Homotopy inverses need not induce inverse homomorphisms Let $f:X \rightarrow Y$ and $g : Y \rightarrow X$ be homotopy inverses, ie. $f \circ g$ and $g\circ f$ are homotopic to the identities on $X$ and $Y$. We know that $f_*$ and $g_*$ are isomorphisms on the fundamental groups of $X$ and $Y$. However, it is my understanding that they need not be inverse isomorphisms. Is there an explicit example where they are not?

If $f,g$ are pointed maps (which is necessary so that $f_*,g_*$ make sense): No, they induce inverse homomorphsism.
Homotopic maps induce the same maps on homotopy groups, in particular fundamental groups. This means that we have a functor $\pi_1 : \mathrm{hTop}_* \to \mathrm{Grp}$. Every functor maps two inverse isomorphisms to the corresponding two inverse isomorphisms.
