Anton Savin

Elliptic Theory on Manifolds with Corners: II. Homotopy Classification and K-Homology

We establish the stable homotopy classification of elliptic pseudo-differential operators on manifolds with corners and show that the set of elliptic operators modulo stable homotopy is isomorphic to the K-homology group of some stratified manifold. By way of application, generalizations of some recent results due to Monthubert and Nistor are given.

Boundary value problems on manifolds with fibered boundary

We define a class of boundary value problems on manifolds with fibered boundary. This class is in a certain sense a deformation between the classical boundary value problems and the Atiyah–Patodi–Singer problems in subspaces (it contains both as special cases). The boundary conditions in this theory are taken as elements of the C *-algebra generated by pseudodifferential operators and families of pseudodifferential operators in the fibers. We prove the Fredholm property for elliptic boundary value problems and compute a topological obstruction (similar to Atiyah–Bott obstruction) to the existence of elliptic boundary conditions for a given elliptic operator. Geometric operators with trivial and nontrivial obstruction are given. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Index defects in the theory of spectral boundary value problems

We study index defects in spectral boundary value problems for elliptic operators. Explicit analytic expressions for index defects in various situations are given. The corresponding topological indices are computed as homotopy invariants of the principal symbol.

The eta invariant and parity conditions

Abstract We give a formula for the η -invariant of odd-order operators on even-dimensional manifolds and even-order operators on odd-dimensional manifolds. Second-order operators with nontrivial η -invariants are found. This solves a problem posed by Gilkey.

Elliptic Theory on Manifolds with Corners: I. Dual Manifolds and Pseudodifferential Operators

In this first part of the paper, we define a natural dual object for manifolds with corners and show how pseudodifferential calculus on such manifolds can be constructed in terms of the localization principle in C*-algebras. In the second part, these results will be applied to the solution of Gelfand’s problem on the homotopy classification of elliptic operators for the case of manifolds with corners.

Pseudodifferential Subspaces and Their Applications in Elliptic Theory

The aim of this paper is to explain the notion of subspace defined by means of pseudodifferential projection and give its applications in elliptic theory. Such subspaces are indispensable in the theory of well-posed boundary value problems for an arbitrary elliptic operator, including the Dirac operator, which has no classical boundary value problems. Pseudodifferential subspaces can be used to compute the fractional part of the spectral Atiyah73-Patodi— Singer eta invariant, when it defines a homotopy invariant (Gilkey’s problem). Finally, we explain how pseudodifferential subspaces can be used to give an analytic realization of the topological K-group with finite coefficients in terms of elliptic operators. It turns out that all three applications are based on a theory of elliptic operators on closed manifolds acting in subspaces.