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Let $f:X\to Y$ be a map of path connected spaces such that for all $n>0$ the homorphism $f_* :H_n(X;G)\to H_n(Y;G)$ is an isomorphism for $G=\mathbb{Q}$ and for $G=Z_p$ for any prime number $p$. show that the homorphism $f_* :H_n(X;\mathbb{Z})\to H_n(Y;\mathbb{Z})$ is an isomorphism for all $n>0$.

I think consider cone map long exact $\tilde{H_n}(X)\rightarrow\tilde{H_n}(Y)\rightarrow\tilde{H}(C_f)\rightarrow\tilde{H}_{n-1}(X)$, then tensor them by $Q$ and $Z_p$,then I do not know how to do next?

noname1014
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  • The only thing left is when a homology with $\mathbb{Q}$ and $\mathbb{Z}/p\mathbb{Z}$ coefficients are 0, then it also vanishes with $\mathbb{Z}$, and this follows from UCT for homology and the two exact sequence $0\to \mathbb{Z}\to \mathbb{Z}\to \mathbb{Z}/p\mathbb{Z}\to 0$ and $0\to \mathbb{Z}\to\mathbb{Q}\to\mathbb{Q}/\mathbb{Z}\to 0$. See the corresponding chapter in Hatcher's Algebraic topology. – cjackal Apr 20 '17 at 21:36

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