The Unapologetic Mathematician

Mathematics for the interested outsider

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


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 | Differential Topology, Topology | 2 Comments