On pseudo-collarability and Z-compactifiablity of manifolds

Abstract：

Siebenmann's landmark 1965 dissertation established conditions for compactifying open high-dimensional manifolds by adding boundaries, a process termed 'completion' or 'collaring'. Nearly six decades later, Gu-Guilbault broadened this scope to include noncompact manifolds with boundaries, offering a complete characterization of completable manifolds. However, the emergence of exotic universal covering spaces and shifts towards topics like the Borel and Novikov conjectures and geometric group theory, where fundamental groups at infinity are unstable, has necessitated extensions to the completion of manifolds. Key among these are pseudo-collarability, introduced by Guilbault in 2000, and Z-compactifiability, dating back to Anderson's 1967 work on infinite-dimensional manifolds. This talk aims to address the implication between these two concepts. Specifically, we will show that a well-established set of conditions proposed by Chapman and Siebenmann in 1976 assures Z-compactifiability of manifolds. Time permitting, we will also discuss some applications to the Borel conjecture.