On the geometric simple connectivity of open manifolds

TL;DR

The paper establishes an obstruction to certain open simply connected manifolds being geometrically simply connected, revealing many such manifolds are not, and explores conditions under which proper homotopy equivalence implies geometric simple connectivity.

Contribution

It introduces a new obstruction criterion for geometric simple connectivity in high-dimensional manifolds and analyzes the special case of four-dimensional manifolds related to Poénaru's conjecture.

Findings

Existence of uncountably many simply connected manifolds not geometrically simply connected.

Proper homotopy equivalence to a w.g.s.c. polyhedron implies w.g.s.c. for n≠4.

Special analysis of 4-manifolds and Poénaru's conjecture.

Abstract

One proves that there exists an obstruction to an open simply connected $n$-manifold of dimension $n\geq 5$ being geometrically simply connected. In particular there exist uncountably many simply connected $n$-manifolds which are not w.g.s.c. One proves that for $n\neq 4$ an $n$-manifold proper homotopy equivalent to a w.g.s.c. polyhedron is w.g.s.c. (for $n=4$ it is only end compressible). We analyze further the case $n=4$ and Po\'enaru's conjecture.