#### Volume 5, issue 3 (2005)

 Recent Issues
 The Journal About the Journal Editorial Board Subscriptions Editorial Interests Editorial Procedure Submission Guidelines Submission Page Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Author Index To Appear Other MSP Journals
Hopf algebra structure on topological Hochschild homology

### Vigleik Angeltveit and John Rognes

Algebraic & Geometric Topology 5 (2005) 1223–1290
 arXiv: math.AT/0502195
##### Abstract

The topological Hochschild homology $THH\left(R\right)$ 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 Bökstedt spectral sequence converging to the mod $p$ homology of $THH\left(R\right)$ 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 $THH\left(ku\right)$ and $THH\left(ko\right)$ 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, Bökstedt spectral sequence, Steenrod operations, Dyer–Lashof operations
##### Mathematical Subject Classification 2000
Primary: 55P43, 55S10, 55S12, 57T05
Secondary: 13D03, 55T15