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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.
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Cimasoni, David “Dimers on graphs in non-orientable surfaces", · 2009
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Acta Mathematica Hungarica, 137(4), 272-281, (2012)
Eccles, Peter & Mark Grant “Self-intersections of immersions and Steenrod operations", · 2012
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Gu, Zheng-Cheng & Xiao-Gang Wen · 2014
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“Spin TQFTs and fermionic phases of matter",
Gaiotto, Davide & Anton Kapustin, · 2016
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“Combinatorial Topology and Applications to Quantum Field Theory",
Thorngren, Ryan (2018), · 2018
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Algebr. Geom. Topol. 18 (2018) 1259-1279
Budney, Ryan “Combinatorial spin structures on triangulated manifolds", · 2018
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Tachikawa, Yuji & Kazuya Yonekura “Why are fractional charges of orientifolds compatible with Dirac quantization?", · 2019
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Combinatorial spin structures, URL (version: 2013-09-24):
MathOverflow Thread: ‘Combinatorial spin structures’, · 2013
Cited alongside, same era.
“Quadratic Functions of Cocycles and Pin Structures",
Brumfiel, Greg & John Morgan,
Cited in the paper.
Dover Books on Mathematics Series
Mosher, Robert & Martin Tangora “Cohomology operations and application in homotopy theory",
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In S. Donaldson & C. Thomas (Eds.), Geometry of Low-Dimensional Manifolds: Symplectic Manifolds and Jones-Witten Theory (London Mathematical Society Lecture Note Series, pp. 177-242)
Kirby, Robion & Laurence Taylor “Pin Strucrtures on Low-dimensional Manifolds",
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Journal of the London Mathematical Society, s2-22: 365-373
Johnson, Dennis “Spin Structures and Quadratic forms on Surfaces",
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