Efficient multisections of odd-dimensional tori

Kindred, Thomas

doi:10.2140/agt.2023.23.3997

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Efficient multisections of odd-dimensional tori

Thomas Kindred

Algebraic & Geometric Topology 23 (2023) 3997–4056

DOI: 10.2140/agt.2023.23.3997

Abstract

Rubinstein and Tillmann generalized the notions of Heegaard splittings of $3$ –manifolds and trisections of $4$ –manifolds by defining multisections of PL $n$ –manifolds, which are decompositions into $k = ⌊ \frac{1}{2} n ⌋ + 1$ $n$ –dimensional $1$ –handlebodies with nice intersection properties. For each odd-dimensional torus $T^{n}$ , we construct a multisection which is efficient in the sense that each $1$ –handlebody has genus $n$ , which we prove is optimal; each multisection is symmetric with respect to both the permutation action of $S_{n}$ on the indices and the $ℤ_{k}$ translation action along the main diagonal. We also construct such a trisection of $T^{4}$ , lift all symmetric multisections of tori to certain cubulated manifolds, and obtain combinatorial identities as corollaries.

Keywords

multisection, piecewise-linear, PL, handle decomposition, cubulation, trisection

Mathematical Subject Classification

Primary: 57K50, 57M99, 57N99, 57R10, 57R15

Secondary: 05A10

References

Publication

Received: 5 November 2020

Revised: 9 March 2022

Accepted: 26 July 2022

Published: 23 November 2023

Authors

Thomas Kindred
Department of Mathematics Wake Forest University Winston-Salem, NC United States
https://www.thomaskindred.com

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Volume 23, issue 9 (2023)