On the stabilization of embedded thickenings

TL;DR

This paper introduces a space of relative embedded thickenings for maps into Poincare duality spaces and proves a stabilization theorem generalizing smooth stabilization results to the Poincare setting.

Contribution

It defines a new space of embedded thickenings and establishes a highly connected stabilization map, extending smooth stabilization theorems to Poincare duality spaces.

Findings

Existence of a highly connected stabilization map induced by fiberwise suspension.

Generalization of Connolly and Williams' smooth stabilization theorem to Poincare duality spaces.

Framework for studying embedded thickenings in the Poincare duality category.

Abstract

We define a space of relative embedded thickenings of a given map from a finite complex to a Poincare Duality space, and show that there is a highly connected stabilization map between such spaces induced by fiberwise suspension. As a result, we obtain a generalization to the Poincare Duality category of a smooth stabilization theorem of Connolly and Williams.