On the Homotopy Classification of Elliptic Operators on Manifolds with Edges

TL;DR

The paper extends the Atiyah–Singer paradigm to manifolds with edges by developing an edge calculus and showing that elliptic operators on such spaces are classified up to stable homotopy by the K-homology of the singular space. It constructs a precise cycle in analytic K-homology from zero-order elliptic operators, reduces the general problem to order-zero via order reduction, and then identifies Ell_*(𝓜) with K_*(𝓜) by leveraging a cone-algebra framework and semiclassical quantization. Central contributions include a detailed computation of boundary maps, a three-part proof structure culminating in a K-theory–K-homology isomorphism, and an explicit edge-theory obstruction analysis. The results significantly generalize the classical smooth-case classification to stratified manifolds with nonisolated singularities, providing explicit obstructions and a robust topological framework for edge problems.

Abstract

We obtain a classification of elliptic operators modulo stable homotopy on manifolds with edges (this is in some sense the simplest class of manifolds with nonisolated singularities). We show that the operators are classified by the K-homology group of the manifold. To this end, we construct exact sequence in elliptic theory isomorphic to the K-homology exact sequence. The main part of the proof is the computation of the boundary map using semiclassical quantization.

Theorems & Definitions (36)

• Proposition 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 1.4
• Definition 1.5
• Theorem 1.6
• Definition 1.7
• Definition 1.8
• Theorem 1.9
• Corollary 1.10