Geometry & Topology

Homology operations in the topological cyclic homology of a point

Håkon Schad Bergsaker and John Rognes

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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 SG of the sphere spectrum and the algebraic K–theory spectrum of finite G–sets, as structured ring spectra.

Article information

Geom. Topol., Volume 14, Number 2 (2010), 755-772.

Received: 18 November 2008
Revised: 11 December 2009
Accepted: 6 December 2009
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55S12: Dyer-Lashof operations 55P43: Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.)
Secondary: 19D55: $K$-theory and homology; cyclic homology and cohomology [See also 18G60] 55P92: Relations between equivariant and nonequivariant homotopy theory 19D10: Algebraic $K$-theory of spaces

topological cyclic homology homology operation algebraic $K$–theory


Bergsaker, Håkon Schad; Rognes, John. Homology operations in the topological cyclic homology of a point. Geom. Topol. 14 (2010), no. 2, 755--772. doi:10.2140/gt.2010.14.755.

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