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?

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.
So it is sufficient to exhibit an example of two simply connected finite CW-complexes with the same homology groups which are not homotopy equivalent. Can you do this?
