Â-genus as the index of the Dirac operator

Remarks

The theorem is the case of the Atiyah–Singer index theorem for the spin Dirac operator. In the cohomological formula, the Chern character of the spinor symbol combined with the Todd class leaves exactly the $\hat A$-class. The first terms are $\hat A_1 = -\tfrac{1}{24} p_1$ and $\hat A_2 = \tfrac{1}{5760}(-4 p_2 + 7 p_1^2)$. Consequence: the $\hat A$-genus of a closed spin manifold is an integer, while for general manifolds it is only rational. The spin hypothesis is necessary: $\C P^2$ is not spin, and $\hat A(\C P^2) = -\tfrac{1}{24} \cdot 3 = -\tfrac18$. Example: a K3 surface is spin with $p_1 = -48$, so $\hat A = 2$, and the Dirac operator has index $2$. With the Lichnerowicz formula $D^2 = \nabla^*\nabla + \tfrac{s}{4}$, where $s$ is the scalar curvature, the theorem shows that a closed spin manifold with $\hat A(M) \ne 0$ has no metric of positive scalar curvature.

Going deeper

For a student

On an even-dimensional spin manifold, the K-theory Thom isomorphism shows that every class in $K(T^*M)$ is a difference of symbol classes of Dirac operators twisted by vector bundles. Twisting by a complex vector bundle $W$ gives $\operatorname{ind} D^+_W = \langle \hat A(TM) \operatorname{ch}(W), [M] \rangle$, where $\operatorname{ch}$ is the Chern character. So on spin manifolds this special case already contains the general index theorem. The vanishing of $\hat A(M)$ under positive scalar curvature, due to Lichnerowicz, began the study of obstructions to positive scalar curvature.