Global analysis, analysis on manifolds Definition

Index theory

Builds on Analytic index Topological index Elliptic operator on a compact manifold

Definition

Let $M$ be a closed smooth manifold, let $E$ and $F$ be complex vector bundles over $M$ , and let $P \colon C^\infty(M; E) \to C^\infty(M; F)$ be an elliptic differential operator. Its analytic index $\operatorname{ind} P = \dim \ker P - \dim \operatorname{coker} P$ is a finite integer. Index theory is the part of global analysis that computes such indices, and their refinements, in terms of topological data. Its main tool is the topological index, which depends only on the homotopy class of the principal symbol of $P$ .

Remarks

The Atiyah–Singer index theorem states that the analytic index and the topological index of $P$ are equal. Refinements replace the integer index by an element of a representation ring (equivariant index) or of a K-theory group of a parameter space (families index). Other refinements add correction terms on manifolds with boundary.

Example: on the circle $S^1 = \R / 2\pi\Z$ , the operator $P = d/d\theta$ on complex functions is elliptic. Its kernel is the constants and its image is the functions of mean zero, so $\operatorname{ind} P = 1 - 1 = 0$ .

Non-example: the dimension of the kernel alone is not an index quantity, because it is not stable. For $P_c = d/d\theta + c$ with $c \in \C$ , $\dim \ker P_c$ is $1$ when $c \in i\Z$ and $0$ otherwise, while $\operatorname{ind} P_c = 0$ for all $c$ .

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Analytic index concept
Elliptic operator on a compact manifold concept
Topological index concept
Index theory definition

Tag	`MBQS4B0TPA`
URI	`m2://MBQS4B0TPA`
Kind	definition
Writing	W0 stub
MSC	58J20, 19K56, 47A53
Defines	Index theory
Uses	Analytic index, Topological index, Elliptic operator on a compact manifold

Index theory

Definition

Remarks

Reading path

Further reading

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