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Let us assume we have a chain complex $(X_\bullet,\partial_\bullet)$ of vector spaces and a subcomplex $(Y_\bullet,\partial_\bullet)$.

Let us furthermore assume that there exists a morphism $f_\bullet : (X_\bullet,\partial_\bullet) \rightarrow (Y_\bullet,\partial_\bullet)$ and a homotopy operator $\Psi_k : X_k \rightarrow X_{k-1}$ such that $\Psi$ gives a homotopy between $f$ and the identity. We have

$x - f_{k} x = \Psi_{k+1} \partial_{k} x + \partial_{k-1} \Psi_{k} x$

for $x \in X_k$.

Then $f_\bullet$ induces isomorphisms on homology. I have read the claim that if $f_\bullet$ takes image in the subcomplex $(Y_\bullet,\partial_\bullet)$, then it induces a isomorphism on homology between the subcomplexes.

But I do not see that - I believe you need additional assumptions, such that $\Psi_k ( Y_k ) \subseteq Y_{k-1}$. Can you prove the claim without such an additional assumption.

shuhalo
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    With the current definition of $f_\bullet$, it always takes image in the subcomplex you mention. Moreover, where do your $A_i$'s live? Could it be that there should be two subcomplexes involved in this question or am I just missing something? – HSN Jun 12 '15 at 17:14
  • @HSN Thank you for pointing out this typo. There is only one subcomplex involved. – shuhalo Jun 12 '15 at 18:39

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