M\"obius transform, moment-angle complexes and Halperin-Carlsson conjecture

TL;DR

This paper develops an algebraic-combinatorial formula for the M"obius transform of simplicial complexes, uses it to estimate Betti number sums, and applies these results to verify the Halperin-Carlsson conjecture for moment-angle complexes.

Contribution

It introduces a new formula linking the M"obius transform to Betti numbers and applies it to confirm the conjecture for specific topological spaces.

Findings

The formula relates the M"obius transform to Betti numbers of Stanley-Reisner rings.

A method to estimate Betti number sums via complex compression.

Verification of the Halperin-Carlsson conjecture for moment-angle complexes.

Abstract

In this paper, we give an algebra-combinatorics formula of the M\"obius transform for an abstract simplicial complex $K$ on $[m]=\{1, ..., m\}$ in terms of the Betti numbers of the Stanley-Reisner face ring of $K$. Furthermore, we employ a way of compressing $K$ to estimate the lower bound of the sum of those Betti numbers by using this formula. As an application, associating with the moment-angle complex $\mathcal{Z}_K$ (resp. real moment-angle complex ${\Bbb R}\mathcal{Z}_K$) of $K$, we show that the Halperin-Carlsson conjecture holds for $\mathcal{Z}_K$ (resp. ${\Bbb R}\mathcal{Z}_K$) under the restriction of the natural $T^m$-action on $\mathcal{Z}_K$ (resp. $({\Bbb Z}_2)^m$-action on ${\Bbb R}\mathcal{Z}_K$).