Volume 18, issue 3 (2018)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Comparing $4$–manifolds in the pants complex via trisections

Gabriel Islambouli

Algebraic & Geometric Topology 18 (2018) 1799–1822
Abstract

Given two smooth, oriented, closed 4–manifolds, M1 and M2, we construct two invariants, DP(M1,M2) and D(M1,M2), coming from distances in the pants complex and the dual curve complex, respectively. To do this, we adapt work of Johnson on Heegaard splittings of 3–manifolds to the trisections of 4–manifolds introduced by Gay and Kirby. Our main results are that the invariants are independent of the choices made throughout the process, as well as interpretations of “nearby” manifolds. This naturally leads to various graphs of 4–manifolds coming from unbalanced trisections, and we briefly explore their properties.

Keywords
trisection, pants complex, dual curve complex, $4$–manifold
Mathematical Subject Classification 2010
Primary: 57M15, 57M99
References
Publication
Received: 24 July 2017
Revised: 4 December 2017
Accepted: 13 February 2018
Published: 3 April 2018
Authors
Gabriel Islambouli
Department of Mathematics
University of Virginia
Charlottesville, VA
United States