Efficient multisections of odd-dimensional tori

TL;DR

This paper constructs and analyzes efficient, symmetric multisections of odd-dimensional tori, demonstrating their optimal genus and symmetry properties, and extends these constructions to cubulated manifolds with combinatorial implications.

Contribution

It introduces the first explicit construction of optimal, symmetric multisections for odd-dimensional tori and explores their properties and extensions to cubulated manifolds.

Findings

Constructed efficient multisections of odd-dimensional tori with optimal genus.

Proved symmetry properties under permutation and translation actions.

Derived combinatorial identities from lifted multisections.

Abstract

Rubinstein--Tillmann generalized the notions of Heegaard splittings of 3-manifolds and trisections of 4-manifolds by defining {\it multisections} of PL $n$-manifolds, which are decompositions into $k=\lfloor n/2\rfloor+1$ $n$-dimensional 1-handlebodies with nice intersection properties. For each odd-dimensional torus $T^n$, we construct a multisection which is {\it efficient} in the sense that each 1-handlebody has genus $n$, which we prove is optimal; each multisection is {\it 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.