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Generation of sutured manifolds

Yi Ni

TL;DR

The paper proves that the set of connected sutured manifolds with fixed boundary type $R_{\pm}(\gamma)\cong F$ is generated, via surgery triads, by the product sutured manifold $(F,\partial F)\times[0,1]$, extending folklore results and enabling Floer-theoretic applications. The authors develop a framework based on generalized sutured Heegaard splittings (GSHS) and a Heegaard complexity $HC(M,\gamma)$, and they implement an induction on $HC$ by performing framed surgeries along pairs of non-separating attaching curves (NSACs) and Dehn twists to produce lower-$HC$ instances. The core technical steps show that any non-product case yields a surgery triad $(M,M',M'')$ with $HC(M',\gamma')<HC(M,\gamma)$ or $HC(M'',\gamma'')<HC(M,\gamma)$, allowing the construction of generators from the product. The results have implications for the use of Floer exact triangles in low-dimensional topology and connect to finite-generation phenomena for bordered sutured manifolds, with folklore special cases for $F=D^2$ or $F=S^2$.

Abstract

Given a compact, oriented, connected surface $F$, we show that the set of connected sutured manifolds $(M,γ)$ with $R_{\pm}(γ)\cong F$ is generated by the product sutured manifold $(F,\partial F)\times[0,1]$ through surgery triads. This result has applications in Floer theories of $3$--manifolds. The special case when $F=D^2$ or $S^2$ has been a folklore theorem, which has already been used by experts before.

Generation of sutured manifolds

TL;DR

The paper proves that the set of connected sutured manifolds with fixed boundary type is generated, via surgery triads, by the product sutured manifold , extending folklore results and enabling Floer-theoretic applications. The authors develop a framework based on generalized sutured Heegaard splittings (GSHS) and a Heegaard complexity , and they implement an induction on by performing framed surgeries along pairs of non-separating attaching curves (NSACs) and Dehn twists to produce lower- instances. The core technical steps show that any non-product case yields a surgery triad with or , allowing the construction of generators from the product. The results have implications for the use of Floer exact triangles in low-dimensional topology and connect to finite-generation phenomena for bordered sutured manifolds, with folklore special cases for or .

Abstract

Given a compact, oriented, connected surface , we show that the set of connected sutured manifolds with is generated by the product sutured manifold through surgery triads. This result has applications in Floer theories of --manifolds. The special case when or has been a folklore theorem, which has already been used by experts before.

Paper Structure

This paper contains 3 sections, 12 theorems, 29 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of the main theorem

Key Result

Theorem 1.4

The set of closed, oriented, connected $3$--manifolds is generated by $\{S^3\}$ through surgery triads.

Figures (5)

  • Figure 1: The local picture near $K$.
  • Figure 2: The local picture near $x$. The four angles are labeled with $1,2,3,4$.
  • Figure 3: In each part of the picture, two ovals are glued together by a vertical reflection to form a tube in the surface.
  • Figure 4: The two black dots are intersection points between $\alpha$ and $\beta$ of the same sign. The arc $b$ may also intersect $\alpha$ in its interior. In this case, $\alpha_1$ is obtained from $\alpha$ by a left-handed Dehn twist along $K$.
  • Figure 5: The local picture in a neighborhood of an arc in $\beta$.

Theorems & Definitions (34)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Definition 1.5
  • Remark 1.6
  • Example 1.7
  • Theorem 1.8
  • Remark 1.9
  • Definition 2.1
  • ...and 24 more