Geometry & Topology

Homology operations in the topological cyclic homology of a point

Håkon Schad Bergsaker and John Rognes

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

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732205

Digital Object Identifier
doi:10.2140/gt.2010.14.755

Mathematical Reviews number (MathSciNet)
MR2602850

Zentralblatt MATH identifier
1204.55014

Subjects
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

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

Citation

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. https://projecteuclid.org/euclid.gt/1513732205


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