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

The motivation for this paper comes from the Halperin–Carlsson conjecture for (real) moment-angle complexes. We first give an algebraic combinatorics formula for the Möbius transform of an abstract simplicial complex K on [m] = {1, . . . ,m} in terms of the Betti numbers of the Stanley–Reisner face ring k(K) of K over a field k. We then employ a way of compressing K to provide the lower bound on the sum of those Betti numbers using our formula. Next we consider a class of generalized moment-angle complexes Z K , including the moment-angle complex ZK and the real moment-angle complex RZK as special examples. We show that H ∗(Z K ;k) has the same graded k-module structure as Tork[v](k(K),k). Finally we show that the Halperin–Carlsson conjecture holds for ZK (resp. RZK ) under the restriction of the natural T -action on ZK (resp. (Z2)-action on RZK ).