Commutative free loop space models at large primes
type: Item
instance of: scholarly article
title: Commutative free loop space models at large primes
zbMATH Open document ID: 1028.55006
author: Nicolas Dupont
author: Kathryn Hess
published in: Mathematische Zeitschrift
publication date: 2003-07-01T00:00:00Z
review text: Free loop spaces have gained considerable interest among topologists in the last years, especially in view of their applications to string theory, geometry and non-linear analysis. There are by now fairly well understood techniques for calculating the homology of a ``free loop space'' (or the space of all continuous maps from the circle into a target space that is assumed to be simply connected unless otherwise stated). With general commutative ring coefficients, the standard approach is through the Hochschild complex. For rational calculations, one might instead use the pretty optimal model constructed by D. Sullivan and M. Vigué. When the target space is a suspension, there are in addition combinatorial models for the free loop space that give better conceptual understanding of its homology. Both authors here have addressed in a series of papers the problem of constructing an algebraic model for the free loop space over some field of prime characteristic. These models are sufficiently precise to determine the cohomology algebra structure as well. The present paper seems like a culmination of their previous techniques and gives their simplest and most explicit model to date for computing the free loop space cohomology of a class of spaces known as \(p\)-Anick spaces (\(p\) a prime). These consist in particular of \(r\)-connected finite CW complexes of dimension at most \(pr\) (like spheres, projective spaces and wedges or products of those). The main theorem under review exhibits an explicit algebraic model for the free loop space cohomology, coefficients in \(F_p\), \(p\) prime larger than 2, of a \(p\)-Anick space \(E\), expressed in terms of an (enriched) Adams-Hilton model for \(E\). This data consists (roughly) of a homotopy cocommutative and coassociative Hopf algebra weakly equivalent to the chains on the based Moore loops of \(E\). Their main model gives an analog of the Sullivan-Vigué model in characteristic zero. The subject is quite technical in nature a…
reviewed by: Sadok Kallel
Mathematics Subject Classification ID: 55N10
Mathematics Subject Classification ID: 55P35
zbMATH DE Number: 1944230
zbMATH Keywords: free loop spaces
zbMATH Keywords: algebraic models
zbMATH Keywords: Adams-Hilton models
zbMATH Keywords: Anick spaces
MaRDI profile type: Publication
full work available at URL: s00209-003-0206-y
OpenAlex ID: W2035344448
DOI: 10.1007/S00209-003-0206-Y
description: scientific article; zbMATH DE number 1944230