An effective proof of the Cartan formula: the even prime

2019/07/28 by Medina-Mardones, Anibal M.
#55S10 (Primary) 55S05 #55S12 (Secondary) #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.1907.12113

Abstract

The Cartan formula encodes the relationship between the cup product and the action of the Steenrod algebra in \mathbb F_p-cohomology. In this work, we present an effective proof of the Cartan formula at the cochain level when the field is \mathbb F₂. More explicitly, for an arbitrary pair of cocycles and any non-negative integer, we construct a natural coboundary that descends to the associated instance of the Cartan formula. Our construction works for general algebras over the Barratt-Eccles operad, in particular, for the singular cochains of spaces.

Related

• An effective proof of the Cartan formula: Odd primes
• A Cartan formula for the cohomology of polyhedral products and its application to the ring structure
• A Cochain Level Proof of Adem Relations in the Mod 2 Steenrod Algebra
• Adem-Cartan operads
• Cup product of inhomogeneous Tate cochains, and application to tori over local fields that split over cyclic extensions