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?
Example of a quasi-isomorphism in $\operatorname{Top}$ which is not a homotopy equivalence
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algebraic-topology
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2Related: http://en.wikipedia.org/wiki/Whitehead_theorem – 2012-12-10
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1@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