A Topological Splitting Theorem for Poincare Duality Groups and High-dimensional Manifolds

TL;DR

This paper proves that certain -dimensional manifold pairs with -dimensional differences have -injective maps that factor through finite covers, using algebraic splitting theorems for Poincare9 duality groups and surgery techniques.

Contribution

It introduces a new algebraic splitting theorem for Poincare9 duality groups and applies it to topological manifold maps, extending Waldhausen's torus theorem.

Findings

-injective maps factor through finite covers as embeddings.

New algebraic splitting theorem for Poincare9 duality groups.

Derived obstructions to -injective map existence.

Abstract

We show that for a wide class of manifold pairs N, M satisfying dim(M) = dim(N) + 1, every \pi_1-injective map f : N --> M factorises up to homotopy as a finite cover of an embedding. This result, in the spirit of Waldhausen's torus theorem, is derived using Cappell's surgery methods from a new algebraic splitting theorem for Poincare duality groups. As an application we derive a new obstruction to the existence of \pi_1-injective maps.