Topological Hochschild homology of X(n)

Beardsley, Jonathan

doi:10.48550/arxiv.1708.09486

Cited by 1 publication

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“…The Lie group SU (n) has trivial tangent bundle, so this is equivalent to Σ −d SU (n) + ∧ X(n), where d = dim SU (n). Notice that this is indeed a shift, or twist by a trivial tangent bundle, of T HH(X(n)); as in [8], T HH(X(n)) X(n) ∧ SU (n) + .…”

Section: Then Our New Model Formentioning

confidence: 97%

The factorization theory of Thom spectra and twisted nonabelian Poincaré duality

Klang

2018

Algebr. Geom. Topol.

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We give a description of the factorization homology and En topological Hochschild cohomology of Thom spectra arising from n-fold loop maps f : A → BO, where A = Ω n X is an n-fold loop space. We describe the factorization homology M T h(f ) as the Thom spectrum associated to a certain map M A → BO, where M A is the factorization homology of M with coefficients in A. When M is framed and X is (n − 1)-connected, this spectrum is equivalent to a Thom spectrum of a virtual bundle over the mapping space Map c (M, X); in general, this is a Thom spectrum of a virtual bundle over a certain section space. This can be viewed as a twisted form of the non-abelian Poincaré duality theorem of Segal, Salvatore, and Lurie, which occurs when f : A → BO is nullhomotopic. This result also generalizes the results of Blumberg-Cohen-Schlichtkrull on the topological Hochschild homology of Thom spectra, and of Schlichtkrull on higher topological Hochschild homology of Thom spectra. We use this description of the factorization homology of Thom spectra to calculate the factorization homology of the classical cobordism spectra, spectra arising from systems of groups, and the Eilenberg-MacLane spectra HZ/p, HZ (p) , and HZ. We build upon the description of the factorization homology of Thom spectra to study the (n = 1 and higher) topological Hochschild cohomology of Thom spectra, which enables calculations and a description in terms of sections of a parametrized spectrum. If X is a closed manifold, Atiyah duality for parametrized spectra allows us to deduce a duality between En topological Hochschild homology and En topological Hochschild cohomology, recovering string topology operations when f is nullhomotopic. In conjunction with the higher Deligne conjecture, this gives En+1 structures on a certain family of Thom spectra, which were not previously known to be ring spectra.

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Section: Then Our New Model Formentioning

confidence: 97%

The factorization theory of Thom spectra and twisted nonabelian Poincaré duality

Klang

2018

Algebr. Geom. Topol.

View full text Add to dashboard Cite

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Topological Hochschild homology of X(n)

Cited by 1 publication

References 3 publications

The factorization theory of Thom spectra and twisted nonabelian Poincaré duality

The factorization theory of Thom spectra and twisted nonabelian Poincaré duality

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