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Efficient multisections of odd-dimensional tori

Thomas Kindred

Algebraic & Geometric Topology 23 (2023) 3997–4056
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 =1 2n + 1 n–dimensional 1–handlebodies with nice intersection properties. For each odd-dimensional torus Tn, 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 Sn on the indices and the k translation action along the main diagonal. We also construct such a trisection of T4, 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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