The Poincaré Lemma (setup)

Now we’ve seen that differentiable manifolds, smooth maps, and homotopies form a 2-category, but it’s not the only 2-category around. The algebra of differential forms — together with the exterior derivative — gives us a chain complex. Since pullbacks of differential forms commute with the exterior derivative, they define a chain map between two chain complexes.

And now I say that a homotopy $H:f_0\to f_1$ between two maps $f_0,f_1:M\to N$ induces a chain homotopy between the two chain maps $f_0^*$ and $f_1^*$ . And, indeed, if the homotopy is given by a smooth map $H:M\times[0,1]\to N$ then we can write $f_i=H\circ\iota_i$ , where $\iota_0(p)=(p,0)$ and $\iota_1(p)=(p,1)$ are the two boundary inclusions of $M$ into the “homotopy cylinder” $M\times[0,1]$ , and we will work with these inclusions first.

Since $\iota_i:M\to M\times[0,1]$ , we have chain maps $\iota_i^*:\Omega^k(M\times[0,1])\to\Omega^k(M)$ , and we’re going to construct a chain homotopy $I:\Omega^k(M\times[0,1])\to\Omega^{k-1}(M)$ . That is, for any differential form $\omega$ we will have the equation

$\displaystyle\iota_1^*\omega-\iota_0^*\omega=d(I\omega)+I(d\omega)$

Given this, we can write

$\displaystyle\begin{aligned}f_1^*\omega-f_0^*\omega&=\iota_1^*H^*\omega-\iota_1^*H^*\omega\\&=d(I(H^*\omega))+I(d(H^*\omega))\\&=d([I\circ H^*]\omega)+[I\circ H^*](d\omega)\end{aligned}$

which shows that $I\circ H^*$ is then a chain homotopy from $f_0$ to $f_1$ .

Sometimes the existence of the chain homotopy $I$ is referred to as the Poincaré lemma; sometimes it’s the general fact that a homotopy $H$ induces the chain homotopy $I\circ H^*$ ; sometimes it’s a certain corollary of this fact, which we will get to later. Given my categorical bent, I take it to be the general assertion that we have a 2-functor between the homotopy 2-category and that of chain complexes, chain maps, and chain homotopies.

As a side note: now we can finally understand what the name “chain homotopy” means.

December 2, 2011 - Posted by John Armstrong | Differential Topology, Topology

2 Comments »

[…] can now prove the Poincaré lemma by proving its core assertion: there is a chain homotopy between the two chain maps and induced […]

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[…] first and most important implication of the Poincaré lemma is actually the most […]

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