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

Abstract

We show that any natural construction of cup-$i$ products on the normalized cochains of simplicial sets, parameterized by the minimal free resolution of the trivial ${\mathbb{F}}_2[{\mathbb{S}}_2]$-module, is isomorphic---not just homotopic---to Steenrod's original construction if it is non-zero, irreducible, and free. We show that these three conditions are independent, and use them to prove that all cup-$i$ constructions in the literature represent the same isomorphism class.

Add independency of axioms and semi-simplicial naturality

References in corpus (1)

• Quadratic Functions of Cocycles and Pin Structures

Cited by in corpus (5)

• New formulas for cup-$i$ products and fast computation of Steenrod squares
• A finitely presented $E_{\infty }$-prop II: cellular context
• Flowing from intersection product to cup product
• An algebraic representation of globular sets
• A finitely presented $E_{\infty }$-prop I: algebraic context