2023/04/17 by Steven Amelotte, Amelotte, Steven, Benjamin Briggs +1 · 1 citation
Mathematics · #13F55 #55U10 #57S12 #Advanced Combinatorial Mathematics #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2304.08476
openalex publication_date 2023/04/17 · openalex created_date 2023/04/20 · openalex updated_date 2026/07/28
A fundamental result in toric topology identifies the cohomology ring of the moment-angle complex ZK associated to a simplicial complex K with the Koszul homology of the Stanley--Reisner ring of K. By studying cohomology operations induced by the standard torus action on the moment-angle complex, we extend this to a topological interpretation of the minimal free resolution of the Stanley-Reisner ring. The exterior algebra module structure in cohomology induced by the torus action recovers the linear part of the minimal free resolution, and we show that higher cohomology operations induced by the action (in the sense of Goresky-Kottwitz-MacPherson) can be assembled into an explicit differential on the resolution. Describing these operations in terms of Hochster's formula, we recover and extend a result due to Katthän. We then apply all of this to study the equivariant formality of torus actions on moment-angle complexes. For these spaces, we obtain complete algebraic and combinatorial characterisations of which subtori of the naturally acting torus act equivariantly formally.