Hopf algebra structure on topological Hochschild homology

By Vigleik Angeltveit, John Rognes. 2005

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The topological Hochschild homology THH(R) of a commu- tative S-algebra (E1 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 Bokstedt 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 computa- tions of the algebraic K-theory of S-algebras, by means of the cyclotomic trace map to topological cyclic homology. AMS Classification 55P43, 55S10, 55S12, 57T05; 13D03, 55T15

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

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