Homotopy inverse and retracts

Question

I was thinking about the following.

Let $f:X \rightarrow Y$ and $g:Y \rightarrow X$ be a homotopy equivalence.

I was wondering about the induced maps on the fundamentalgroups and whether we have that $g_{*}^{-1} = f_{*}$? (So I am asking if the canonical group isomorphisms are somehow related to each other?). I cannot show this, so I suspect that it is false?- Despite, this is maybe true for deformation retracts. So is the isomorphism induced by the inclusion the inverse of the one induced by the retraction?

Answer 1

The point is that $f$ may not be a base point preserving homotopy equivalence. So $g_*^{-1}$ may differ from $f_*$ by a conjugation of the fundamental group at a given point. This result arises from considering "change of base point" in the fundamental group.

There is a generalisation of this method in Chapter 7 of Topology and Groupoids. In the above argument the fundamental group of $(X,x)$ can be regarded as homotopy classes of maps $(S^1,1) \to (X,x)$. One can instead consider a pair $(E,A)$ of spaces which satisfies the HEP (Homotopy Extension Property) and homotopy classes rel $A$ of maps $E \to X$ which extend a given map $u: A \to X$. Write this set $[E,X;u]$. Then one considers change of $u$ by a homotopy. Now suppose one is given a homotopy equivalence $f: X \to Y$. The above argument generalises and leads to a gluing theorem for homotopy equivalences!

Answer 2

First off: Very good observation sir!

The answer is: yes by functoriality and homotopy invariance! So those maps can look ugly but this still will be true. This is an interesting fact, since sometimes it is the rigorous argument behind your intuition saying "yeaah.. actually the induced map should be an isomorphism".

Details: by functoriality $f_*g_* =(fg)_*$, by homotopy invariance $(fg)_* = id_*$, by functoriality $id_* = id_{\pi_1}$. The same for $g_*f_*$ yields that they are inverse to each other!

And note that we only used that this is a homotopy equivalence. If you have a retraction $r$ you can use a similar argument for injectivity of the induced map since $ri =id$.