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Let $f,g:C_\bullet \rightarrow D_\bullet$ be chain maps. Let $T$ be a generator (possibly not the only one) of $C_n$ and assume $H_n(C)=0$. I'm trying to prove that for every $T$ there's a solution $z$ to the equation $$\partial z=(g-f-s_{n-1}\partial)(T)$$ The proof is easy if $T$ is a cycle, but what if it isn't?

Update: Could it be that there's a typo in the following excerpt and $H_n(C)=0$ should in fact be $H_n(D)=0$? (The excerpt is from Selick's Introduction to Homotopy Theory.)

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  • May be I do not understand something but if $T$ is a generator of $C_{n}$ then any element of $C_{n}$ is of the form $k.T$ where $k \in \mathbb{Z}$. so in particular if $kT$ is a cycle then shouldnt we have $kd(T) = 0$ so $d(T) = 0$? – DBS Mar 17 '15 at 10:40
  • @DBS $C_n$ may have more than one generator. I'll edit the question to emphasize that. –  Mar 17 '15 at 10:45

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