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

In this paper, we give an algebra-combinatorics formula of the M"obius transform for an abstract simplicial complex $K$ on $[m]=\{1,\mathrm{.}\mathrm{.}\mathrm{.},m\}$ 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. ${\Bbb R}$\mathcal{Z}$_K$)undertherestrictionofthenatural$T^m$-actionon$$\mathcal{Z}$_K$(resp\mathrm{.}$({\Bbb Z}_2)^m$-actionon${\Bbb R}$\mathcal{Z}$_K$).