2012/12/17 by Nickolai Erokhovets, Erokhovets, Nickolai
Mathematics · #52Bxx #55Nxx #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #math.AT #math.CO #msc:52Bxx #msc:55Nxx
paper · pdf · doi:10.48550/arxiv.1212.3970
8 pages
arxiv created 2012/12/17 · arxiv updated 2012/12/18
In this paper we study the Buchstaber invariant of simplicial complexes, which comes from toric topology. With each simplicial complex K on m vertices we can associate a moment-angle complex \mathcal ZK with a canonical action of the compact torus Tm. Then s(K) is the maximal dimension of a toric subgroup that acts freely on \mathcal ZK. We develop the Buchstaber invariant theory from the viewpoint of the set of minimal non-simplices of K. It is easy to show that s(K)=1 if and only if any two and any three minimal non-simplices intersect. For K=∂ P^*, where P is a simple polytope, this implies that P is a simplex. The case s(P)=2 is such more complicated. For example, for any k\geqslant 2 there exists an n-polytope with n+k facets such that s(P)=2. Our main result is the criterion for the Buchstaber invariant of a simplicial complex K to be equal to two.