Does the cohomology Bockstein homomorphism map to the homology Bockstein homomorphism under Poincarè duality?

Question

Given a manifold $X$ and short exact sequence of abelian groups $$ 1\rightarrow A_1\overset{\iota}{\rightarrow} A_2\overset{\pi}{\rightarrow} A_3\rightarrow 1 $$ we get the Bockstein map in cohomology $\beta : H^p(X,A_3)\rightarrow H^{p+1}(X,A_1)$. On cochains this is defined as follows. Choose a section $s:A_3\rightarrow A_2$, then given $f\in Z^p(X,A_3)$, $s(f)\in C^{p}(X,A_2)$ is not closed, but $\pi(\delta (s(f)))=0$ and thus there exists $\beta(f)\in Z^{p+1}(X,A_1)$ such that $\iota(\beta(f))=\delta s(f)$. In a similar way one can define the Bockstein map in homology $b : H_k(X,A_3)\rightarrow H_{k-1}(X,A_1)$. On a chain $c\in Z_k(X,A_3)$ this is defined such that $\iota(b(c))=\partial s(c)$. In the last equation for any map $m : A\rightarrow B$ (even not a homomorphism) of abelian groups and $c\in C_k(X,A)$ I denoted by $m(c)\in C_k(X,B)$ the chains obtained by acting with $m$ on the coefficients.

On the other hand, if $X$ is orientable, by using Poincarè duality $\Phi_A :H_p(X,A)\rightarrow H^{d-p}(X,A)$ for homology/cohomology we induce a map in homology $b': H_p(X,A_3)\rightarrow H_{p-1}(X,A_1)$ as follows. Given $c\in H_p(X,A_3)$, construct $\Phi _{A_3}(c)\in H^{d-p}(X,A_3)$, and then by using the Bockstein $\beta(\Phi_{A_3}(c))\in H^{d-p+1}(X,A_1)$. Then define $$ b'(c)=\Phi_{A_1}^{-1}\left(\beta(\Phi_{A_3}(c))\right) $$

It is natural to expect that $b'=b$ but this does not seem to me totally obvious. I tried to prove it but at some point I need to use $\Phi$ on non-closed cycles (which I don't know how much is legal) and I would need to prove that this action commutes with the section $s:A_3\rightarrow A_2$, and I don't know how to do this...

Answer 1

I just found another proof you might prefer as Lemma 2.4 of this paper: https://arxiv.org/pdf/1708.03754.pdf. It's stated only for one case of groups, but I don't see why it wouldn't extend more generally. Note that the Poincar'e duality isomorphism is given by the cap product with a fundamental class. That's the inverse of the map you denote $\Phi$.

Answer 2

I'm not sure I quite understand your formulation of the question (what, for example, is a section $s\colon A_3 \to A_1$?) but the following is probably what you're looking for. Given a closed oriented manifold $X$ of dimension $d$ there is a quasi-isomorphism at the level of cochains and chains $C^*(X,\mathbb{Z}) \to \Sigma^{-d}C_*(X,\mathbb{Z})$. These are complexes of free abelian groups, so if you tensor this with your short exact sequence of abelian groups, you get the statement that the Bockstein homomorphisms agree in cohomology and homology.