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

We introduce the relative -invariant r(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 -invariant for smooth, orientable, closed 4-manifolds by Kirby and Thompson. We show that if X is a rational homology ball, then r(X) = 0 if and only if XB4. This is analogous to the case for closed 4-manifolds: Kirby and Thompson showed that if X is a rational homology sphere, then (X) = 0 if and only if XS4.

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.

trisection, curve complex, arc complex, relative trisection, open book decomposition
Mathematical Subject Classification 2010
Primary: 57M99, 57R15
Secondary: 57M15
Received: 26 October 2019
Revised: 1 August 2021
Accepted: 21 August 2021
Published: 19 January 2022
Nickolas A. Castro
Department of Mathematics
Rice University
Houston, TX 77005-1892
United States
Gabriel Islambouli
Department of Mathematics
University of California
Davis, CA 95616
United States
Maggie Miller
Department of Mathematics
Stanford University
Stanford, CA 94305
United States
Maggy Tomova
Department of Mathematics
University of Iowa
Iowa City, IA 52240
United States