# A question about homology group

For a pair of spaces $X,Y$ we have $H_*(X)=H_*(Y)$. Can we necessarily find a continuous function $f$ from $X$ to $Y$ or from $Y$ to $X$, such that $f_*$ induces the isomorphism of homology group?

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A more difficult version of this was asked on MO here mathoverflow.net/questions/53399/… (more difficult because there Dylan wanted spaces with the same homotopy and homology groups) The lens spaces I mentioned below also work for this variant. – Mariano Suárez-Alvarez Jul 27 '12 at 5:17

No, we cannot. ${}{}{}{}{}{}{}{}$
Suppose, for example, that $X$ and $Y$ are spaces which have the homotopy type of CW-complexes which have the same homology groups and which are simply connected. If there is a map $f:X\to Y$ which induces an isomorphism in homology, then by the homology version of Whitehead's theorem, $f$ is in fact an homotopy equivalence.