Higher topological Hochschild homology of periodic complex K-theory

2017/12/30 by Stonek, Bruno · 1 citation
#55N15 #55P20 (Secondary) #55P43 (Primary) #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.1801.00156

Abstract

We describe the topological Hochschild homology of the periodic complex K-theory spectrum, THH(KU), as a commutative KU-algebra: it is equivalent to KU[K(ℤ,3)] and to F(ΣKU_ℚ), where F is the free commutative KU-algebra functor on a KU-module. Moreover, F(ΣKU_ℚ)≃ KU \vee ΣKU_ℚ, a square-zero extension. In order to prove these results, we first establish that topological Hochschild homology commutes, as an algebra, with localization at an element. Then, we prove that THHⁿ(KU), the n-fold iteration of THH(KU), i.e. Tⁿ⊗ KU, is equivalent to KU[G] where G is a certain product of integral Eilenberg-Mac Lane spaces, and to a free commutative KU-algebra on a rational KU-module. We prove that Sⁿ ⊗ KU is equivalent to KU[K(ℤ,n+2)] and to F(Σⁿ KU_ℚ). We describe the topological André-Quillen homology of KU.

Cited by

• On the Residue Sequence in Logarithmic Topological Cyclic Homology

Related

• A computation of THH_*(ku) using a gathered spectral sequence
• Integral topological Hochschild homology of connective complex K-theory
• Topological Hochschild homology of l and ko
• Bass' NK groups and cdh-fibrant Hochschild homology
• Topological Hochschild homology of the image of j