$\require{AMScd}$

**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]. 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.