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The algebraic topology of 4-manifolds multisections

Delphine Moussard, Trenton Schirmer

Abstract

A multisection of a 4-manifold is a decomposition into 1-handlebodies intersecting pairwise along 3-dimensional handlebodies or along a central closed surface; this generalizes the Gay-Kirby trisections. We show how to compute the twisted absolute and relative homology, the torsion and the twisted intersection form of a 4-manifold from a multisection diagram. The homology and torsion are given by a complex of free modules defined by the diagram and the intersection form is expressed in terms of the intersection form on the central surface. We give efficient proofs, with very few computations, thanks to a retraction of the (possibly punctured) 4-manifold onto a CW-complex determined by the multisection diagram. Further, a multisection induces an open book decomposition on the boundary of the 4-manifold; we describe the action of the monodromy on the homology of the page from the multisection diagram.

The algebraic topology of 4-manifolds multisections

Abstract

A multisection of a 4-manifold is a decomposition into 1-handlebodies intersecting pairwise along 3-dimensional handlebodies or along a central closed surface; this generalizes the Gay-Kirby trisections. We show how to compute the twisted absolute and relative homology, the torsion and the twisted intersection form of a 4-manifold from a multisection diagram. The homology and torsion are given by a complex of free modules defined by the diagram and the intersection form is expressed in terms of the intersection form on the central surface. We give efficient proofs, with very few computations, thanks to a retraction of the (possibly punctured) 4-manifold onto a CW-complex determined by the multisection diagram. Further, a multisection induces an open book decomposition on the boundary of the 4-manifold; we describe the action of the monodromy on the homology of the page from the multisection diagram.
Paper Structure (11 sections, 35 theorems, 40 equations, 6 figures)

This paper contains 11 sections, 35 theorems, 40 equations, 6 figures.

Table of Contents

  1. Introduction and main results
  2. Algebraic Preliminaries
  3. Algebraic torsion
  4. Twisted homology and Reidemeister torsion
  5. The equivariant intersection form
  6. The twisted absolute homology groups and torsion
  7. The twisted relative homology groups and torsion
  8. Intersection forms
  9. The case of closed $4$--manifolds
  10. The boundary: monodromy and homology
  11. Sample calculations

Key Result

Theorem 1

If $\partial X\neq\emptyset$, the twisted homology of $X$ is given by the chain complex of free $R$--modules where $\partial_2((x_i)_{1\leq i\leq n})=((x_i-x_{i+1})_{1\leq i\leq n})$ and $\partial_1((x_i)_{1\leq i\leq n})=\sum_{i=1}^n x_i$. Moreover, if $R$ is a field, there are complex bases $c$ of complexforX such that $\tau^\varphi(X;h)=\tau(\mathcal{C};c,h)$ for any homology basis $h$ of $X$

Figures (6)

  • Figure 1: Schematic of a multisection
  • Figure 2: Heegaard diagram for $C_{i-1}\cup C_i$ In this example, $C_{i-1}$ and $C_i$ are constructed with eight $1$--handles and $X_i$ with six $1$--handles. The manifold $X$ has four boundary components. The components of the page $\Sigma_\partial$ have a pair (genus,number of boundary components) equal to $(1,2)$, $(2,1)$, $(1,1)$ and $(0,2)$.
  • Figure 3: Pushing the relative $2$--skeleton
  • Figure 4: Curves on $\Sigma$ for the compression body $C_i$ The curves of $c_i$ are in red. The homology classes of the blue and violet curves form bases of $J_i$ and $\mathcal{J}_i$ respectively.
  • Figure 5: A trisection diagram of a disk bundle over $S^2$ with Euler number $-2$
  • ...and 1 more figures

Theorems & Definitions (71)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem : Theorem \ref{['thm:XHomology']}
  • Theorem : Theorem \ref{['thm:RelXHomology']}
  • Theorem : Theorem \ref{['thm:XClosedHomology']}
  • Theorem : Theorem \ref{['thm:IntersectionForm']}
  • Theorem : Theorem \ref{['thIntFormOdd']}
  • Definition 2.1
  • ...and 61 more