Connected sum at infinity and Cantrell-Stallings hyperplane unknotting

TL;DR

This paper introduces and clarifies the Connected Sum at Infinity operation on manifolds, providing elementary proofs of key theorems like the Hyperplane Linearization Theorem, and classifies hyperplane embeddings in Euclidean spaces across different categories.

Contribution

It offers a clear, elementary treatment of CSI, proves the Hyperplane Linearization Theorem uniformly across categories, and classifies hyperplane embeddings via simplicial trees.

Findings

Elementary proof of the Hyperplane Linearization Theorem (HLT)

Classification of hyperplane embeddings by simplicial trees

Connections between CSI, HLT, and Schoenflies theorem

Abstract

We give a general treatment of the somewhat unfamiliar operation on manifolds called Connected Sum at Infinity, or CSI for short. A driving ambition has been to make the geometry behind the well definition and basic properties of CSI as clear and elementary as possible. CSI then yields a very natural and elementary proof of a remarkable theorem of J. C. Cantrell and J. R. Stallings. It asserts unknotting of proper embeddings of euclidean (m-1)-space in euclidean m-space with m not equal to 3, for all three classical manifold categories: topological, piecewise linear, and differentiable. It is one of the few major theorems whose statement and proof can be the same for all three categories. We give it the acronym HLT, which is short for Hyperplane Linearization Theorem. The topological version of the HLT immediately implies B. Mazur's topological Schoenflies theorem. We can thus claim that the Cantrell-Stallings theorem, as we present it, is an enhancement of the topological Schoenflies theorem that has exceptional didactic value. We also prove a classification of multiple codimension 1 hyperplane embeddings in eucldiean m-space for m not equal to 3. Namely, they are classified by countable simplicial trees with one edge for each hyperplane (planar trees for m=2). This result is called the Multiple Hyperplane Linearization Theorem, or MHLT for short. We give an exposition of C. Greathouse's Slab Theorem, and in conclusion some possibly novel proofs of the 2-dimensional MHLT and related results classifying contractible 2-manifolds with boundary.