# at.algebraic topology – Induced map in homology for a map to a loop space

Suppose $$Y$$ is an $$(n-1)$$-connected space, $$n>2$$, so we have Hurewicz isomorphisms $$pi_n(Y)cong H_n(Y)$$ and $$pi_{n-1}(Omega Y)cong H_{n-1}(Omega Y)$$. Let a map $$alphacolon XtoOmega Y$$ be given. Naturally it induces a map $$betacolon Xtimes S^1to Y$$. I want to show the following diagram is commutative:
$$require{AMScd} begin{CD} H_{n-1}(X) @>times(S^1)>> H_n(Xtimes S^1)\ @Valpha_*VV @Vbeta_*VV \ H_{n-1}(Omega Y) @

Since $$Omega Y$$ is path connected, we may homotope $$alpha$$ to a based map. Then $$beta$$ factors though the reduced suspension $$Sigma X$$. If $$X=S^{n-1}$$ is a sphere, the commutativity would then follow from tracking down the definition of $$pi_n(Y)xrightarrow{cong}pi_{n-1}(Omega Y)$$. However I don’t know how this helps for the general case.

One can also phrase the question in cohomology in the obvious way. (In particular the cross product $$times(S^1)$$ will be replaced by the slant product $$/(S^1)$$.)