Homotopic Maps Induce Identical Maps On Homology
The first and most important implication of the Poincaré lemma is actually the most straightforward.
Now if is a homotopy, then the Poincaré lemma gives us a chain homotopy from to as chain maps, which tells us that the maps they induce on homology are identical. That is, passing to homology “decategorifies” the 2-categorical structure we saw before and makes two maps “the same” if they’re homotopic.
As a great example of this, let’s say that is a contractible manifold. That is, the identity map and the constant map for some are homotopic. These two maps thus induce identical maps on homology. Clearly, by functoriality, is the identity map on . Slightly less clearly, is the trivial map sending everything in to . But this means that the identity map on is the same thing as the zero map, and thus must be trivial for all .
The upshot is that contractible manifolds have trivial homology. And — as an immediate corollary — we see that any compact, oriented manifold without boundary cannot be contractible, since we know that they have some nontrivial homology!