$$E_n$$ E n -cell attachments and a local-to-global principle for homological stability

Kupers, Alexander; Miller, Jeremy

doi:10.1007/s00208-017-1533-3

Cited by 21 publications

(28 citation statements)

References 45 publications

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“…We will consider the θ-framed little discs operad, D θ n , whose k th space D θ n (k) = Emb θ ( k R n , R n ) is roughly the space of embeddings k R n → R n preserving θ-framing. For a precise definition, see Section 2 of [34]. Composition is by multicomposition of embeddings, and the unit is the identity map of R n .…”

Section: Operads and Monadsmentioning

confidence: 99%

“…. This is equivalent to the May model B(Σ n , D n , A); see Section 5 of [34] for more details about this model. Definition 2.9.…”

Section: The Two-sided Monadic Bar Constructionmentioning

confidence: 99%

“…Remark 2.11. In [34], factorization homology of a D θ n -algebra A in topological spaces is defined as…”

Section: The Two-sided Monadic Bar Constructionmentioning

confidence: 99%

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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: Operads and Monadsmentioning

confidence: 99%

“…. This is equivalent to the May model B(Σ n , D n , A); see Section 5 of [34] for more details about this model. Definition 2.9.…”

Section: The Two-sided Monadic Bar Constructionmentioning

confidence: 99%

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The factorization theory of Thom spectra and twisted nonabelian Poincaré duality

Klang

2018

Algebr. Geom. Topol.

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“…2. For general X, Theorem 1.6 gives a simultaneous generalization of stability for configuration spaces [Chu12], [RW13], [BM14], [Knu] (the pm, nq " p1, 2q case, proved only recently), for bounded symmetric powers [Vas92,KM] (the pm, nq " p1, nq, n ą 2 case), and for spaces of rational maps [Seg79] (the pm, nq " pm, 1q, m ě 2 case).…”

Section: Introductionmentioning

confidence: 99%

Coincidences between homological densities, predicted by arithmetic

Farb

Wolfson

Wood

2019

Advances in Mathematics

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Motivated by analogies with basic density theorems in analytic number theory, we introduce a notion (and variations) of the homological density of one space in another. We use Weil's number field/ function field analogy to predict coincidences for limiting homological densities of various sequences Z pd1,...,dmq n pXq of spaces of 0-cycles on manifolds X. The main theorem in this paper is that these topological predictions, which seem strange from a purely topological viewpoint, are indeed true.One obstacle to proving such a theorem is the combinatorial complexity of all possible "collisions" of points. This problem does not arise in the simplest (and classical) case pm, nq " p1, 2q of configuration spaces. To overcome this obstacle we apply the Björner-Wachs theory of lexicographic shellability from algebraic combinatorics.

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“…In this setting, however, one cannot construct free factorization algebras, due to, as we shall see, the lack of the proper-pushforward functor in general. Free factorization algebras are nonetheless one of the most basic objects in the topological setting, and they are used extensively, for example, in [Knu14,KM14].…”

mentioning

confidence: 99%

Free factorization algebras and homology of configuration spaces in algebraic geometry

2017

Sel. Math. New Ser.

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$$E_n$$ E n -cell attachments and a local-to-global principle for homological stability

Cited by 21 publications

References 45 publications

The factorization theory of Thom spectra and twisted nonabelian Poincaré duality

The factorization theory of Thom spectra and twisted nonabelian Poincaré duality

Coincidences between homological densities, predicted by arithmetic

Free factorization algebras and homology of configuration spaces in algebraic geometry

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