# homology with integral coefficients vs homology with field coefficients

Suppose that $$f:X\rightarrow Y$$ is a continuous mapping between connected topological spaces such that for any field $$k$$, $$H_{\ast}(X,k)\rightarrow H_{\ast}(Y,k)$$ is an isomorphism. Does it follow that $$H_{\ast}(X,\mathbf{Z})\rightarrow H_{\ast}(Y,\mathbf{Z})$$ is an isomorphism in integral homolgy ? Do we need to assume that $$X$$ and $$Y$$ are simply connected ?

• I think the universal coefficient theorem implies it has to be surjective. – ronno Jan 14 at 15:58

In fact, you only need it for $$\mathbb{Q}$$ and $$\mathbb{Z}/p$$. There is a very useful short exact sequence $$0 \rightarrow \mathbb{Z}\rightarrow \mathbb{Q} \rightarrow \oplus _p \mathbb{Z}/p^\infty \rightarrow 0$$. In this case, it gives us a long exact sequence in homology $$\dots \rightarrow H_{n+1} (X;\oplus _p \mathbb{Z}/p^\infty) \rightarrow H_n(X; \mathbb{Z}) \rightarrow H_n(X; \mathbb{Q}) \rightarrow H_n(X;\oplus _p \mathbb{Z}/p^\infty) \rightarrow \dots$$ . If we knew that an isomorphism with coefficients in $$\mathbb{Z}/p$$ gave rise to an isomorphism with coefficients in $$\mathbb{Z}/p ^\infty$$ we would be done by the five lemma. So let's prove that it does.
For all $$k$$ we have a short exact sequence $$0 \rightarrow \mathbb{Z}/p^{k-1} \rightarrow \mathbb{Z}/p^k \rightarrow \mathbb{Z}/p \rightarrow 0$$, so by induction, the long exact sequence on homology coming from this short exact sequence, and the five lemma we have that the induced map on homology with coefficients in $$\mathbb{Z}/p^m$$ is an isomorphism for all $$m$$. Now we want to show that the homology with coefficients in $$\lim\limits_{k \rightarrow \infty}\mathbb{Z}/p^k=\mathbb{Z}/p^\infty$$ is the same as the colimit of the homology with coefficients in $$\mathbb{Z}/p^k$$. This follows directly from the fact that on the chain complex level the colimit of the $$C_n(X ; \mathbb{Z}/p^k)$$ is $$C_n(X ; \lim\limits_{k \rightarrow \infty}\mathbb{Z}/p^k)$$ and that taking homology commutes with direct limits. So we have an isomorphism on homology with coefficients in $$\mathbb{Z}/p^\infty$$. Direct summing over all $$p$$ we can then apply the five lemma to the initial long exact sequence, and we are done.