Algebraic & Geometric Topology

Cyclotomic structure in the topological Hochschild homology of $DX$

Cary Malkiewich

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Let X be a finite CW complex, and let DX be its dual in the category of spectra. We demonstrate that the Poincaré/Koszul duality between THH(DX) and the free loop space Σ+LX is in fact a genuinely S1–equivariant duality that preserves the Cn–fixed points. Our proof uses an elementary but surprisingly useful rigidity theorem for the geometric fixed point functor ΦG of orthogonal G–spectra.

Article information

Algebr. Geom. Topol., Volume 17, Number 4 (2017), 2307-2356.

Received: 17 May 2016
Revised: 21 January 2017
Accepted: 16 February 2017
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 19D55: $K$-theory and homology; cyclic homology and cohomology [See also 18G60] 55P43: Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.)
Secondary: 55P25: Spanier-Whitehead duality 55P91: Equivariant homotopy theory [See also 19L47]

topological Hochschild homology cyclotomic spectra multiplicative norm geometric fixed points of orthogonal spectra


Malkiewich, Cary. Cyclotomic structure in the topological Hochschild homology of $DX$. Algebr. Geom. Topol. 17 (2017), no. 4, 2307--2356. doi:10.2140/agt.2017.17.2307.

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  • V Angeltveit, A,J Blumberg, T Gerhardt, M,A Hill, T Lawson, Interpreting the Bökstedt smash product as the norm, Proc. Amer. Math. Soc. 144 (2016) 5419–5433
  • V Angeltveit, A Blumberg, T Gerhardt, M Hill, T Lawson, M Mandell, Topological cyclic homology via the norm, preprint (2014)
  • G Arone, A generalization of Snaith-type filtration, Trans. Amer. Math. Soc. 351 (1999) 1123–1150
  • D Ayala, J Francis, Poincaré/Koszul duality, preprint (2014)
  • C Barwick, S Glasman, Cyclonic spectra, cyclotomic spectra, and a conjecture of Kaledin, preprint (2016)
  • C Berger, I Moerdijk, On an extension of the notion of Reedy category, Math. Z. 269 (2011) 977–1004
  • A,J Blumberg, R,L Cohen, C Schlichtkrull, Topological Hochschild homology of Thom spectra and the free loop space, preprint (2008)
  • A,J Blumberg, M,A Mandell, The homotopy theory of cyclotomic spectra, preprint (2013)
  • A,M Bohmann, A comparison of norm maps, Proc. Amer. Math. Soc. 142 (2014) 1413–1423
  • M B ökstedt, Topological Hochschild homology, preprint, Bielefeld (1985)
  • M Bökstedt, W,C Hsiang, I Madsen, The cyclotomic trace and algebraic $K$–theory of spaces, Invent. Math. 111 (1993) 465–539
  • M Brun, B,I Dundas, M Stolz, Equivariant structure on smash powers, preprint (2016)
  • J,A Campbell, Derived Koszul duality and topological Hochschild homology, preprint (2014)
  • R,L Cohen, J,D,S Jones, A homotopy theoretic realization of string topology, Math. Ann. 324 (2002) 773–798
  • A Connes, Cohomologie cyclique et foncteurs ${\rm Ext}\sp n$, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983) 953–958
  • W,G Dwyer, M,J Hopkins, D,M Kan, The homotopy theory of cyclic sets, Trans. Amer. Math. Soc. 291 (1985) 281–289 \goodbreak
  • 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
  • J,D,S Jones, Cyclic homology and equivariant homology, Invent. Math. 87 (1987) 403–423
  • D Kaledin, Motivic structures in non-commutative geometry, from “Proceedings of the International Congress of Mathematicians, II” (R Bhatia, A Pal, G Rangarajan, V Srinivas, M Vanninathan, editors), Hindustan Book Agency, New Delhi (2010) 461–496
  • N,J Kuhn, The McCord model for the tensor product of a space and a commutative ring spectrum, from “Categorical decomposition techniques in algebraic topology” (G Arone, J Hubbuck, R Levi, M Weiss, editors), Progr. Math. 215, Birkhäuser, Basel (2004) 213–236
  • L,G,J Lewis, The stable category and generalized Thom spectra, PhD thesis, The University of Chicago (1978) Available at \setbox0\makeatletter\@url {\unhbox0
  • L,G Lewis, Jr, J,P May, M Steinberger, J,E McClure, Equivariant stable homotopy theory, Lecture Notes in Mathematics 1213, Springer (1986)
  • I Madsen, Algebraic $K$–theory and traces, from “Current developments in mathematics, 1995” (R Bott, M Hopkins, A Jaffe, I Singer, D Stroock, S-T Yau, editors), Int. Press, Cambridge, MA (1994) 191–321
  • C Malkiewich, Duality and linear approximations in Hochschild homology, $K$–theory, and string topology, PhD thesis, Stanford University (2014) Available at \setbox0\makeatletter\@url {\unhbox0
  • C Malkiewich, A tower connecting gauge groups to string topology, J. Topol. 8 (2015) 529–570
  • 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
  • R Schwänzl, R,M Vogt, Strong cofibrations and fibrations in enriched categories, Arch. Math. $($Basel$)$ 79 (2002) 449–462
  • M Stolz, Equivariant structure on smash powers of commutative ring spectra, PhD thesis, University of Bergen (2011) Available at \setbox0\makeatletter\@url {\unhbox0