$\require{AMScd}$


**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]](https://www.maths.ed.ac.uk/~v1ranick/papers/cf1.pdf). 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 $I$.
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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 $f\_\*[\Sigma] = 0$. 
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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?
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**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]](https://www.sciencedirect.com/science/article/pii/0040938366900024). 


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.



null-bordant vs null-homologous sub-manifolds of $\infty$-d spaces/CW complexes
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 $[Z,f] = 0 \in \Omega\_\*(X;\mathbb{Z}\_2)$.
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**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 $f\_\*[Z] = 0 \in H\_\*(A;\mathbb{Z}\_2)$.
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**Proof:** A very convenient tool for this is the stratifold homology theory of [[1]](https://www.him.uni-bonn.de/fileadmin/user_upload/kreck-DA.pdf), 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]](https://www.him.uni-bonn.de/fileadmin/user_upload/kreck-DA.pdf)) 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]](https://www.him.uni-bonn.de/fileadmin/user_upload/kreck-DA.pdf)). 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]](https://www.him.uni-bonn.de/fileadmin/user_upload/kreck-DA.pdf)), 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 $sH\_\*(A;\mathbb{Z}\_2) \simeq H\_\*(A;\mathbb{Z}\_2)$.


