Here are three results where the conclusion is not exactly an equality in the topological category, but is something quite close to that. All three are of fundamental importance.

• If $f\colon X\to Y$ is a map of simply connected CW complexes or of connective CW spectra, and $H_*(f)$ is an isomorphism, then $f$ is a homotopy equivalence.
• If $f\colon X\to Y$ is a map of finite spectra, and $H_*(f;\mathbb{Q})=0$, then $nf=0\colon X\to X$ for some $n>0$.
• If $f\colon\Sigma^d X\to X$ is a map of finite spectra, and $MU_*(f)=0$, then $f^n=0\colon\Sigma^{nd}X\to X$ for some $n>0$.