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 M and bundles E,F→M, together with an elliptic differential operator D:Γ(E)→Γ(F). Then kerD and cokerD are finite-dimensional, and the index of D to be
indexD=dimkerD−dimcokerD.
D has a formal adjoint D∗, and cokerD≅kerD∗. So
indexD=dimkerD−dimkerD∗.
Usually, the operator D∗D is more recognizable. If ψ∈kerD∗D, then
0=(ψ,D∗Dψ)=(Dψ,Dψ).
So kerD=kerD∗D. Then we can write
indexD=dimkerD∗D−dimkerDD∗.
The Hodge decomposition theorem allows us to relate the kernel of differential operators to something more topological.
Theorem
A.1
(Hodge decomposition theorem)
Let M be a Riemannian manifold. Recall that the Laplacian on p-forms is defined by
Δ=dd∗+d∗d=(d+d∗)2.
If ψ∈Ωp(M) is such that Δψ=0, then, as above, dψ=d∗ψ=0. So ψ is in particular a closed form. This defines a map
kerΔ→H∗(M).
The Hodge decomposition theorem states that this map is an isomorphism.
Note that d+d∗:Ω→Ω is self-adjoint, so its index is just zero. However, by varying its domain and codomain, we can get interesting indices.
Example
A.2
Write
Ωeven=⨁Ω2k,Ωodd=⨁Ω2k+1.
Then we have a map
d+d∗:Ωeven→Ωodd.
The index is then
dimkerΔ∣Ωeven−dimkerΔ∣Ωodd=χ(M),
the Euler characteristic.
Example
A.3
We can play the same game with the signature. Suppose dimM=4k. Recall that the Hodge star operator is an endomorphism Ω∗→Ω∗ that squares to 1. Write Ω± for the ±1 eigenspaces. One can show that d+d∗ anti-commutes with the Hodge star, so induces a map
d+d∗:Ω+→Ω−.
We claim the index of this is exactly the signature of M.
We focus on the kernel of this map; the cokernel is similar. The kernel is the subspace of H∗(M) that is invariant under the Hodge star operator. This consists of the +1 eigenspace in H2k(M) plus the subspace spanned by ψ+∗ψ for ψ∈H2k−ε(M) with 0≤ε<2k.
Similarly, the kernel of d+d∗:Ω−→Ω+ consists of the −1 eigenspace in H2k(M) plus the subspace spanned by ψ−∗ψ for ψ∈H2k−ε(M) with 0≤ε<2k.
When we subtract the two, we are left with the difference between the ±1 eigenspaces of H2k(M), i.e. the signature.