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On the isotopy classes of embeddings of surfaces in 5-manifolds

Ruoyu Qiao

Abstract

Let f, g be two homotopic smooth embeddings of a closed surface in a closed oriented 5-dimensional manifold. We show that if f admits a common algebraic dual 3-sphere, or if the fundamental group of the ambient space is trivial, then f and g must be isotopic. This generalizes a result of Kosanovic, Schneiderman, and Teichner. The proof is based on the construction of an invariant that classifies the isotopy classes of smooth embeddings of surfaces in ambient 5-dimensional manifolds within a homotopy class, which may be of independent interest. The invariant is defined in terms of the homotopy groups of the 5-dimensional manifold.

On the isotopy classes of embeddings of surfaces in 5-manifolds

Abstract

Let f, g be two homotopic smooth embeddings of a closed surface in a closed oriented 5-dimensional manifold. We show that if f admits a common algebraic dual 3-sphere, or if the fundamental group of the ambient space is trivial, then f and g must be isotopic. This generalizes a result of Kosanovic, Schneiderman, and Teichner. The proof is based on the construction of an invariant that classifies the isotopy classes of smooth embeddings of surfaces in ambient 5-dimensional manifolds within a homotopy class, which may be of independent interest. The invariant is defined in terms of the homotopy groups of the 5-dimensional manifold.
Paper Structure (6 sections, 24 theorems, 23 equations, 6 figures)

This paper contains 6 sections, 24 theorems, 23 equations, 6 figures.

Table of Contents

  1. Introduction and Results
  2. Intersection Number of Homotopies up to Homotopy
  3. Affine Action of Self-homotopies
  4. Action of Self-homotopies on $\mathscr{H}^f$
  5. Action of Self-homotopies on $\mathbb{A}_{[f]}$
  6. The invariant map and applications

Key Result

Theorem 1.1

There exists the following commutative diagram: where $\mu$ is the intersection number (Definition def2.5), $\mathscr{H}^f_0$ acts on $\mathscr{H}^f$ and $\mathbb{A}_{[f]}$ inducing the map $\overline\mu$ (Proposition prop3.25), which is a bijection. And there is a bijection $p^{-1}([f])\rightarrow\mathscr{H}^f/\mathscr{H}^f_0$ and $\mathbb{A}_{f is a bijection.

Figures (6)

  • Figure 1: A cusp homotopy
  • Figure 2: The action of $g$ on $i$
  • Figure 3: The relation $1=0$ realized by isotopies
  • Figure 4: The relation $0=g\in G$ realized by isotopies
  • Figure 5: An example that the homotopy can be extended
  • ...and 1 more figures

Theorems & Definitions (57)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Lemma 2.4
  • proof : Sketch of proof
  • Definition 2.5
  • ...and 47 more