Möbius transform, moment-angle complexes and Halperin-Carlsson conjecture

Abstract

In this paper, we give an algebra-combinatorics formula of the Möbius 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 $\mathbb{R}{\mathcal{Z}}_K$) of $K$, we show that the Halperin-Carlsson conjecture holds for ${\mathcal{Z}}_K$ (resp. $\mathbb{R}{\mathcal{Z}}_K$) under the restriction of the natural $T^m$-action on ${\mathcal{Z}}_K$ (resp. $({\mathbb{Z}}_2{)}^m$-action on $\mathbb{R}{\mathcal{Z}}_K$).

16 pages. Section 4 of v1 is expanded significatively in v2

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