2019/12/25 by Anton Ayzenberg, Ayzenberg, Anton, Mikiya Masuda +1 · 1 citation
Mathematics · #06A07 (Primary) 55R20 #06A11 (Secondary) #18G10 #18G35 #55N25 #55N30 #55N91 #57N65 #57S25 #Advanced Combinatorial Mathematics #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT)
paper · pdf · doi:10.48550/arxiv.1912.11696
openalex publication_date 2019/12/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let a compact torus T=Tn-1 act on an orientable smooth compact manifold X=X2n effectively, with nonempty finite set of fixed points, and suppose that stabilizers of all points are connected. If Hodd(X)=0 and the weights of tangent representation at each fixed point are in general position, we prove that the orbit space Q=X/T is a homology (n+1)-sphere. If, in addition, π1(X)=0, then Q is homeomorphic to Sn+1. We introduce the notion of j-generality of tangent weights of torus action. For any action of Tk on X2n with isolated fixed points and Hodd(X)=0, we prove that j-generality of weights implies (j+1)-acyclicity of the orbit space Q. This statement generalizes several known results for actions of complexity zero and one. In complexity one, we give a criterion of equivariant formality in terms of the orbit space. In this case, we give a formula expressing Betti numbers of a manifold in terms of certain combinatorial structure that sits in the orbit space.