# is it true that surjective chain map induce surjective homomorphism between homology groups?

Need surjective chain map induce surjective homomorphism between homology groups? Since $$f_*[x]\to[f(x)]$$?

## 1 Answer

No, consider the morphism of chain complexes of vector spaces over a field $$k$$ $$C:= ... \to 0\to k \to k \to 0\to ...$$ and $$C':= ... \to 0 \to k \to 0 \to 0\to ...$$ where the first $$k$$ is in degree $$0$$ for both complexes, and consider $$f: C \to C'$$ as the obvious non zero morphism between those two ($$id$$ in degree $$0$$ and $$0$$ otherwise). Then this defines a surjective morphism of chain complexes, but since $$H^*(C)=0$$ and $$H^0(C')=k$$ we get that $$H^*(f)$$ can't be surjective.

The problem with your reasoning is that your preimage might die in homology, i.e not even be represented.

• vvery thankful! – Daniel Xu Nov 7 '18 at 14:16