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 
between two maps 
induces a chain homotopy between the two chain maps 
and 
. And, indeed, if the homotopy is given by a smooth map ![H:M\times[0,1]\to N](https://s0.wp.com/latex.php?latex=H%3AM%5Ctimes%5B0%2C1%5D%5Cto+N&bg=e6e6e6&fg=333333&s=0 "H:M\times[0,1]\to N")
then we can write 
, where 
and 
are the two boundary inclusions of 
into the “homotopy cylinder” ![M\times[0,1]](https://s0.wp.com/latex.php?latex=M%5Ctimes%5B0%2C1%5D&bg=e6e6e6&fg=333333&s=0 "M\times[0,1]")
, and we will work with these inclusions first.
Since ![\iota_i:M\to M\times[0,1]](https://s0.wp.com/latex.php?latex=%5Ciota_i%3AM%5Cto+M%5Ctimes%5B0%2C1%5D&bg=e6e6e6&fg=333333&s=0 "\iota_i:M\to M\times[0,1]")
, we have chain maps ![\iota_i^*:\Omega^k(M\times[0,1])\to\Omega^k(M)](https://s0.wp.com/latex.php?latex=%5Ciota_i%5E%2A%3A%5COmega%5Ek%28M%5Ctimes%5B0%2C1%5D%29%5Cto%5COmega%5Ek%28M%29&bg=e6e6e6&fg=333333&s=0 "\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)](https://s0.wp.com/latex.php?latex=I%3A%5COmega%5Ek%28M%5Ctimes%5B0%2C1%5D%29%5Cto%5COmega%5E%7Bk-1%7D%28M%29&bg=e6e6e6&fg=333333&s=0 "I:\Omega^k(M\times[0,1])\to\Omega^{k-1}(M)")
. That is, for any differential form 
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}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7Df_1%5E%2A%5Comega-f_0%5E%2A%5Comega%26%3D%5Ciota_1%5E%2AH%5E%2A%5Comega-%5Ciota_1%5E%2AH%5E%2A%5Comega%5C%5C%26%3Dd%28I%28H%5E%2A%5Comega%29%29%2BI%28d%28H%5E%2A%5Comega%29%29%5C%5C%26%3Dd%28%5BI%5Ccirc+H%5E%2A%5D%5Comega%29%2B%5BI%5Ccirc+H%5E%2A%5D%28d%5Comega%29%5Cend%7Baligned%7D&bg=e6e6e6&fg=333333&s=0 "\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 
is then a chain homotopy from 
to 
.
Sometimes the existence of the chain homotopy 
is referred to as the Poincaré lemma; sometimes it’s the general fact that a homotopy 
induces the chain homotopy ; 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.
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December 2, 2011 -
Posted by John Armstrong |
Differential Topology, Topology
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