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Question: Is it true that, for a chain complex of vector spaces, two chain maps induce the same homomorphisms on homology if and only if they are chain homotopic?

More importantly, as a sanity check for myself, isn't it true that only one direction holds for general chain complexes?

I.e., for a general chain complex, we only have that two chain homotopic chain maps induce the same homomorphisms on homology, but the converse is not necessarily true?

A pointer to a reference, or even just a yes/no will suffice as an answer.

Context: The following problem from Greub's Linear Algebra (4th edition), p.184 suggests that both directions hold for vector spaces:

  1. Let $(E,\partial_E)$ and $(F,\partial_F)$ be two differential spaces and let $\varphi, \psi$ be homomorphisms of differential spaces. Prove that $\varphi_{\#}=\psi_{\#}$ if and only if there exists a linear mapping $h: E \to F$ such that $$h \partial_E + \partial_F h = \varphi - \psi \,; $$ $h$ is called a homotopy operator connecting $\varphi$ and $\psi$.

I remember, however, having had some confusion in the past because I tried to show that one direction holds for general chain complexes when it doesn't. So, before solving this problem and possibly confusing myself further, I want to confirm that this result does not generalize completely to arbitrary chain complexes. In particular I do not want a hint or solution for this problem.

This would seemingly be analogous to how every exact sequence of vector spaces is split exact, whereas not every arbitrary/general exact sequence is split exact. I.e. it would not be too surprising to me if this were just another instance of vector spaces having special properties. But Greub doesn't mention this explicitly, so again I want to make sure before confusing myself again.

Definitions:

A differential operator in a vector space $E$ is a linear mapping $\partial: E \to E$ such that $\partial^2 = 0$... A vector space $E$ together with a fixed differential operator $\partial_E$ is called a differential space.

A linear mapping $\varphi$ of a differential space $(E, \partial_E)$ into a differential space $(F,\partial_F)$ is called a homomorphism (of differential spaces) if $$\partial_F \circ \varphi = \varphi \circ \partial_E \,.$$

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Chill2Macht
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1 Answers1

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Yes, that's correct.

Also, you say "This would seemingly be analogous to how every exact sequence of vector spaces is split exact, whereas not every arbitrary/general exact sequence is split exact." In fact, that is precisely what makes this true for complexes of vector spaces.

Jeremy Rickard
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  • That's interesting. I guess I will start with showing the result about split exactness first then. Thank you for your help! – Chill2Macht Apr 04 '17 at 09:02