The Double n-Space Property for Contractible n-Manifolds

TL;DR

This paper explores how certain contractible n-manifolds, especially in four dimensions, can be decomposed into simpler pieces, revealing new insights into their structure and splitting properties.

Contribution

It introduces new examples of 4-manifolds that split into parts homeomorphic to Euclidean space or n-balls, expanding understanding of their topological decompositions.

Findings

Constructed large classes of 4-manifolds with splitting properties.

Extended Whitehead manifold concepts to higher dimensions.

Provided new examples of contractible manifolds with specific decompositions.

Abstract

Motivated by a recent paper of Gabai on the Whitehead contractible 3-manifold, we investigate contractible manifolds $M^n$ which decompose or split as $M^n = A \cup_C B$ where $A,B,C \approx \mathbb{R}^n$ or $A,B,C \approx \mathbb{B}^n$. Of particular interest to us is the case $n=4.$ Our main results exhibit large collections of $4$-manifolds that split in this manner.