2020/08/24 by Sri Tata, Tata, Sri · 1 citation
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Geometric Topology (math.GT) #High Energy Physics - Theory (hep-th) #Strongly Correlated Electrons (cond-mat.str-el) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2008.10170
openalex publication_date 2020/08/24 · openalex created_date 2020/09/01 · openalex updated_date 2026/07/28
We provide a geometric interpretation of the formulas for Steenrod's ∪i products, giving an explicit construction for a conjecture of Thorngren. We construct from a simplex and a branching structure a special frame of vector fields inside each simplex that allow us to interpret cochain-level formulas for the ∪i as a generalized intersection product on the dual cellular decomposition. It can be thought of as measuring the intersection between a collection of dual cells and thickened, shifted version of another collection, where the vector field frame determines the thickening and shifting. Defining this vector field frame in a neighborhood of the dual 1-skeleton of a simplicial complex allows us to combinatorially define Spin and Pin^± structures on triangulated manifolds. We use them to geometrically interpret the `Grassmann Integral' of Gu-Wen/Gaiotto-Kapustin, without using Grassmann variables. In particular, we find that the `quadratic refinement' property of Gaiotto-Kapustin can be derived geometrically using our vector fields and interpretation of ∪i, together with a certain trivalent resolution of the dual 1-skeleton. This lets us extend the scope of their function to arbitrary triangulations and explicitly see its connection to spin structures. Vandermonde matrices play a key role in all constructions.