An axiomatic characterization of Steenrod's cup-$i$ products

Abstract

We introduce new formulae for the cup-i products on the cochains of spaces. We prove that any set of choices for the cup-i products is isomorphic to the one defined by our formulae if it is natural, minimal, non-degenerate, and free. We also show that Steenrod's original set of choices, as well as all of those induced from operadic and prop theoretic constructions known to the author satisfy these axioms.

Higher Steenrod squares for Khovanov homology

“…Acknowledgments. The author is especially grateful to Aníbal Medina-Mardones for the inspiration received while reading his paper [MM18]. He is also grateful to Fernando Muro and David Chataur, and to Javier Gutiérrez, Carles Casacuberta, Joana Cirici and Marithania Silvero from the Topology group at Barcelona.…”

“…The proof is given in the next section. Our presentation is parallel to the presentation of the symmetric multiplication on the cochain complex of a simplicial set given in [MM18].…”

“…which is the formula given in [MM18] to define the symmetric comultiplication on the chain complex of a simplicial set.…”

A Cochain Level Proof of Adem Relations in the Mod 2 Steenrod Algebra

“…In fact, the preferred choice of the ∪ i is Steenrod's original definition. Medina has emphasized the important properties of Steenrod's definition of the ∪ i , and has given axioms characterizing that choice, [20].…”

An effective proof of the Cartan formula: the even prime

“…enforcing its derived commutativity. (Axioms for these and connections with higher-dimensional category theory can be found in [MM18a] and [MM19]). Then, with coefficients in F 2 , Steenrod defined…”