Periodicity and the values of the real Buchstaber invariants

Abstract

The Buchstaber invariant s(K) is defined to be the maximum integer for which there is a subtorus of dimension s(K) acting freely on the moment-angle complex associated with a finite simplicial complex K. Analogously, its real version s_ℝ(K) can also be defined by using the real moment-angle complex instead of the moment-angle complex. The importance of these invariants comes from the fact that s(K) and s_ℝ(K) distinguish two simplicial complexes and are the source of nontrivial and interesting combinatorial tasks. The ultimate goal of this paper is to compute the real Buchstaber invariants of skeleta K = Δ_m−p−1^m−1 of the simplex Δ^m−1 by making a formula. In fact, it can be solved by integer linear programming. We also give a counterexample to the conjecture which is proposed in [6] and we provide an adjusted formula which can be thought of as a preperiodicity of some numbers related to the real Buchstaber invariants.