Elliptic symbols, elliptic operators and Poincar\'e duality on conical pseudomanifolds

TL;DR

This paper extends the concept of symbols and Poincaré duality to conical pseudomanifolds, providing a geometric interpretation of duality as a principal symbol map in singular spaces.

Contribution

It generalizes the notion of symbols to conical pseudomanifolds and interprets Poincaré duality as a principal symbol map in this singular setting.

Findings

Symbols on manifolds extend to conical pseudomanifolds

Poincaré duality interpreted as a principal symbol map

Provides a geometric framework for duality in singular spaces

Abstract

In a earlier work of Claire Debord and the author, a notion of noncommutative tangent space isdefined for a conical pseudomanifold and the Poincar\'e duality in $K$-theory is proved between this space and the pseudomanifold. The present paper continues this work. We show that an appropriate and natural presentation of the notion of symbols on a manifold generalizes right away to conical pseudomanifolds and that it enables us to interpret the Poincar\'e duality in the singular setting as a principal symbol map.