Elliptic Quasicomplexes on Compact Closed Manifolds

TL;DR

This paper studies elliptic quasicomplexes of pseudodifferential operators on compact manifolds, establishing their Fredholm property, defining an Euler characteristic, and generalizing the Atiyah-Singer index theorem.

Contribution

It introduces the concept of elliptic quasicomplexes, proves their Fredholm property, and extends the Atiyah-Singer index theorem to this setting.

Findings

Elliptic quasicomplexes are shown to be Fredholm.

An Euler characteristic for elliptic quasicomplexes is defined.

A generalized Atiyah-Singer index theorem is proved.

Abstract

We consider quasicomplexes of pseudodifferential operators on a smooth compact manifold without boundary. To each quasicomplex we associate a complex of symbols. The quasicomplex is elliptic if this symbol complex is exact away from the zero section. We prove that elliptic quasicomplexes are Fredholm. Moreover, we introduce the Euler characteristic for elliptic quasicomplexes and prove a generalization of the Atiyah-Singer index theorem.