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I have the same question posted here (namely, if $f:\mathcal{C}\to \mathcal{D}$ is a surjective chain map between chain complexes then can I be sure that the induced in homology $f_\ast:H(\mathcal{C})\to H(\mathcal{D})$ is surjective too?)

There, @Enkidu gives a counter-example. However, is it really correct? What is the operator $\partial:k\to k$ in that case?

  1. If it is $Id$ the identity, then $f\partial =Id$ while $\partial f = 0$ and then $f$ is not a chain map.
  2. If it is $0$, then $H^0(C)$ is also $k$, instead of $0$.

I'm considering that he refers to the complex chains as below: $$\begin{array}{} \mathcal{C}:\cdots &0 & \rightarrow & 0 & \rightarrow & k & \rightarrow &k&\rightarrow& 0\\ &\downarrow & & \downarrow & & \downarrow& & \downarrow & & \downarrow\\ \mathcal{D}:\cdots &0 & \rightarrow & 0 & \rightarrow & 0 & \rightarrow &k&\rightarrow& 0\\ \end{array}$$

I still totally agree with the fact that, if $f$ is surjective and you take $z+B_n(\mathcal{D})\in H_n(\mathcal{D})$ then there is some $c\in C_n$ such that $f(c) = z$ BUT $c$ may not be a $n$-cicle, so I may not be able to write $f_\ast(c+B_n(\mathcal{C})) = f(c)+B_n(\mathcal{D}) = z+B_n(\mathcal{D})$ (i.e. we possibly lose pre-images when we pass to homology. Need a explicit counter-example though).

What am I missing? Sorry if I'm duplicating for no reason...

Derso
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    In his example the complex $\mathcal D$ is shifted to the left in comparison to your configuration (so that the nonzero term in $\mathcal D$ corresponds to the first nonzero term in $\mathcal C$). – Matsmir Aug 29 '20 at 12:54
  • Thank you for your comment! The phrase "where the first k is in degree $0$ for both complexes" confused me. Also, with your correction, we should look at $H_1(\mathcal{C}) = 0$ and $H_1(\mathcal{D}) = k$, right? (not at the degree $0$, I mean, as he says) – Derso Aug 29 '20 at 13:04
  • Yes, that's right. – Matsmir Aug 29 '20 at 13:16
  • That isn't like a chain map: move the $k$ in the second row one place left. – Angina Seng Aug 29 '20 at 14:35

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