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Abstract
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We introduce the
relative -invariant
of a smooth, orientable,
compact 4-manifold
with boundary. This invariant is defined by measuring the lengths
of certain paths in the cut complex of a trisection surface for
. This is motivated by the
definition of the
-invariant
for smooth, orientable, closed 4-manifolds by Kirby and Thompson. We show that if
is a rational
homology ball, then
if and only if
.
This is analogous to the case for closed 4-manifolds: Kirby and Thompson showed that if
is a rational homology
sphere, then
if and only if
.
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
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
are
related by interior stabilizations.
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Keywords
trisection, curve complex, arc complex, relative
trisection, open book decomposition
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Mathematical Subject Classification 2010
Primary: 57M99, 57R15
Secondary: 57M15
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Milestones
Received: 26 October 2019
Revised: 1 August 2021
Accepted: 21 August 2021
Published: 19 January 2022
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