Suppose $(C,d)$ and $(D,\delta)$ are two chain complexes over a field and $f:C\to D$ is a chain map.
- We say $f$ is a quasi-isomorphism if it induces an isomorphism of the homology groups $H(C,d)\to H(D,\delta)$.
- We say $f$ is a chain homotopy equivalence if there is a chain map $g:D\to C$ so that $gf$ and $fg$ are chain homotopic to the identity maps.
It seems that the item 2 is stronger and can imply the item 1, but my question is that, conversely, under what additional conditions the item 1 can also imply the item 2? Thank you.
$\endgroup$ 4 Reset to default
