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THE RELATIVE L-INVARIANT OF A COMPACT 4-MANIFOLD
Journal article   Peer reviewed

THE RELATIVE L-INVARIANT OF A COMPACT 4-MANIFOLD

Nickolas A Castro, Gabriel Islambouli, Maggie Miller and Maggy Tomova
Pacific journal of mathematics, Vol.315(2), pp.305-346
12/01/2021
DOI: 10.2140/pjm.2021.315.305

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Abstract

We introduce the relative L-invariant r 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 L-invariant for smooth, orientable, closed 4-manifolds by Kirby and Thompson. We show that if X is a rational homology ball, then r L(X) = 0 if and only if X congruent to 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 L(X) = 0 if and only if X congruent to 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.
 Mathematics Physical Sciences Science & Technology

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