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Smoothing 3-manifolds in 5-manifolds

Michelle Daher, Mark Powell

Abstract

We show that every locally flat topological embedding of a 3-manifold in a smooth 5-manifold is homotopic, by a small homotopy, to a smooth embedding. We deduce that topologically locally flat concordance implies smooth concordance for smooth surfaces in smooth 4-manifolds.

Smoothing 3-manifolds in 5-manifolds

Abstract

We show that every locally flat topological embedding of a 3-manifold in a smooth 5-manifold is homotopic, by a small homotopy, to a smooth embedding. We deduce that topologically locally flat concordance implies smooth concordance for smooth surfaces in smooth 4-manifolds.
Paper Structure (10 sections, 15 theorems, 34 equations, 3 figures)

This paper contains 10 sections, 15 theorems, 34 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Topological concordance implies smooth concordance for surfaces in 4-manifolds
  3. Comparison with other dimensions
  4. Outline of the proof of Theorem A
  5. Smoothing theory
  6. Recap of smoothing theory
  7. Lashof's nonsmoothable 3-knot
  8. Smoothing the complement of an embedding
  9. Comparing with the standard smooth structure on N
  10. Conditions for smoothing up to isotopy

Key Result

Theorem A

Let $f\colon Y\to N$ be a locally flat proper topological embedding that is smooth near $\partial Y$. Then $f$ is homotopic rel. boundary, via an arbitrarily small homotopy, to a smooth embedding.

Figures (3)

  • Figure 1: The picture on the left is a schematic of $K$. The middle picture illustrates adding the collar $\partial_1 K\times[0,1)$ to obtain $K'$. The picture on the right illustrates adding $\partial K'\times[0,1)$ to obtain $K"$.
  • Figure 2: A schematic diagram of $N$ decomposed as $N= W_f \cup_E \overline{\nu} M$, where $M = f(Y)$, showing the case that $Y=Y_1\sqcup Y_2$ has two connected components with nonempty boundary.
  • Figure 3: A schematic diagram of $W_g$ when one Lashof knot is attached.

Theorems & Definitions (30)

  • Theorem A
  • Definition 1.1
  • Corollary 1.2
  • Theorem 1.3: Schultz
  • Theorem 1.4: Schultz
  • Remark 1.5
  • Theorem 2.1
  • proof
  • Definition 2.2
  • Proposition 2.3
  • ...and 20 more