Algebraic and Geometric Topology 5 (2005), paper no. 49, pages 1223-1290.

Hopf algebra structure on topological Hochschild homology

Vigleik Angeltveit, John Rognes

Abstract. The topological Hochschild homology THH(R) of a commutative S-algebra (E_infty ring spectrum) R naturally has the structure of a commutative R-algebra in the strict sense, and of a Hopf algebra over R in the homotopy category. We show, under a flatness assumption, that this makes the Boekstedt spectral sequence converging to the mod p homology of THH(R) into a Hopf algebra spectral sequence. We then apply this additional structure to the study of some interesting examples, including the commutative S-algebras ku, ko, tmf, ju and j, and to calculate the homotopy groups of THH(ku) and THH(ko) after smashing with suitable finite complexes. This is part of a program to make systematic computations of the algebraic K-theory of S-algebras, by means of the cyclotomic trace map to topological cyclic homology.

Keywords. Topological Hochschild homology, commutative S-algebra, coproduct, Hopf algebra, topological K-theory, image-of-J spectrum, Boekstedt spectral sequence, Steenrod operations, Dyer-Lashof operations.

AMS subject classification. Primary: 55P43, 55S10, 55S12, 57T05. Secondary: 13D03, 55T15.

E-print: arXiv:math.AT/0502195

DOI: 10.2140/agt.2005.5.1223

Submitted: 16 July 2004. (Revised: 21 September 2005.) Accepted: 29 September 2005. Published: 5 October 2005.

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Vigleik Angeltveit, John Rognes
Department of Mathematics, Massachusetts Institute of Technology
Cambridge, MA 02139-4307, USA
and
Department of Mathematics, University of Oslo
Blindern NO-0316, Norway
Email: vigleik@math.mit.edu, rognes@math.uio.no

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