On the various notions of Poincar\'e duality pair

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.