Geometry & Topology

Detecting periodic elements in higher topological Hochschild homology

Torleif Veen

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

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

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

THH K-theory spectral sequences Morava K-theory


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.

Export citation


  • V Angeltveit, M A Hill, T Lawson, Topological Hochschild homology of $\ell$ and $\mathit{ko}$, Amer. J. Math. 132 (2010) 297–330
  • V Angeltveit, J Rognes, Hopf algebra structure on topological Hochschild homology, Algebr. Geom. Topol. 5 (2005) 1223–1290
  • C Ausoni, Topological Hochschild homology of connective complex $K$–theory, Amer. J. Math. 127 (2005) 1261–1313
  • C Ausoni, J Rognes, Algebraic $K$–theory of topological $K$–theory, Acta Math. 188 (2002) 1–39
  • C Ausoni, J Rognes, The chromatic red-shift in algebraic $K$–theory, Enseign. Math. 54 (2008) 13–15
  • N A Baas, B I Dundas, J Rognes, Two-vector bundles and forms of elliptic cohomology, from “Topology, geometry and quantum field theory” (U Tillmann, editor), London Math. Soc. Lecture Note Ser. 308, Cambridge Univ. Press (2004) 18–45
  • J M Boardman, Conditionally convergent spectral sequences, from “Homotopy invariant algebraic structures” (J-P Meyer, J Morava, W S Wilson, editors), Contemp. Math. 239, Amer. Math. Soc., Providence, RI (1999) 49–84
  • M Bökstedt, Topological Hochschild homology of $\mathbb{F}_p$ and $\mathbb{Z}$, preprint (1986)
  • M Bökstedt, W C Hsiang, I Madsen, The cyclotomic trace and algebraic $K$–theory of spaces, Invent. Math. 111 (1993) 465–539
  • M Brun, G Carlsson, B I Dundas, Covering homology, Adv. Math. 225 (2010) 3166–3213
  • M Brun, B I Dundas, M Stolz, Equivariant structure on smash powers, preprint (2016)
  • R R Bruner, J Rognes, Differentials in the homological homotopy fixed point spectral sequence, Algebr. Geom. Topol. 5 (2005) 653–690
  • G Carlsson, C L Douglas, B I Dundas, Higher topological cyclic homology and the Segal conjecture for tori, Adv. Math. 226 (2011) 1823–1874
  • H Cartan, Algèbres d'Eilenberg–Mac Lane et homotopie, 2nd edition, Séminaire Henri Cartan de l'Ecole Normale Supérieure 7, Secrétariat math., Paris (1956)
  • B I Dundas, T G Goodwillie, R McCarthy, The local structure of algebraic K-theory, Algebra and Applications 18, Springer (2013)
  • A D Elmendorf, I Kriz, M A Mandell, J P May, Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs 47, Amer. Math. Soc., Providence, RI (1997)
  • L Hesselholt, On the $p$–typical curves in Quillen's $K$–theory, Acta Math. 177 (1996) 1–53
  • L Hesselholt, I Madsen, On the $K$–theory of finite algebras over Witt vectors of perfect fields, Topology 36 (1997) 29–101
  • M A Hill, M J Hopkins, D C Ravenel, On the nonexistence of elements of Kervaire invariant one, Ann. of Math. 184 (2016) 1–262
  • T J Hunter, On the homology spectral sequence for topological Hochschild homology, Trans. Amer. Math. Soc. 348 (1996) 3941–3953
  • D C Johnson, W S Wilson, $BP$ operations and Morava's extraordinary $K$–theories, Math. Z. 144 (1975) 55–75
  • M A Mandell, J P May, Equivariant orthogonal spectra and $S$–modules, Mem. Amer. Math. Soc. 755, Amer. Math. Soc., Providence, RI (2002)
  • M A Mandell, J P May, S Schwede, B Shipley, Model categories of diagram spectra, Proc. London Math. Soc. 82 (2001) 441–512
  • J McCleary, A user's guide to spectral sequences, 2nd edition, Cambridge Studies in Advanced Mathematics 58, Cambridge Univ. Press (2001)
  • J McClure, R Schwänzl, R Vogt, $\mathit{THH}(R)\cong R\otimes S^1$ for $E_\infty$ ring spectra, J. Pure Appl. Algebra 121 (1997) 137–159
  • J E McClure, R E Staffeldt, On the topological Hochschild homology of $b{\rm u}$, I, Amer. J. Math. 115 (1993) 1–45
  • J Milnor, The Steenrod algebra and its dual, Ann. of Math. 67 (1958) 150–171
  • J W Milnor, J C Moore, On the structure of Hopf algebras, Ann. of Math. 81 (1965) 211–264
  • J Rognes, Algebraic K-theory of strict ring spectra, preprint (2014)
  • B Shipley, Symmetric spectra and topological Hochschild homology, $K$-Theory 19 (2000) 155–183
  • M Stolz, Equivariant structures on smash powers of commutative ring spectra, PhD thesis, University of Bergen (2011) Available at \setbox0\makeatletter\@url {\unhbox0