A Index and Geometry
In this appendix, we briefly demonstrate how interesting topological invariants can be expressed as the index of a differential operator.
We recall some basics. Fix a manifold and bundles , together with an elliptic differential operator . Then and are finite-dimensional, and the index of to be
has a formal adjoint , and . So
Usually, the operator is more recognizable. If , then
So . Then we can write
The Hodge decomposition theorem allows us to relate the kernel of differential operators to something more topological.
(Hodge decomposition theorem)
Let be a Riemannian manifold. Recall that the Laplacian on -forms is defined by
If is such that , then, as above, . So is in particular a closed form. This defines a map
The Hodge decomposition theorem states that this map is an isomorphism.
Note that is self-adjoint, so its index is just zero. However, by varying its domain and codomain, we can get interesting indices.
Then we have a map
The index is then
the Euler characteristic.
We can play the same game with the signature. Suppose . Recall that the Hodge star operator is an endomorphism that squares to . Write for the eigenspaces. One can show that anti-commutes with the Hodge star, so induces a map
We claim the index of this is exactly the signature of .
We focus on the kernel of this map; the cokernel is similar. The kernel is the subspace of that is invariant under the Hodge star operator. This consists of the eigenspace in plus the subspace spanned by for with .
Similarly, the kernel of consists of the eigenspace in plus the subspace spanned by for with .
When we subtract the two, we are left with the difference between the eigenspaces of , i.e. the signature.