On the index of elliptic operators on manifolds with conical singularities.

*(English. Russian original)*Zbl 0959.58034
Dokl. Math. 59, No. 2, 212-215 (1999); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 365, No. 2, 170-173 (1999).

From the introduction: The goal of this paper is to derive an index formula for elliptic differential operators on manifolds with conical singularities. Such operators have been investigated from the index theoretic point of view for almost twenty years, and this brief paper can hardly provide a complete review of studies on this subject. For this reason, we briefly treat only basic publications devoted to the subject of this work.

They may be divided into two directions. The first is related to the calculation of the \(L_2\)-index of the so-called Dirac-type operators and its use for the analysis of differential-geometric and spectral invariants of manifolds with singularities. The second direction covers the studies that derive an index formula for general elliptic operators on singular manifolds and, in particular, make a homotopic classification of the set of elliptic operators on such manifolds.

They may be divided into two directions. The first is related to the calculation of the \(L_2\)-index of the so-called Dirac-type operators and its use for the analysis of differential-geometric and spectral invariants of manifolds with singularities. The second direction covers the studies that derive an index formula for general elliptic operators on singular manifolds and, in particular, make a homotopic classification of the set of elliptic operators on such manifolds.

##### MSC:

58J20 | Index theory and related fixed-point theorems on manifolds |