]$$ were as in Corollary 2, but $$X$$ can now be infinite. Clearly $$[\Sigma,f] = 0$$ implies $$f_*[\Sigma] = 0$$ under no hypotheses at all, so we need to show the other way. By cellular approximation, we can choose a representative $$(\Sigma,f)$$ of $$[\Sigma,f]$$ such that $$f(\Sigma) \subset A$$ with $$A \subset X$$ a finite sub-complex. If we could choose A so that the map $$\subset_*:H_*(A;\mathbb{Z}_2) \to H_*(X;\mathbb{Z}_2)$$ were injective, then we would get a map into a finite complex with $$f_*[\Sigma] = 0$$ and Corollary 2 applies.

I got a bit stuck while pursuing some inductive proof that such an $$A$$ can be chosen, and now I'm not so convinced either way that you can find such an $$A$$.

Motivation: This discussion is pertinent to the homology theory of $$\infty$$-dimensional spaces, e.g. spaces of diffeomorphisms, symplectomorphisms, embeddings etc from a closed manifold $$M$$ to a closed manifold $$N$$. In particular, every metrizable Banach manifold is homotopy equivalent to a (possibly infinite) CW complex by a result of Palais, see [2].

When one can show that a sub-manifold $$\Sigma$$ of such a space $$\mathcal{X}$$ is non-null-bordant, it would be nice to have criteria to check that $$\Sigma$$ is actually non-null-homologous.

• The point of the weak topology is that any map from a compact space factors through a finite subcomplex. So if $f_* \Sigma = 0$ in $X$, pick a map from a pseudomanifold bounding it. It factors through some $A$, and therefore $f_*\Sigma \in H_*(A) = 0$. This is the general argument that homotopy/homology of an infinite complex is the colimit of those of its finite subcomplexes, and works for anything defined by mappings out of compact things. – Mike Miller Oct 21 '18 at 5:28
• @MikeMiller Hi Mike. I see, it's a good observation that you don't need a sub-complex $A \subset X$ such that $H_*(A) \to H_*(X)$ is injective to get $f_*[\Sigma] = 0 \in H_*(A)$. That seems to prove Corollary 2, thanks! It would still be nice to know something about the theorem. – Julian Chaidez Oct 21 '18 at 5:39
• I think the following formal argument works. Your Stiefel-Whitney numbers define a map $\Omega_n(X) \to \left(H^*(X;\Bbb Z/2)^\vee\right)^{p(n)} = \left(H_*(X;\Bbb Z/2)\right)^{p(n)}$. Theorem 1 is that if $X$ is a finite complex, this is injective. If $X$ is an infinite complex, $\Omega_n(X) = \colim_A \Omega_n(A)$, the colimit running over every finite subcomplex $A$, and you still have the map to $\left(H^*(X;\Bbb Z/2)^\vee\right)^{p(n)} = \left(\lim H^*(A;\Bbb Z/2)^\vee\right)^{p(n)} = \left(\colim H_*(A;\Bbb Z/2)\right)^{p(n)}$. Finally, a colimit of injective maps of v.s. is injective. – Mike Miller Oct 21 '18 at 9:53
• there are some unclear parentheses in the last line: $(\lim H^*(A;\Bbb Z/2))^\vee = \text{colim }(H^*(A;\Bbb Z/2))^\vee = \text{colim } H_*(A;\Bbb Z/2)$ – Mike Miller Oct 21 '18 at 9:58
• @MikeMiller Maybe that could work. My qualms are: (1) $H^*(X;\mathbb{Z}_2)$ is not finitely dimensional as a $\mathbb{Z}_2$-vectorspace, so $H^*(X;\mathbb{Z}_2)^{\vee} \neq H_*(X;\mathbb{Z}_2)$and (2) it isn't clear to me that the Stieffel-Whitney numbers define a linear map $\Omega_n(X) \to (H^*(X;\mathbb{Z}_2)^\vee)^{p(n)}$ even for finite $X$ because of the dependence of the Stieffel-Whitney numbers non-linearly on the Stieffel-Whitney classes, which depend on your choice of $[\Sigma,f] \in \Omega_n(X)$. I would need to check that both of these things are actually problems though. – Julian Chaidez Oct 21 '18 at 10:11

## 1 Answer

I would like to share a careful proof of the generalized Corollary 2 from my question.

The essential idea is to factor the map $$f:Z \to X$$ through a map to a finite sub-complex of $$X$$, like I originally had in mind. However, I use Mike Miller's observation that you only need to know that $$f_*[Z]$$ is 0 in the homology of the finite sub-complex. Also I use a stratifold as my compact bounding object instead of pseudo-manifolds (Mike suggested the latter in his comment).

Lemma 1: Let $$X$$ homotopy equivalent to a CW complex, and let $$f:Z \to X$$ be a continuous map from a closed manifold $$Z$$ with Stieffel-Whitney class $$w(Z) = 1 \in H^*(Z;\mathbb{Z}_2)$$. Then $$f_*[Z] = 0 \in H_*(X;\mathbb{Z}_2)$$ if and only $$[Z,f] = 0 \in \Omega_*(X;\mathbb{Z}_2)$$.

Proof: ($$\Rightarrow$$) Suppose that $$f_*[Z] = 0 \in H_2(Z;\mathbb{Z}_2)$$. Pick a homotopy equivalence $$\varphi:X \simeq X'$$ with a CW complex $$X'$$. Such an equivalence induces an isomorphism of unoriented bordism groups $$\Omega_*(X;\mathbb{Z}_2) \simeq \Omega_*(X';\mathbb{Z}_2)$$, so it suffices to show that the pair $$(Z,\varphi \circ f)$$ is null-bordant, or equivalently to assume that $$X$$ is a CW-complex to begin with.

So assume that $$X$$ is a CW-complex. By Lemma 2, we can find a finite sub-complex $$A \subset X$$ such that $$f(Z) \subset A$$ and $$f_*[Z] = 0 \in H_*(A;\mathbb{Z}_2)$$. By Theorem 17.2 of \cite{conner1964}, $$[Z,f] = 0 \in \Omega_*(A;\mathbb{Z}_2)$$ if and only if the Stieffel-Whitney numbers $$\text{sw}_{\alpha,I}[Z,f]$$ are identically $$0$$. Recall that the Stieffel-Whitney number $$\text{sw}_{\alpha,I}[Z,f]$$ associated to $$[Z,f]$$, a cohomology class $$\alpha \in H_k(A;\mathbb{Z}_2)$$ and a partition $$I = (i_1,\dots,i_k)$$ of $$\text{dim}(Z) - k$$ is defined to be: $$\text{sw}_{\alpha,I}[Z,f] = \langle w_{i_1}(Z)w_{i_2}(Z) \dots w_{i_k}(Z) f^*\alpha,[Z]\rangle \in \mathbb{Z}_2$$ Here $$w_j(Z) \in H^j(Z;\mathbb{Z}_2)$$ denotes the $$j$$-th Stieffel-Whitney class of $$Z$$. By assumption, $$w(Z) = 1$$ and so $$w_j(Z) = 0$$ for all $$j \neq 0$$. In particular, the only possible non-zero Stieffel-Whitney numbers have $$I = (0)$$. But we see that: $$\text{sw}_{\alpha,(0)}[Z,f] = \langle f^*\alpha,[Z]\rangle = \langle \alpha,f_*[Z]\rangle = 0$$ Therefore, $$\text{sw}_{\alpha,I}[Z,f] \equiv 0$$ and $$[Z,f]$$ must be null-bordant.

($$\Leftarrow$$) This direction is completely obvious, since the map $$\Omega_*(X) \to H_*(X;\mathbb{Z}_2)$$ given by $$[Z,f] \mapsto f_*[Z]$$ is well-defined.

Lemma 2: Let $$X$$ be a CW complex, and let $$f:Z \to X$$ be a map from a closed manifold $$Z$$ with $$f_*[Z] = 0 \in H_*(X;\mathbb{Z}_2)$$. Then there exists a finite sub-complex $$A \subset X$$ with $$f(Z) \subset A$$ and $$f_*[Z] = 0 \in H_*(A;\mathbb{Z}_2)$$.

Proof: A very convenient tool for this is the stratifold homology theory of [1], which we now review briefly.

Given a space $$M$$, the $$n$$-th stratifold group $$sH_n(M;\mathbb{Z}_2)$$ with $$\mathbb{Z}_2$$-coefficients (see Proposition 4.4 in [1]) is generated by equivalence classes of pairs $$(S,g)$$ of a compact, regular stratifold $$S$$ and a continuous map $$g:S \to M$$. Two pairs $$(S_i,g_i)$$ for $$i \in \{0,1\}$$ are equivalent if they are bordant by a $$c$$-stratifold, i.e. if there is a pair $$(T,h)$$ of a compact, regular $$c$$-stratifold and a continuous map $$g:T \to M$$ such that $$(\partial T,h|_{\partial T}) = (S_0 \sqcup S_1,g_0 \sqcup g_1)$$ (see Chapter 3 and Section 4.4 of [1]). Given a map $$\varphi:M \to N$$ of spaces, the pushforward map $$\varphi_*:sH(M;\mathbb{Z}_2) \to sH(M;\mathbb{Z}_2)$$ on stratifold homology is given (on generators) by $$[S,g] \mapsto [S,\varphi \circ g] = \varphi_*[\Sigma,g]$$.

Stratifold homology satisfies the Eilenberg-Steenrod axioms (see Chapter 20 of [1]), and thus if $$M$$ is a CW complex then there is a natural isomorphism $$sH_*(M;\mathbb{Z}_2) \simeq H_*(M;\mathbb{Z}_2)$$. If $$M$$ is a closed manifold of dimension $$n$$, the fundamental class $$[M] \in sH_n(M;\mathbb{Z}_2)$$ is given by the tautological equivalence class $$[M] = [M,\text{Id}]$$.

The proof of the lemma is simple with the above machinery in place. Since $$f_*[Z] = 0$$, the pair $$(Z,f)$$ must be null-bordant via some compact $$c$$-stratifold $$(Y,g)$$. Since $$Y$$ and its image $$g(Y)$$ are both compact, we can choose a sub-complex $$A \subset X$$ such that $$g(T) \subset A \subset X$$. Then the pair $$(Z,f)$$ are null-bordant by $$(Y,g)$$ in $$A$$ as well, so that $$[Z,f] = 0 \in sH_*(A;\mathbb{Z}_2)$$ and thus $$f_*[Z] = 0 \in H_*(A;\mathbb{Z}_2)$$ via the isomorphism $$sH_*(A;\mathbb{Z}_2) \simeq H_*(A;\mathbb{Z}_2)$$.

• Stratifolds are very nice, and deserve to be better known. I'm glad you wrote this up! I still think that Theorem (1) should hold more generally, but I got stuck in my initial attempts to write down a cleaner proof attempt than the comments above, and got distracted with other things. – Mike Miller Oct 24 '18 at 20:09
• No problem, thanks for giving this some thought! I'll accept this answer, but if you come up with something for the general case and post it, I'll accept that one instead :) – Julian Chaidez Oct 25 '18 at 18:21