Geometry & Topology

Detecting periodic elements in higher topological Hochschild homology

Torleif Veen

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Abstract

Given a commutative ring spectrum R, let ΛXR be the Loday functor constructed by Brun, Carlson and Dundas. Given a prime p5, we calculate π(ΛSnHFp) and π(ΛTnHFp) for np, and use these results to deduce that vn1 in the (n1)st connective Morava K-theory of (ΛTnHFp)hTn is nonzero and detected in the homotopy fixed-point spectral sequence by an explicit element, whose class we name the Rognes class.

To facilitate these calculations, we introduce multifold Hopf algebras. Each axis circle in Tn gives rise to a Hopf algebra structure on π(ΛTnHFp), and the way these Hopf algebra structures interact is encoded with a multifold Hopf algebra structure. This structure puts several restrictions on the possible algebra structures on π(ΛTnHFp) and is a vital tool in the calculations above.

Article information

Source
Geom. Topol., Volume 22, Number 2 (2018), 693-756.

Dates
Received: 6 March 2014
Revised: 2 May 2017
Accepted: 4 June 2017
First available in Project Euclid: 1 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.gt/1517454107

Digital Object Identifier
doi:10.2140/gt.2018.22.693

Mathematical Reviews number (MathSciNet)
MR3748678

Zentralblatt MATH identifier
1384.55007

Subjects
Primary: 55P42: Stable homotopy theory, spectra 55P91: Equivariant homotopy theory [See also 19L47] 55T99: None of the above, but in this section

Keywords
THH K-theory spectral sequences Morava K-theory

Citation

Veen, Torleif. Detecting periodic elements in higher topological Hochschild homology. Geom. Topol. 22 (2018), no. 2, 693--756. doi:10.2140/gt.2018.22.693. https://projecteuclid.org/euclid.gt/1517454107


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