# M721: Index Theory

Ruminations of a Graduate Class

## Statement of the index theorem

Atiyah-Singer Index Theorem (1968). Let E and F be smooth vector bundles over a compact manifold X. Let $D : C^\infty(E) \to C^\infty(F)$ be an elliptic differential operator. Then Index D = t-Index D.

Remarks:

• The Fundamental Theorem of Elliptic Operators implies that ker D and cok D are finite-dimensional. Index D := dim ker D – dim cok D is called the analytic index.
• t-Index D is called the topological index.
• Clearly the analytic index is an integer. Thus the index theory says that one integer expression equals another. But the topological index has three forms and is not clearly an integer.

1. Cohomology form of the topological index $(-1)^n (ch (\sigma D) td (X))[TX]$
2. K-theory form of the topological index $i_! (\sigma D) \in K^0(\text{pt}) \cong {\mathbb Z}$
3. Geometric form of the topological index $\int_X K_{AS}$ where $K_{AS}$ is an explicitly defined polynomial in the curvatures of X, E, and F.

Here is a glossary for the terms above.
${E,F}$ are smooth complex vector bundles over a closed smooth manifold ${X}$.
${C^\infty E}$ and ${C^\infty F}$ are vector spaces of smooth sections.
${\sigma D}$ is the symbol of ${D}$. This is a certain virtual bundle over the Thom space of the tangent bundle of X.
${ch}$ and ${td}$ are the Chern character and the the Todd class; they are characteristic classes.

Four classical elliptic operators

• DeRham operator. Here the index is the Euler characteristic of X.
• Signature operator, defined for oriented manifolds. Here the index is the signature of X and the index theorem implies the the Hirzebruch Signature Theorem
• Dolbeault operator, defined for complex manifolds. Implies the Riemann-Roch Theorem.
• Dirac operator, defined for Spin manifolds.