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Can we have a continous map $f: X \longrightarrow Y$ such that $f$ induces an isomorphism on all homology groups i.e. $f_* : H_n(X) \longrightarrow H_n(Y)$ is an isomorphism of abelian groups for all $n \geq 0$ but $f$ itself is not a homotopy equivalence?

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    Related: http://en.wikipedia.org/wiki/Whitehead_theorem2012-12-10
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    @Neal I think that more than Whitehead's theorem is the fact that if a map $f:X\to Y$ induces an isomorphism on all homology groups, and $X$ and $Y$ are simply connected CW complexes, then $f$ is actually a homotopy equivalence. This is in Chapter Three of Hatcher somewhere.2012-12-10

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