Global analysis, analysis on manifolds Index theory Problem

Gelfand's problem on the index

Builds on Analytic index Elliptic operator on a compact manifold

Problem

Let $M$ be a closed smooth manifold and let $P$ be an elliptic differential operator between vector bundles over $M$ . The analytic index $\operatorname{ind} P = \dim \ker P - \dim \operatorname{coker} P$ is unchanged under continuous deformations of $P$ through elliptic operators. Find a formula for $\operatorname{ind} P$ in terms of topological invariants of $M$ and of the principal symbol of $P$ .

Remarks

Gelfand raised the question in an article on elliptic equations published in 1960. Atiyah and Singer solved it with their index theorem, announced in 1963: the analytic index equals the topological index. Earlier special cases were known, such as the Riemann–Roch theorem for curves and its generalization by Hirzebruch, and the Hirzebruch signature theorem.

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Analytic index concept
Elliptic operator on a compact manifold concept
Gelfand's problem on the index problem

Tag	`25G4AP9P4R`
URI	`m2://25G4AP9P4R`
Kind	problem
Writing	W0 stub
MSC	58J20, 35J99
Status	This copy of the site has no build data. Run `m2 status 25G4AP9P4R`.
About	Analytic index
Uses	Analytic index, Elliptic operator on a compact manifold

Gelfand's problem on the index

Problem

Remarks

Reading path

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