Prove that if there is a homotopy between $f_∗$ and $g_∗$, then the homomorphisms induced by $f_∗$ and $g_∗$ in homology coincide.

Question

Suppose that $f_*, g_*: C_* → D_*$ are two homomorphisms of chain complexes. A homotopy between the homomorphisms of chain complexes $f*, g*$ is a collection of homomorphisms $T_n: C_n → D_{n+1}$ for all $n∈\mathbb{Z}$ such that

$$f_n-g_n=∂_{n+1}\circ T_n + T_{n-1}\circ ∂_n.$$

Prove that if there is a homotopy between $f_*$ and $g_*$, then the homomorphisms induced by $f_*$ and $g_*$ in homology coincide, that is, demonstrate that

$$g_* = f_*: H_n (C_*) → H_n (D_*)$$

for all $n ∈ \mathbb{Z}$.

Let $H_n(C_*)$, I want to prove that $(f_n-g_n)(\overline{w})=0$ to conclude that $f_*=g_*$, but I have not been able to reach any conclusion then: $(f_n-g_n)(\overline{w})=∂_{n+1}(T_n(\overline{w})) + T_{n-1}(∂_n(\overline{w}))=f_n(\overline{w})+g_n(\overline{w})=2f_n(\overline{w})=2g_n(\overline{w})$, but I do not know what else to do, could someone help me please?

This question already has an answer here Homotopy between two homomorphisms and homology but I do not understand the answer.

Answer 1

If $T:C_n\rightarrow D_{n-1}$, then assume that $\partial w=0$. Then $fw-gw=\partial Tw+T\partial w=\partial Tw $ so that $fw,\ gw$ are in same class.