#### Volume 14, issue 2 (2010)

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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
##### Abstract

We consider the commutative $\mathbb{S}$–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 ${\mathbb{S}}^{G}$ 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
##### 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