#### Vol. 315, No. 2, 2021

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The relative $\mathcal{L}$-invariant of a compact $4$-manifold

### Nickolas A. Castro, Gabriel Islambouli, Maggie Miller and Maggy Tomova

Vol. 315 (2021), No. 2, 305–346
DOI: 10.2140/pjm.2021.315.305
##### Abstract

We introduce the relative $\mathsc{ℒ}$-invariant $r\mathsc{ℒ}\left(X\right)$ 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 $\mathsc{ℒ}$-invariant for smooth, orientable, closed 4-manifolds by Kirby and Thompson. We show that if $X$ is a rational homology ball, then $r\mathsc{ℒ}\left(X\right)=0$ if and only if $X\cong {B}^{4}$. This is analogous to the case for closed 4-manifolds: Kirby and Thompson showed that if $X$ is a rational homology sphere, then $\mathsc{ℒ}\left(X\right)=0$ if and only if $X\cong {S}^{4}$.

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 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.

##### Keywords
trisection, curve complex, arc complex, relative trisection, open book decomposition
##### Mathematical Subject Classification 2010
Primary: 57M99, 57R15
Secondary: 57M15