A finitely presented $E_{\infty }$-prop II: cellular context

Abstract

We construct, using finitely many generating cell and relations, props in the category of CW-complexes with the property that their associated operads are models for the $E_{\infty }$-operad. We use one of these to construct a cellular $E_{\infty }$-bialgebra structure on the interval and derive from it a natural cellular $E_{\infty }$-coalgebra structure on the geometric realization of a simplicial set which, passing to cellular chains, recovers up to signs the Barratt-Eccles and Surjection coalgebra structures introduced by Berger-Fresse and McClure-Smith. We use another prop, a quotient of the first, to relate our constructions to earlier work of Kaufmann and prove a conjecture of his. This is the second of two papers in a series, the first investigates analogue constructions in the category of chain complexes.

Version after referee revisions

References in corpus (2)

• An axiomatic characterization of Steenrod's cup-$i$ products
• Noncommutative aspects of open/closed strings via foliations

Cited by in corpus (3)

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