Topological Hochschild Homology and Beyond

University Of Chicago, Chicago IL

Investigators

Abstract

Topological Hochschild homology and cohomology are spectrum-level versions of Hochschild homology and cohomology, and come up in a variety of subjects including the calculation of algebraic K-theory, universal deformations, obstruction theory and string topology. Given a ring or S-algebra A, the topological Hochschild homology spectrum THH(A) carries a circle action. The topological cyclic homology spectrum TC(A) is defined in terms of the fixed points of THH(A) under finite subgroups of the circle, and there is a map from the algebraic K-theory spectrum K(A) to TC(A) which is often close to an equivalence. We are especially interested in trying to understand K(ku) using this approach. Here ku is the connective spectrum for complex K-theory, and the spectrum K(ku) models algebraic K-theory of the category of 2-vector bundles. There is a corresponding spectrum K(ko) for real 2-vector bundles, and we can ask if the canonical map from K(ko) to K(ku) is close to being a Galois extension. This is a higher chromatic analogue of the Lichtenbaum-Quillen conjecture. We also want to take the S^1-action on THH(A) more seriously and study the RO(S^1)-graded homotopy groups of the various fixed points of THH(A). These groups enter in the calculation of algebraic K-theory of truncated polynomial rings. A good understanding of the de Rham-Witt complex has been very useful in calculations, and we would like to be able to define the RO(S^1)-graded version of the de Rham-Witt complex. In algebraic topology there is a world of ``brave new rings'', technically known as S-algebras, which combines the worlds of algebra and topology. In this world we can use techniques from topology, a branch of geometry, and from algebra. Topological Hochschild homology is the generalization of Hochschild homology to this setting. Hochschild homology is a standard construction in algebra which has proven to be very useful, and topological Hochschild homology has already been used to solve some previously unsolved questions in algebra. For example, given a ring A, the algebraic K-theory K(A) captures much of the arithmetic and number theory of A. The Hochschild homology HH(A) gives a (poor) approximation to K(A), while the topological Hochschild homology THH(A) together with some extra structure gives a much better approximation to K(A). Hence the study of topological Hochschild homology ultimately gives insight into arithmetic and number theory, as well as geometry and topology. The project studies several aspects of topological Hochschild homology, some foundational and some calculational. We also go further, studying generalizations of topological Hochshcild homology and its relative, topological Hochschild cohomology. These generalizations have no algebraic analogue.