Let us consider two simply-connected CW complexes. Combining the theorems of Whitehead and Hurewicz we have that a map between them is an equivalence if and only if its induced map on integral chains is a quasi-isomorphism.

Is it true that two such maps are homotopic if and only if their induced maps are chain homotopic?

  • 7
    $\begingroup$ No, the Hopf map $S^3\rightarrow S^2$ induces a nullhomotopic map on chains, but it is not nullhomotopic. $\endgroup$ – Fernando Muro Dec 21 '18 at 20:55

Your Answer


Browse other questions tagged or ask your own question.