On the genera of moment-angle manifolds associated to dual-neighborly polytopes, combinatorial formulas and sequences

TL;DR

This paper derives formulas for the number of sphere product terms in the connected sum decompositions of moment-angle manifolds associated with dual-neighborly polytopes, expanding understanding of their topological structure.

Contribution

It provides explicit combinatorial formulas for the number of summands in the connected sum decompositions of these manifolds, including for new polytopes generated by combinatorial operations.

Findings

Formulas for the number of sphere product terms in the connected sum.

Extension of formulas to polytopes obtained via combinatorial operations.

Application to a wide class of simple and odd-dimensional polytopes.

Abstract

For a family of polytopes of even dimension $2p$, known as \textit{dual-neighborly}, it has been shown for $p\ne 2$ that the associated intersection of quadrics is a connected sum of sphere products $S^p\times S^p$. In this article we give formulas for the number of terms in that connected sum. Certain combinatorial operations produce new polytopes whose associated intersections are also connected sums of sphere products and we give also formulas for their number. These include a large amount of simple polytopes, including many odd-dimensional ones.