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 $[m]=\{1, ..., m\}$ 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 ${\Bbb R}\mathcal{Z}_K$).