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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 http://mathoverflow.net/questions/53399/spaces-with-same-homotopy-and-homology-groups-that-are-not-homotopy-equivalent (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.2012-07-27

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