Topological cyclic homology (TC)

Topological cyclic homology (TC)
Name	Topological cyclic homology
Field	Algebraic topology
Introduced	1990s

Topological cyclic homology (TC) is an invariant in algebraic topology connected to Algebraic K-theory and homotopy theory, developed to provide calculable approximations to Quillen, Waldhausen, and Thomason style K-theory constructions. It arose from interactions among researchers at institutions such as Princeton University, Massachusetts Institute of Technology, University of Chicago, and Max Planck Institute for Mathematics and has been shaped by work of figures including Waldhausen, Bökstedt, Hsiang, Madsen, Goodwillie, and Dundas. TC connects to objects studied at Institute for Advanced Study, Harvard University, Stanford University, and in collaborations involving Simons Foundation programs.

Introduction

Topological cyclic homology was introduced in response to computational challenges in Algebraic K-theory and combines tools from Stable homotopy theory, Equivariant homotopy theory, and Cyclic homology. Key milestones include the Bökstedt construction, the development of cyclotomic spectra by researchers at University of Copenhagen and contributions by teams at University of Oxford and École Normale Supérieure. TC serves as a bridge between classical invariants studied by Connes and modern approaches championed by Lurie and Voevodsky.

Definitions and construction

The construction of TC begins with the formulation of Topological Hochschild homology (THH) for a ring spectrum or structured ring object associated to contexts at Princeton University Press–style seminars. THH is defined via bar constructions and geometric realizations used in Bökstedt’s original work and later formalized in frameworks developed by Elmendorf, Mandell, and May. From THH one builds cyclotomic spectra equipped with Frobenius and restriction maps inspired by ideas from Frobenius endomorphism studies in Weil conjectures-era algebraic geometry. The cyclotomic structure incorporates actions of finite cyclic groups like C_n and uses fixed-point and Tate constructions seen in work at Brown University and Columbia University. TC is obtained by assembling fixed points across these maps, organizing data into homotopy limits reminiscent of techniques used by Dundas, Goodwillie, and McCarthy.

Properties and computational tools

Topological cyclic homology inherits structural features from stable categories studied at Princeton University and categorical frameworks advanced by Grothendieck-inspired schools. It admits spectral sequences analogous to the Hesselholt–Madsen spectral sequences and interacts with trace methods originating in research by Bökstedt, Hsiang, and Madsen. Tools include the cyclotomic trace map, Tate spectral sequences, and descent sequences used in calculations at Imperial College London and University of Cambridge. Comparisons with Topological André–Quillen homology and techniques from Morava K-theory have been developed in collaborations involving University of California, Berkeley and Institut des Hautes Études Scientifiques. Computational frameworks often invoke equivariant homotopy fixed-point spectral sequences and models from Model category theory advanced by Quillen.

Relationships with algebraic K-theory

The cyclotomic trace map from Algebraic K-theory to TC, elaborated by researchers at Max Planck Institute for Mathematics and Columbia University, provides a central link that enabled breakthrough calculations in cases studied by Hesselholt and Madsen. The Dundas–Goodwillie–McCarthy theorem compares relative K-theory and TC in excisive situations, a result originated in circles involving University of Chicago and Northwestern University. These relationships have been deployed in work on the K-theory of local fields, schemes considered in seminars at University of Michigan and University of Illinois Urbana–Champaign, and in projects related to the Beilinson conjectures and methods influenced by Beilinson and Bloch.

Examples and calculations

Classical calculations include TC of discrete valuation rings, finite fields studied by teams at Copenhagen Business School-adjacent groups, and truncated polynomial algebras computed in collaborations involving Technische Universität Berlin and University of Göttingen. Results for perfect fields, p-adic rings, and schemes such as those analyzed in seminars at École Polytechnique feature in the literature of Hesselholt, Madsen, and later contributors like Bhatt, Morrow, and Scholze. Computational case studies often exploit comparisons with crystalline cohomology techniques associated to work at IHÉS and ideas from Fontaine-style p-adic Hodge theory.

Variants and generalizations

Generalizations of TC include periodic and completed variants studied in projects at Massachusetts Institute of Technology and Harvard University, and refinements such as logarithmic TC and relative TC developed by researchers affiliated with University of California, San Diego and Yale University. Cyclotomic spectra have been recast in higher-categorical languages by groups around Lurie and used in synthetic approaches promoted at Institute for Advanced Study. Other extensions interact with motivic homotopy theory investigated by teams at University of Copenhagen and Princeton University.

Applications in arithmetic and geometry

Topological cyclic homology has been applied to problems in arithmetic geometry, including calculations relevant to the K-theory of schemes considered by Grothendieck-inspired seminars, investigations into local-global principles explored at University of Oxford, and contributions to understanding special values of L-functions in programs associated with Institute for Advanced Study initiatives. It has informed work on p-adic Hodge theory connected to Fontaine and Fargues research, and fed into progress on conjectures influenced by Beilinson, Bloch, and Kato through collaborations spanning European Research Council-funded groups and national institutes.

Category:Algebraic topology