New formulas for cup-$i$ products and fast computation of Steenrod squares

Abstract

Operations on the cohomology of spaces are important tools enhancing the descriptive power of this computable invariant. For cohomology with mod 2 coefficients, Steenrod squares are the most significant of these operations. Their effective computation relies on formulas defining a cup-$i$ construction, a structure on (co)chains which is important in its own right, having connections to lattice field theory, convex geometry and higher category theory among others. In this article we present new formulas defining a cup-$i$ construction, and use them to introduce a fast algorithm for the computation of Steenrod squares on the cohomology of finite simplicial complexes. In forthcoming work we use these formulas to axiomatically characterize the cup-$i$ construction they define, showing additionally that all other formulas in the literature define the same cup-$i$ construction up to isomorphism.

arXiv commentPublished in Comput. Geom

References in corpus (11)

• Ripser: efficient computation of Vietoris-Rips persistence barcodes
• Pore-geometry recognition: on the importance of quantifying similarity in nanoporous materials
• Fermionic SPT phases in higher dimensions and bosonization
• Combinatorial operad actions on cochains
• State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter
• Classification of (2+1)D invertible fermionic topological phases with symmetry
• giotto-tda: A Topological Data Analysis Toolkit for Machine Learning and Data Exploration
• A Combinatorial Method for Computing Steenrod Squares
• Cochain level May-Steenrod operations
• An axiomatic characterization of Steenrod's cup-$i$ products
• A finitely presented $E_{\infty }$-prop II: cellular context

Cited by in corpus (1)

• Steenrod operations via higher Bruhat orders