## Statement of the index theorem

*Atiyah-Singer Index Theorem (1968). Let E and F be smooth vector bundles over a compact manifold X. Let be an elliptic differential operator. Then Index D = t-Index D.*

Remarks:

*analytic index*.

*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

2. K-theory form of the topological index

3. Geometric form of the topological index where is an explicitly defined polynomial in the curvatures of X, E, and F.

Here is a glossary for the terms above.

are smooth complex vector bundles over a closed smooth manifold .

and are vector spaces of smooth sections.

is the symbol of . This is a certain virtual bundle over the Thom space of the tangent bundle of X.

and are the Chern character and the the Todd class; they are characteristic classes.

**Four classical elliptic operators**

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