Mathematics Asked by zero2infinity on December 24, 2020

Suppose that $f:X to Y$ is a continuous map. Suppose it induces an isomorphism $f_*:H_*(X) to H_*(Y)$. Does that imply $f:X to Y$ is a homotopy equivalence?

I think that this is not the case. This is because the following is not True.

Let $f_*:C_*to D_*$ be a chain map such that $f_*$ induces an isomorphism in homology. Then $f_*$ is a chain homotopy equivalence,

Can anyone give me a simple example.

In case when $Y={*}$ is a singleton your question can be stated as:

Suppose $X$ has all homology groups trivial. Is it contractible?

Unfortunately this is false. The double comb space is famously known to have all homotopy, homology and cohomology groups trivial, even though it is not contractible.

Answered by freakish on December 24, 2020

No; for example, there are non-contractible spaces that are acyclic (have trivial reduced homology), such as the Poincare homology sphere minus a point. You can take $X$ to be any such space and $Y$ to be a point.

However, this is true if $X$ and $Y$ are simply connected CW complexes, by a combination of the relative Hurewicz theorem and Whitehead's theorem. This is "Whitehead's theorem for homology."

Answered by Qiaochu Yuan on December 24, 2020

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