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Complements of locally flat submanifolds are finite CW complexes

Andrew Ho

Abstract

We show that if $Y$ is a compact topological manifold and $X$ is a locally flat submanifold, then the complement $Y - X$ is homotopy equivalent to a finite CW complex. This is a direct proof, and does not rely on much of the theory of topological manifolds.

Complements of locally flat submanifolds are finite CW complexes

Abstract

We show that if is a compact topological manifold and is a locally flat submanifold, then the complement is homotopy equivalent to a finite CW complex. This is a direct proof, and does not rely on much of the theory of topological manifolds.
Paper Structure (10 sections, 12 theorems, 8 equations, 1 figure)

This paper contains 10 sections, 12 theorems, 8 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof for $n - d \leq 2$ and preparation for $n - d \geq 3$
  3. Proof of Theorem 1.1 when $1 \leq n - d \leq 2$
  4. Proof of Theorem 1.1 when $n - d = 0$
  5. $n - d \geq 3$: first facts
  6. $Y - X$ is finitely dominated
  7. Reduction to $(Y \times \mathbb{R}^{N}) - (X \times 0)$
  8. Assembly of Proof for \ref{['thm:maintheorem']}
  9. Proof of \ref{['prop:diskbundle']}
  10. Consideration of Boundary

Key Result

Theorem 1.1

Suppose $Y^{n}$ is a compact topological manifold while $X^{d}$ is a compact locally flat submanifold. Then $Y - X$ is homotopy equivalent to a finite CW complex.

Figures (1)

  • Figure 1: The homotopy $H_{t}$.

Theorems & Definitions (21)

  • Theorem 1.1
  • Proposition 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • ...and 11 more