Does a map of spaces inducing an isomorphism on homology induce an isomorphism between the homologies of the loop spaces?

Question

That is, let $f:X \rightarrow Y$ be a map of spaces such that $f_*: H_*(X) \rightarrow H_*(Y)$ induces an isomorphism on homology. We get an induced map $\tilde{f}: \Omega X \rightarrow \Omega Y$, where $\Omega X$ is the loop space of $x$. Does $\tilde{f}$ also induce an isomorphism on homology?

Answer 1

It's not true in general.

Say, take a ring $R$ and consider the map $BGL(R)\to BGL(R)^+$. It always induces an isomorphism on homology, but $$H_1(\Omega BGL(R))=H_1(GL(R))=0$$ ($GL(R)$ has discrete topology) and $$H_1(\Omega BGL(R)^+)=\pi_1(\Omega BGL(R)^+)=\pi_2(BGL(R)^+)=K_2(R)$$ is often non-trivial.