Geometrically Interpreting Higher Cup Products, and Application to Combinatorial Pin Structures

Abstract

We provide a geometric interpretation of the formulas for Steenrod's ${\cup }_i$ 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 ${\cup }_i$ 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 $Pi{n}^{\pm }$ 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 ${\cup }_i$, 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.

49 pages, 18 figures

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Cited by in corpus (6)

• Classification of (2+1)D invertible fermionic topological phases with symmetry
• Codimension-2 defects and higher symmetries in (3+1)D topological phases
• Higher cup products on hypercubic lattices: application to lattice models of topological phases
• Symmetry-preserving boundary of (2+1)D fractional quantum Hall states
• Lattice construction of exotic invertible topological phases
• Flowing from intersection product to cup product