On the various notions of Poincar\'e duality pair

TL;DR

This paper explores foundational aspects of Poincaré spaces, proving new results including a conjecture resolution, and constructing examples that distinguish different notions of Poincaré duality.

Contribution

It establishes several fundamental results on Poincaré spaces, including resolving an old conjecture and constructing finite CW pairs with specific duality properties.

Findings

Resolved C.T.C. Wall's conjecture affirmatively

Constructed finite CW pairs with relative Poincaré duality where the subspace fails duality

Proved a relative version of Gottlieb's result on Poincaré duality and fibrations

Abstract

We establish a number of foundational results on Poincar\'e spaces which result in several applications. One application settles an old conjecture of C.T.C. Wall in the affirmative. Another result shows that for any natural number n, there exists a finite CW pair $(X,Y)$ satisfying relative Poincar\'e duality in dimension n with the property that $Y$ fails to satisfy Poincar\'e duality. We also prove a relative version of a result of Gottlieb about Poincar\'e duality and fibrations.