This article was accepted into the corpus but its outbound wikilinks were never NER-processed — typical at the deepest BFS hop or when the run's entity cap was reached. No expansion funnel to show.
| Topological Hochschild homology | |
|---|---|
| Name | Topological Hochschild homology |
| Field | Algebraic topology; Homotopy theory; Algebraic K-theory |
| Introduced | 1980s |
| Contributors | Bökstedt; Goodwillie; Hesselholt; Madsen; McCarthy; Schwede; Shipley |
Topological Hochschild homology is a homotopy-theoretic invariant that refines classical Hochschild homology by encoding multiplicative and higher coherence data of ring spectra in a topological setting. It plays a central role in modern computations in algebraic K-theory, trace methods, and equivariant stable homotopy theory, and interacts deeply with ∞-category approaches pioneered by authors associated with the Institute for Advanced Study, Max Planck Institute for Mathematics, and departments at University of Chicago and Massachusetts Institute of Technology.
Topological Hochschild homology is defined for associative or E_n ring spectra arising in contexts studied by Bökstedt and developed by researchers associated with Goodwillie, Madsen, Hesselholt, and McCarthy. Its core properties include a circle action used by authors at Stanford University and Princeton University to define topological cyclic homology, and homotopy invariance under Morita-type equivalences as in work by Schwede and Shipley. It exhibits multiplicative structures captured by operads studied at Institut des Hautes Études Scientifiques and behaves compatibly with base change phenomena analyzed by groups at University of Oxford and École Normale Supérieure.
Constructions of Topological Hochschild homology use several models: the cyclic bar construction adapted to symmetric spectra developed by researchers at University of Bonn and University of Illinois, Bökstedt's original point-set model associated with University of Copenhagen, and ∞-categorical formulations using stable ∞-categories promoted by groups at Harvard University and Columbia University. Model-categorical approaches employ symmetric monoidal model structures studied by teams at University of California, Berkeley and University of Edinburgh, while equivariant refinements invoke genuine equivariant spectra techniques advanced at Rutgers University and University of California, Los Angeles.
Topological Hochschild homology categorifies classical Hochschild homology appearing in algebraic contexts handled by scholars at University of Cambridge and Yale University, and it maps to Hochschild homology via spectral algebra comparisons used in seminars at Northwestern University and University of Michigan. The cyclotomic trace connects Topological Hochschild homology to topological cyclic homology, a bridge developed in collaborations involving Goodwillie, Hesselholt, and Madsen, and studied alongside cyclic homology frameworks associated with research groups at University of Bonn and Brown University.
Computations of Topological Hochschild homology for familiar inputs draw on methods from equivariant homotopy theory used at University of Toronto and cohomology operations techniques from Princeton University. Notable examples include calculations for the sphere spectrum investigated by teams at University of Chicago and Stanford University, for truncated polynomial algebras pursued at University of California, San Diego and University of Pennsylvania, and for rings of integers in number fields treated by collaborations including Max Planck Institute for Mathematics and University of Cambridge. Computational tools include spectral sequences popularized in lectures at ETH Zurich and Adams-type approaches from Johns Hopkins University.
The cyclotomic trace from algebraic K-theory to topological cyclic homology, central to work by Goodwillie, Hesselholt, and Madsen, leverages Topological Hochschild homology as an intermediate invariant and has driven advances in computations of algebraic K-theory for schemes studied at Cornell University and University of Warwick. These trace methods underpin recent progress on conjectures associated with researchers at Institute for Advanced Study and have informed approaches to descent and localization problems investigated at University of Oxford and Massachusetts Institute of Technology.
Variants include relative Topological Hochschild homology developed in collaborations linked to University of California, Santa Barbara and University of Notre Dame, TR and TC theories refined by groups at Technical University of Munich and University of Copenhagen, and noncommutative or spectral generalizations studied within the ∞-categorical frameworks advanced at Institut des Hautes Études Scientifiques and Harvard University. Higher versions interacting with E_n-algebras connect to work at Kavli Institute for Theoretical Physics and categorical traces examined by researchers at Max Planck Institute for Mathematics.
Applications of Topological Hochschild homology appear in questions about structured ring spectra arising in algebraic geometry programs at Princeton University and University of Cambridge, in the study of manifold invariants pursued at University of Bonn and University of California, Berkeley, and in string topology contexts explored by groups at Yale University and Columbia University. It also informs computations in chromatic homotopy theory developed at Northwestern University and has influenced arithmetic geometry projects affiliated with Max Planck Institute for Mathematics and University of Chicago.