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We provide a geometric interpretation of the formulas for Steenrod's $\cup_i$products, giving an explicit construction for a conjecture of Thorngren. Weconstruct from a simplex and a branching structure a special frame of vectorfields inside each simplex that allow us to interpret cochain-level formulasfor the $\cup_i$ as a generalized intersection product on the dual cellulardecomposition. It can be thought of as measuring the intersection between acollection of dual cells and thickened, shifted version of another collection,where the vector field frame determines the thickening and shifting. Definingthis vector field frame in a neighborhood of the dual 1-skeleton of asimplicial complex allows us to combinatorially define $Spin$ and $Pin^\pm$structures on triangulated manifolds. We use them to geometrically interpretthe `Grassmann Integral' of Gu-Wen/Gaiotto-Kapustin, without using Grassmannvariables. In particular, we find that the `quadratic refinement' property ofGaiotto-Kapustin can be derived geometrically using our vector fields andinterpretation of $\cup_i$, together with a certain trivalent resolution of thedual 1-skeleton. This lets us extend the scope of their function to arbitrarytriangulations and explicitly see its connection to spin structures.Vandermonde matrices play a key role in all constructions.

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