Compact contractible n-manifolds have arc spines (n ≥ 5)

Fredric Davis Ancel and Craig R. Guilbault

Vol. 168 (1995), No. 1, 1–10

DOI: 10.2140/pjm.1995.168.1

Abstract

The following two theorems were motivated by questions about the existence of disjoint spines in compact contractible manifolds.

Theorem 1. Every compact contractible n-manifold (n ≥ 5) is the union of two n-balls along a contractible (n − 1)-dimensional submanifold of their boundaries.

A compactum X is a spine of a compact manifold M if M is homeomorphic to the mapping cylinder of a map from ∂M to X.

Theorem 2. Every compact contractible n-manifold (n ≥ 5) has a wild arc spine.

Also a new proof is given that for n ≥ 6, every homology (n− 1)-sphere bounds a compact contractible n-manifold. The implications of arc spines for compact contractible manifolds of dimensions 3 and 4 are discussed in §5. The questions about the existence of disjoint spines in compact contractible manifolds which motivated the preceding theorems are stated in §6.

Mathematical Subject Classification 2000

Primary: 57Q15

Secondary: 54C99, 57N15, 57N45

Milestones

Received: 18 July 1992

Revised: 13 November 1992

Published: 1 March 1995