Homology operations in the topological cyclic homology of a point

Bergsaker, Håkon Schad; Rognes, John

doi:10.2140/gt.2010.14.755

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Homology operations in the topological cyclic homology of a point

Håkon Schad Bergsaker and John Rognes

Geometry & Topology 14 (2010) 755–772

DOI: 10.2140/gt.2010.14.755

Abstract

We consider the commutative $S$ –algebra given by the topological cyclic homology of a point. The induced Dyer–Lashof operations in mod $p$ homology are shown to be nontrivial for $p = 2$ , and an explicit formula is given. As a part of the calculation, we are led to compare the fixed point spectrum $S^{G}$ of the sphere spectrum and the algebraic $K$ –theory spectrum of finite $G$ –sets, as structured ring spectra.

Keywords

topological cyclic homology, homology operation, algebraic $K$–theory

Mathematical Subject Classification 2010

Primary: 55S12, 55P43

Secondary: 19D55, 55P92, 19D10

References

Publication

Received: 18 November 2008

Revised: 11 December 2009

Accepted: 6 December 2009

Published: 19 February 2010

Proposed: Ralph Cohen

Seconded: Haynes Miller, Paul Goerss

Authors

Håkon Schad Bergsaker
Department of Mathematics University of Oslo Oslo Norway

John Rognes
Department of Mathematics University of Oslo Oslo Norway

Volume 14, issue 2 (2010)