[Paper Review] The relative $\mathcal{L}$-invariant of a compact $4$-manifold

Nickolas A. Castro, Gabriel Islambouli+2 more|arXiv (Cornell University)|Aug 14, 2019

Geometric and Algebraic Topology18 references4 citations

TL;DR

This paper introduces the relative $τ$-invariant $r\mathcal{L}(X)$ for compact, smooth, orientable 4-manifolds with boundary, defined via path lengths in the cut complex of a trisection surface, generalizing Kirby and Thompson's $τ$-invariant for closed 4-manifolds. It proves that for a rational homology ball $X$, $r\mathcal{L}(X) = 0$ if and only if $X \cong B^4$, and establishes a complete stabilization equivalence theorem for relative trisections using interior stabilization, relative stabilization, and a new move called the relative double twist.

ABSTRACT

In this paper, we introduce the relative $\mathcal{L}$-invariant $r\mathcal{L}(X)$ of a smooth, orientable, compact 4-manifold $X$ with boundary. This invariant is defined by measuring the lengths of certain paths in the cut complex of a trisection surface for $X$. This is motivated by the definition of the $\mathcal{L}$-invariant for smooth, orientable, closed 4-manifolds by Kirby and Thompson. We show that if $X$ is a rational homology ball, then $r\mathcal{L}(X)=0$ if and only if $X\cong B^4$. In order to better understand relative trisections, we also produce an algorithm to glue two relatively trisected 4-manifold by any Murasugi sum or plumbing in the boundary, and also prove that any two relative trisections of a given 4-manifold $X$ are related by interior stabilization, relative stabilization, and the relative double twist, which we introduce in this paper as a trisection version of one of Piergallini and Zuddas's moves on open book decompositions. Previously, it was only known (by Gay and Kirby) that relative trisections inducing equivalent open books on $X$ are related by interior stabilizations.

Motivation & Objective

• To define a relative version of the $τ$-invariant for compact 4-manifolds with boundary, extending Kirby and Thompson's invariant for closed 4-manifolds.
• To establish a characterization of the 4-ball $B^4$ among rational homology balls via the vanishing of the relative $τ$-invariant.
• To develop a complete set of moves—interior stabilization, relative stabilization, and relative double twist—that relate any two relative trisections of the same 4-manifold.
• To provide an algorithm for performing Murasugi sums on relative trisection diagrams, enabling the construction of trisections for boundary-connected sums of 4-manifolds.
• To relate the relative $τ$-invariant to the monodromy of the induced open book on the boundary, particularly through displacement distance in the arc complex.

Proposed method

• Define the relative $τ$-invariant $r\mathcal{L}(X)$ as the minimal path length in the cut complex of a trisection surface over all relative trisection diagrams of $X$.
• Introduce three invariants: $r\mathcal{L}(X)$, $r\mathcal{L}^\partial(X)$, and $r\mathcal{L}^\circ(X)$, measuring complexity of the full manifold, its boundary, and its interior, respectively.
• Define the relative double twist move as a trisection-level analog of a Harer twist, enabling manipulation of the monodromy of the induced open book on the boundary.
• Construct an explicit algorithm to glue two relatively trisected 4-manifolds via Murasugi sum, preserving the induced open book structure on the boundary.
• Use subsurface projections and arc complex distances to relate $r\mathcal{L}^\partial(\mathcal{T})$ to the stable translation distance of the monodromy map $\phi$ of the open book on $\partial X$.
• Apply results from the arc complex and Dehn twist factorizations to construct examples of relative trisections with arbitrarily large $r\mathcal{L}^\partial$ in fixed genus.

Experimental results

Research questions

• RQ1 When is the relative $τ$-invariant $r\mathcal{L}(X)$ equal to zero for a rational homology ball $X$?
• RQ2 How can relative trisections of a 4-manifold be related to one another using a finite set of moves?
• RQ3 Can Murasugi sums of 4-manifolds be realized via trisection diagrams, and if so, how?
• RQ4 How does the relative $τ$-invariant relate to the topological complexity of the monodromy of the induced open book on the boundary?
• RQ5 Can relative trisections of fixed genus have arbitrarily large relative $τ$-invariant?

Key findings

• For a rational homology ball $X$, $r\mathcal{L}(X) = 0$ if and only if $X \cong B^4$, generalizing the closed case result of Kirby and Thompson.
• There exist 4-manifolds $X$ with $r\mathcal{L}(X) \geq n$ for any $n \in \mathbb{N}$, showing the invariant can be arbitrarily large.
• If $r\mathcal{L}^\partial(\mathcal{T}) \leq 1$ for a $(g,k;p,b)$-relative trisection $\mathcal{T}$, then $\partial X \cong \#_{2p+b-1} S^1 \times S^2$.
• If $r\mathcal{L}^\partial(\mathcal{T}) < 2(2p+b-1)$, then $\partial X$ admits an $S^1 \times S^2$ summand.
• For any $n \in \mathbb{N}$ and fixed $p,b$ with $2p + b - 1 > 1$, there exists a relative trisection $\mathcal{T}_n$ with $r\mathcal{L}^\partial(\mathcal{T}_n) > n$, even when $H_1(\partial X_n; \mathbb{Z})$ is uniformly bounded.
• The relative $τ$-invariant satisfies $r\mathcal{L}^\partial(\mathcal{T}) \geq \frac{1}{3}(2p + b - 1) d_e(\phi)$, where $d_e(\phi)$ is the displacement distance of the monodromy $\phi$ of the induced open book on $\partial X$.