Commutative free loop space models at large primes
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 and so is the paper. As is usually the case when working with tensor algebras, computations can be forbiddingly difficult. In the case of formal \(p\)-Anick spaces, the authors have a smaller model that allows explicit calculations for spaces such as for example spheres and projective spaces. They recover in particular the known result that the free loop space cohomology mod-\(p\) of the complex \((p-1)\)-projective space \(P\) splits as a tensor product of its based loop cohomology and the cohomology of \(P\).
scientific article; zbMATH DE number 1944230