Riemannian metric | Elliptic Operators and Topology

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 →

In this session we got around to stating the main theorem of the Atiyah-Bott Lefschetz theorem papers. We imagine that we have an elliptic complex (as previously defined) and an endomorphism of that complex. Since an elliptic complex is made up of two (or maybe three) components – manifold, bundles, operators – a geometric endomorphism similarly has several components: a smooth map $f\colon M\to M$ , a bunch of bundle maps from $f^*E\to E$ for all the bundles $E$ appearing in the complex, and the whole to give a (co)chain map and therefore an induced map on cohomology. In these circumstances we can define the Lefschetz number to be the alternating sum of the traces of the induced cohomology maps. The main theorem expresses this Lefschetz number as a sum over the fixed points of $f$ (under the assumption that these fixed points are simple).

We reviewed what this theorem tells us in two cases: when we deal with the de Rham complex and an arbitrary smooth map $M\to M$ (when we recover the classical Lefschetz fixed point theorem), and when we are considering the Dolbeault complex of a complex manifold $M$, and $f$ is a holomorphic map $M\to M$ (this case supplied Atiyah and Bott’s original motivation). There is a very pretty example on page 460 of the second paper in the series where they consider a self-map of projective space induced by multiplying each homogeneous coordinate by a different scalar factor. Then the fixed point theorem gives an algebraic identity

\[1 = \sum_{i=0}^n \frac{\gamma_i^n}{\prod_{j\neq i} (\gamma_i-\gamma_j)}\]

for distinct nonzero complex numbers $\gamma_0,\ldots,\gamma_n $. I wrote a post on my main blog explaining how this identity can be derived from the familiar Lagrange interpolation formula of numerical analysis.

In next class session we will return to the de Rham complex but we will also introduce a Riemannian metric, which makes it possible to get more subtle topological invariants from de Rham theory. It would be appropriate to read section 6 of paper II in preparation for this session.

Elliptic Operators and Topology

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