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Stabilizations of $s$-cobordisms of dimension $5$

Jinzhou Huang

Abstract

It has long been known that the $s$-cobordism theorem fails for $5$-dimensional $s$-cobordisms. In this article we study how many times of "stabilizations" are needed to turn a $5$-dimensional $s$-cobordism to a product cobordism. The question is analogous to asking how many times of stabilizations are needed to turn an exotic pair of four manifolds into diffeomorphic ones. The main tools in this article are Gabai's $4$D light bulb theorem and its applications, and we also use a refinement of $4$D light bulb theorem by Freedman Quinn invariant.

Stabilizations of $s$-cobordisms of dimension $5$

Abstract

It has long been known that the -cobordism theorem fails for -dimensional -cobordisms. In this article we study how many times of "stabilizations" are needed to turn a -dimensional -cobordism to a product cobordism. The question is analogous to asking how many times of stabilizations are needed to turn an exotic pair of four manifolds into diffeomorphic ones. The main tools in this article are Gabai's D light bulb theorem and its applications, and we also use a refinement of D light bulb theorem by Freedman Quinn invariant.

Paper Structure

This paper contains 6 sections, 16 theorems, 12 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tools In $4$-Dimensional Topology
  4. The Geometric Action and Freedman-Quinn Invariant
  5. proof of the main theorem
  6. Remarks of the Main Theorem

Key Result

Theorem 1.0.1

Given an oriented $5$-dimensional s-cobordism $W=W^5$ between oriented, closed $4$-dimensional manifolds $M_0$ and $M_1$ and a Morse function $f:W\to\mathbb{R}$ satisfying that $f$ has only critical points of index $2$, $3$. Assume $f$ has $k$ pairs of critical points of index $2$, $3$, then $W$ bec

Figures (5)

  • Figure 1:
  • Figure 2:
  • Figure 3:
  • Figure 4:
  • Figure 5:

Theorems & Definitions (29)

  • Theorem 1.0.1
  • Theorem 1.0.2
  • Definition 2.1.1: intersection number with values in $\mathbb{Z}\pi_1(M)$
  • Remark 2.1.2
  • Proposition 2.1.3
  • Definition 2.2.1: Freedman-Quinn invariant
  • Remark 2.2.2
  • Lemma 2.2.3
  • Theorem 2.2.4
  • Theorem 2.2.5: The geometric action of $\mathbb{F}_{2}T_M$ on $\mathcal{R}_{[f]}^G$
  • ...and 19 more