In this lecture we discussed the index theorem and the fixed point theorem for the Signature operator on an oriented even-dimensional Riemannian manifold \(M\). The signature operator just “is” as an operator the familiar \(D = d+d^*\) on differential forms, but we make an elliptic complex out of it in a different way. Specifically, one defines an operator \(\epsilon\) on differential forms to equal \(\pm *\), the sign dependent on the dimension, and one shows that \(\epsilon^2=\pm 1\) according to whether the dimension is congruent to 0 or 2 mod 4; moreover, \(D\epsilon+\epsilon D=0\), that is, \(D\) anticommutes with \(\epsilon\). In the classical case where the dimension is a multiple of 4, then, \(D\) interchanges the \(\pm 1\) eigenspaces of the grading operator \(\epsilon\), and its restriction to an operator \(\Omega_+\to\Omega_-\) is the signature operator. Its index is the Hirzebruch signature of the manifold \(M\), that is the signature (in the sense of linear algebra) of the non-degenerate symmetric bilinear form defined on the middle dimensional cohomology by
\[ (\alpha,\beta) \mapsto \int_M \alpha\wedge\beta. \]
I spent some time in class talking about the significance of this invariant in topology and I was asked afterwards if I could write up some of that in a blog post, so here goes. A good reference for this material is the later chapters of Milnor and Stasheff’s book Characteristic classes.
So, here are some facts about the signature. Continue reading