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Problem of the Week #244 - Feb 21, 2017

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Euge

MHB Global Moderator
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Jun 20, 2014
1,892
Here is this week's POTW:

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Let $(\mathcal{C}, \partial)$ be a chain complex of abelian groups. Suppose $f, g : \mathcal{C} \to \mathcal{C}$ are homotopic chain maps. Construct an explicit chain homotopy between the $n$-fold compositions $f^n$ and $g^n$.


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Remember to read the POTW submission guidelines to find out how to submit your answers!
 
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Euge

MHB Global Moderator
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Jun 20, 2014
1,892
No one answered this week's problem. You can read my solution below.



Let $\Delta^1$ be a chain homotopy from $f$ to $g$, and consider the sequence $\Delta_k^1 := f_{k+1}D_{k} + D_k g_k$. Then $\Delta^1 = \{\Delta_n\}_{n \ge 1}$ is a chain homotopy from $f^2$ to $g^2$. Indeed, since $f_k\partial_{k+1} = \partial_{k+1} f_{k+1}$ and $\partial_k g_k = g_{k-1}\partial_{k}$, then

$$f_k^2 - g_k^2 = f_k(f_k - g_k) + (f_k - g_k)g_k $$
$$= f_k(\partial_{k+1} D_k + D_{k-1}\partial_k) + (\partial_{k+1}D_k + D_{k-1}\partial_k)g_k$$
$$= (f_k \partial_{k+1})D_k + f_k D_{k-1}\partial_k + \partial_{k+1}D_k g_k+ D_{k-1}(\partial_k g_k) $$
$$=(\partial_{k+1} f_{k+1}) D_k + f_k D_{k-1}\partial_k + \partial_{k+1}D_k g_k + D_{k-1}(g_{k-1}\partial_k) $$
$$= \partial_{k+1}(f_{k+1}D_k + D_k g_k) + (f_k D_{k-1} + D_{k-1}g_{k-1})\partial_k$$
$$= \partial_{k+1}\Delta_k^1 + \Delta_{k-1}^1 \partial_k$$

Inductively, having defined chain homotopies $\Delta^j: f^{j+1} \cong g^{j+1}$ for $1 \le j < n$, consider the map $\Delta^n$ given by $\Delta_k^n = f_{k+1}\Delta_k^{n-1} + D_k g_k$. Then $\Delta^n$ gives a chain homotopy from $f^{n+1}$ to $g^{n+1}$. The recurrence yields an explicit solution

$$\Delta_k^n = f_{k + 1}^n D_{k+1} + \sum_{1 \le j < n} f_{k + 1}^j D_k g_k$$
 
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