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Smooth functorial field theories from B-fields and D-branes J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2021-01-23 Severin Bunk, Konrad Waldorf
In the Lagrangian approach to 2-dimensional sigma models, B-fields and D-branes contribute topological terms to the action of worldsheets of both open and closed strings. We show that these terms naturally fit into a 2-dimensional, smooth open-closed functorial field theory (FFT) in the sense of Atiyah, Segal, and Stolz–Teichner. We give a detailed construction of this smooth FFT, based on the definition
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Homotopy types of gauge groups of $$\mathrm {PU}(p)$$ PU ( p ) -bundles over spheres J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2021-01-21 Simon Rea
We examine the relation between the gauge groups of \(\mathrm {SU}(n)\)- and \(\mathrm {PU}(n)\)-bundles over \(S^{2i}\), with \(2\le i\le n\), particularly when n is a prime. As special cases, for \(\mathrm {PU}(5)\)-bundles over \(S^4\), we show that there is a rational or p-local equivalence \(\mathcal {G}_{2,k}\simeq _{(p)}\mathcal {G}_{2,l}\) for any prime p if, and only if, \((120,k)=(120,l)\)
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The Segal conjecture for topological Hochschild homology of Ravenel spectra J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2021-01-19 Gabriel Angelini-Knoll, J. D. Quigley
In the 1980’s, Ravenel introduced sequences of spectra X(n) and T(n) which played an important role in the proof of the Nilpotence Theorem of Devinatz–Hopkins–Smith. In the present paper, we solve the homotopy limit problem for topological Hochschild homology of X(n), which is a generalized version of the Segal Conjecture for the cyclic groups of prime order. This result is the first step towards computing
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The Adams spectral sequence for 3-local $$\mathrm {tmf}$$ tmf J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2021-01-06 D. Culver
The purpose of this article is to record the computation of the homotopy groups of 3-local \(\mathrm {tmf}\) via the Adams spectral sequence.
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Groups up to congruence relation and from categorical groups to c-crossed modules J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2020-11-21 Tamar Datuashvili, Osman Mucuk, Tunçar Şahan
We introduce a notion of c-group, which is a group up to congruence relation and consider the corresponding category. Extensions, actions and crossed modules (c-crossed modules) are defined in this category and the semi-direct product is constructed. We prove that each categorical group gives rise to a c-group and to a c-crossed module, which is a connected, special and strict c-crossed module in the
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On the relative K -group in the ETNC, Part II J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2020-11-06 Oliver Braunling
In a previous paper we showed that, under some assumptions, the relative K-group in the Burns–Flach formulation of the equivariant Tamagawa number conjecture (ETNC) is canonically isomorphic to a K-group of locally compact equivariant modules. Our approach as well as the standard one both involve presentations: One due to Bass–Swan, applied to categories of finitely generated projective modules; and
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Homotopy Gerstenhaber algebras are strongly homotopy commutative J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2020-11-01 Matthias Franz
We show that any homotopy Gerstenhaber algebra is naturally a strongly homotopy commutative (shc) algebra in the sense of Stasheff–Halperin with a homotopy associative structure map. In the presence of certain additional operations corresponding to a \(\mathbin {\cup _1}\)-product on the bar construction, the structure map becomes homotopy commutative, so that one obtains an shc algebra in the sense
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Homotopic distance between functors J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2020-10-13 E. Macías-Virgós, D. Mosquera-Lois
We introduce a notion of categorical homotopic distance between functors by adapting the notion of homotopic distance in topological spaces, recently defined by the authors, to the context of small categories. Moreover, this notion generalizes the work on categorical LS-category of small categories by Tanaka.
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Cohomology and deformations of oriented dialgebras J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2020-09-16 Ali N. A. Koam, Ripan Saha
We introduce a notion of oriented dialgebra and develop a cohomology theory for oriented dialgebras by mixing the standard chain complexes computing group cohomology and associative dialgebra cohomology. We also introduce a formal deformation theory for oriented dialgebras and show that cohomology of oriented dialgebras controls such deformations.
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Note on Toda brackets J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2020-08-28 Samik Basu, David Blanc, Debasis Sen
We provide a general definition of Toda brackets in a pointed model category, show how they serve as obstructions to rectification, and explain their relation to the classical stable operations.
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Cyclic homology for bornological coarse spaces J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2020-07-24 Luigi Caputi
The goal of the paper is to define Hochschild and cyclic homology for bornological coarse spaces, i.e., lax symmetric monoidal functors \({{\,\mathrm{\mathcal {X}HH}\,}}_{}^G\) and \({{\,\mathrm{\mathcal {X}HC}\,}}_{}^G\) from the category \(G\mathbf {BornCoarse}\) of equivariant bornological coarse spaces to the cocomplete stable \(\infty \)-category \(\mathbf {Ch}_\infty \) of chain complexes reminiscent
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Bianchi’s additional symmetries J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2020-07-20 Alexander D. Rahm
In a 2012 note in Comptes Rendus Mathématique, the author did try to answer a question of Jean-Pierre Serre; it has recently been announced that the scope of that answer needs an adjustment, and the details of this adjustment are given in the present paper. The original question is the following. Consider the ring of integers \(\mathcal {O}\) in an imaginary quadratic number field, and the Borel–Serre
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Descent theory and mapping spaces J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2020-07-03 Nicholas J. Meadows
The purpose of this paper is to develop a theory of \((\infty , 1)\)-stacks, in the sense of Hirschowitz–Simpson’s ‘Descent Pour Les n–Champs’, using the language of quasi-category theory and the author’s local Joyal model structure. The main result is a characterization of \((\infty , 1)\)-stacks in terms of mapping space presheaves. An important special case of this theorem gives a sufficient condition
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Higher equivariant and invariant topological complexities J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2020-06-21 Marzieh Bayeh, Soumen Sarkar
In this paper we introduce concepts of higher equivariant and invariant topological complexities and study their properties. Then we compare them with equivariant LS-category. We give lower and upper bounds for these new invariants. We compute some of these invariants for moment angle complexes.
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Transfer ideals and torsion in the Morava E -theory of abelian groups J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2020-05-23 Tobias Barthel; Nathaniel Stapleton
Let A be a finite abelian p-group of rank at least 2. We show that \(E^0(BA)/I_{tr}\), the quotient of the Morava E-cohomology of A by the ideal generated by the image of the transfers along all proper subgroups, contains p-torsion. The proof makes use of transchromatic character theory.
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The universal fibration with fibre X in rational homotopy theory J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2020-04-02 Gregory Lupton; Samuel Bruce Smith
Let X be a simply connected space with finite-dimensional rational homotopy groups. Let \(p_\infty :UE \rightarrow B\mathrm {aut}_1(X)\) be the universal fibration of simply connected spaces with fibre X. We give a DG Lie algebra model for the evaluation map \( \omega :\mathrm {aut}_1(B\mathrm {aut}_1(X_\mathbb {Q})) \rightarrow B\mathrm {aut}_1(X_\mathbb {Q})\) expressed in terms of derivations of
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The unit of the total décalage adjunction J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2020-03-19 Viktoriya Ozornova; Martina Rovelli
We consider the décalage construction \({{\,\mathrm{Dec}\,}}\) and its right adjoint \(T\). These functors are induced on the category of simplicial objects valued in any bicomplete category \({\mathcal {C}}\) by the ordinal sum. We identify \(T{{\,\mathrm{Dec}\,}}X\) with the path object \(X^{\Delta [1]}\) for any simplicial object X. We then use this formula to produce an explicit retracting homotopy
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Verifying the Hilali conjecture up to formal dimension twenty J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2020-03-12 Spencer Cattalani; Aleksandar Milivojević
We prove that in formal dimension \(\le 20\) the Hilali conjecture holds, i.e. that the total dimension of the rational homology bounds from above the total dimension of the rational homotopy for a simply connected rationally elliptic space.
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An application of the h -principle to manifold calculus J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2020-03-11 Apurva Nakade
Manifold calculus is a form of functor calculus that analyzes contravariant functors from some categories of manifolds to topological spaces by providing analytic approximations to them. In this paper, using the technique of the h-principle, we show that for a symplectic manifold N, the analytic approximation to the Lagrangian embeddings functor \(\mathrm {Emb}_{\mathrm {Lag}}(-,N)\) is the totally
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Correction to: Representations are adjoint to endomorphisms J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2020-03-06 Gabriel C. Drummond-Cole, Joseph Hirsh, Damien Lejay
The first equation under section “Remark 3” was processed and published incorrectly. The correct equation should read as follows:
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On the capacity and depth of compact surfaces J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2020-02-12 Mahboubeh Abbasi; Behrooz Mashayekhy
K. Borsuk in 1979, at the Topological Conference in Moscow, introduced the concept of capacity and depth of a compactum. In this paper we compute the capacity and depth of compact surfaces. We show that the capacity and depth of every compact orientable surface of genus \(g\ge 0\) is equal to \(g+2\). Also, we prove that the capacity and depth of a compact non-orientable surface of genus \(g>0\) is
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Representations are adjoint to endomorphisms J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-12-30 Gabriel C. Drummond-Cole; Joseph Hirsh; Damien Lejay
The functor that takes a ring to its category of modules has an adjoint if one remembers the forgetful functor to abelian groups: the endomorphism ring of linear natural transformations. This uses the self-enrichment of the category of abelian groups. If one considers enrichments into symmetric sequences or even bisymmetric sequences, one can produce an endomorphism operad or an endomorphism properad
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Formulae in noncommutative Hodge theory J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-11-21 Nick Sheridan
We prove that the cyclic homology of a saturated \(A_\infty \) category admits the structure of a ‘polarized variation of Hodge structures’, building heavily on the work of many authors: the main point of the paper is to present complete proofs, and also explicit formulae for all of the relevant structures. This forms part of a project of Ganatra, Perutz and the author, to prove that homological mirror
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The depth of a Riemann surface and of a right-angled Artin group J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-11-12 Yves Félix; Steve Halperin
We consider two families of spaces, X: the closed orientable Riemann surfaces of genus \(g>0\) and the classifying spaces of right-angled Artin groups. In both cases we compare the depth of the fundamental group with the depth of an associated Lie algebra, L, that can be determined by the minimal Sullivan algebra. For these spaces we prove that$$\begin{aligned} \text{ depth } \,{\mathbb {Q}}[\pi _1(X)]
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Twisting structures and morphisms up to strong homotopy J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-11-08 Kathryn Hess; Paul-Eugène Parent; Jonathan Scott
We define twisted composition products of symmetric sequences via classifying morphisms rather than twisting cochains. Our approach allows us to establish an adjunction that simultaneously generalizes a classic one for algebras and coalgebras, and the bar-cobar adjunction for quadratic operads. The comonad associated to this adjunction turns out to be, in several cases, a standard Koszul construction
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Lie theory for symmetric Leibniz algebras J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-10-05 Mamuka Jibladze; Teimuraz Pirashvili
Lie algebras and groups equipped with a multiplication \(\mu \) satisfying some compatibility properties are studied. These structures are called symmetric Lie \(\mu \)-algebras and symmetric \(\mu \)-groups respectively. An equivalence of categories between symmetric Lie \(\mu \)-algebras and symmetric Leibniz algebras is established when 2 is invertible in the base ring. The second main result of
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Another approach to the Kan–Quillen model structure J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-09-24 Sean Moss
By careful analysis of the embedding of a simplicial set into its image under Kan’s \(\mathop {\mathop {\mathsf {Ex}}^\infty }\) functor we obtain a new and combinatorial proof that it is a weak homotopy equivalence. Moreover, we obtain a presentation of it as a strong anodyne extension. From this description we can quickly deduce some basic facts about \(\mathop {\mathop {\mathsf {Ex}}^\infty }\)
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Correction to: Wrong way maps in uniformly finite homology and homology of groups J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-09-21 Alexander Engel
There is an error in the proof of Theorem 2.16 of Ref. 2. It occured at the end of the second-to-last paragraph of the proof.
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Parallel transport of higher flat gerbes as an extended homotopy quantum field theory J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-07-18 Lukas Müller; Lukas Woike
We prove that the parallel transport of a flat \(n-1\)-gerbe on any given target space gives rise to an n-dimensional extended homotopy quantum field theory. In case the target space is the classifying space of a finite group, we provide explicit formulae for this homotopy quantum field theory in terms of transgression. Moreover, we use the geometric theory of orbifolds to give a dimension-independent
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Enhanced $$A_{\infty }$$A∞ -obstruction theory J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-07-16 Fernando Muro
An \(A_n\)-algebra \(A= (A,m_1, m_2, \ldots , m_n)\) is a special kind of \(A_\infty \)-algebra satisfying the \(A_\infty \)-relations involving just the \(m_i\) listed. We consider obstructions to extending an \(A_{n-1}\) algebra to an \(A_n\)-algebra. We enhance the known techniques by extending the Bousfield–Kan spectral sequence to apply to the homotopy groups of the space of minimal (i.e. \(m_1=0)\)\(A_\infty
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On the cohomology ring and upper characteristic rank of Grassmannian of oriented 3-planes J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-07-12 Somnath Basu; Prateep Chakraborty
In this paper we study the mod 2 cohomology ring of the Grasmannian \(\widetilde{G}_{n,3}\) of oriented 3-planes in \({\mathbb {R}}^n\). We determine the degrees of the indecomposable elements in the cohomology ring. We also obtain an almost complete description of the cohomology ring. This description allows us to provide lower and upper bounds on the cup length of \(\widetilde{G}_{n,3}\). As another
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Weight decompositions of Thom spaces of vector bundles in rational homotopy J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-07-12 Urtzi Buijs; Federico Cantero Morán; Joana Cirici
Motivated by the theory of representability classes by submanifolds, we study the rational homotopy theory of Thom spaces of vector bundles. We first give a Thom isomorphism at the level of rational homotopy, extending work of Félix-Oprea-Tanré by removing hypothesis of nilpotency of the base and orientability of the bundle. Then, we use the theory of weight decompositions in rational homotopy to give
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A model structure via orbit spaces for equivariant homotopy J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-06-26 Mehmet Akif Erdal; Aslı Güçlükan İlhan
Let G be discrete group and \(\mathcal F\) be a collection of subgroups of G. We show that there exists a left induced model structure on the category of right G-simplicial sets, in which the weak equivalences and cofibrations are the maps that induce weak equivalences and cofibrations on H-orbits for all H in \(\mathcal F\). This gives a model categorical criterion for maps that induce weak equivalences
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Cohomology of infinite groups realizing fusion systems J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-06-07 Muhammed Said Gündoğan; Ergün Yalçın
Given a fusion system \({\mathcal {F}}\) defined on a p-group S, there exist infinite group models, constructed by Leary and Stancu, and Robinson, that realize \({\mathcal {F}}\). We study these models when \({\mathcal {F}}\) is a fusion system of a finite group G and prove a theorem which relates the cohomology of an infinite group model \(\pi \) to the cohomology of the group G. We show that for
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Dense products in fundamental groupoids J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-06-03 Jeremy Brazas
Infinitary operations, such as products indexed by countably infinite linear orders, arise naturally in the context of fundamental groups and groupoids. Despite the fact that the usual binary operation of the fundamental group determines the operation of the fundamental groupoid, we show that, for a locally path-connected metric space, the well-definedness of countable dense products in the fundamental
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Minimality in diagrams of simplicial sets J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-05-30 Carles Broto; Ramón Flores; Carlos Giraldo
We formulate the concept of minimal fibration in the context of fibrations in the model category \({\mathbf {S}}^{\mathcal {C}}\) of \({\mathcal {C}}\)-diagrams of simplicial sets, for a small index category \({\mathcal {C}}\). When \({\mathcal {C}}\) is an EI-category satisfying some mild finiteness restrictions, we show that every fibration of \({\mathcal {C}}\)-diagrams admits a well-behaved minimal
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Hearts and towers in stable $$\infty $$ ∞ -categories J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-05-22 Domenico Fiorenza; Fosco Loregian; Giovanni Luca Marchetti
We exploit the equivalence between t-structures and normal torsion theories on a stable \(\infty \)-category to show how a few classical topics in the theory of triangulated categories, i.e., the characterization of bounded t-structures in terms of their hearts, their associated cohomology functors, semiorthogonal decompositions, and the theory of tiltings, as well as the more recent notion of Bridgeland’s
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Comonad cohomology of track categories J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-05-14 David Blanc; Simona Paoli
We define a comonad cohomology of track categories, and show that it is related via a long exact sequence to the corresponding \(({\mathcal {S}}\!,\!\mathcal {O})\)-cohomology. Under mild hypotheses, the comonad cohomology coincides, up to reindexing, with the \(({\mathcal {S}}\!,\!\mathcal {O})\)-cohomology, yielding an algebraic description of the latter. We also specialize to the case where the
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Involutions on surfaces J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-05-14 Daniel Dugger
We use equivariant surgery to classify all involutions on closed surfaces, up to isomorphism. Work on this problem is classical, dating back to the nineteenth century, with a complete classification finally appearing in the 1990s. In this paper we give a different approach to the classification, using techniques that are more accessible to algebraic topologists as well as a new invariant (which we
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Matrix factorizations for quantum complete intersections J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-03-28 Petter Andreas Bergh; Karin Erdmann
We introduce twisted matrix factorizations for quantum complete intersections of codimension two. For such an algebra, we show that in a given dimension, almost all the indecomposable modules with bounded minimal projective resolutions correspond to such factorizations.
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Characteristic classes as complete obstructions J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-03-13 Martina Rovelli
In the first part of this paper, we propose a uniform interpretation of characteristic classes as obstructions to the reduction of the structure group and to the existence of an equivariant extension of a certain homomorphism defined a priori only on a single fiber of the bundle. Afterwards, we define a family of invariants of principal bundles that detect the number of group reductions that a principal
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Homotopy types of SU ( n )-gauge groups over non-spin 4-manifolds J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-03-12 Tseleung So
Let M be an orientable, simply-connected, closed, non-spin 4-manifold and let \({\mathcal {G}}_k(M)\) be the gauge group of the principal G-bundle over M with second Chern class \(k\in {\mathbb {Z}}\). It is known that the homotopy type of \({\mathcal {G}}_k(M)\) is determined by the homotopy type of \({\mathcal {G}}_k({\mathbb {C}}{\mathbb {P}}^2)\). In this paper we investigate properties of \({\mathcal
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Some characterizations of acyclic maps J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-03-04 George Raptis
We discuss two categorical characterizations of the class of acyclic maps between spaces. The first one is in terms of the higher categorical notion of an epimorphism. The second one employs the notion of a balanced map, that is, a map whose homotopy pullbacks along \(\pi _0\)-surjective maps define also homotopy pushouts. We also identify the modality in the homotopy theory of spaces that is defined
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Tate cohomology of connected k-theory for elementary abelian groups revisited J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-01-10 Po Hu; Igor Kriz; Petr Somberg
Tate cohomology (as well as Borel homology and cohomology) of connective K-theory for \(G=({\mathbb {Z}}/2)^n\) was completely calculated by Bruner and Greenlees (The connective K-theory of finite groups, 2003). In this note, we essentially redo the calculation by a different, more elementary method, and we extend it to \(p>2\) prime. We also identify the resulting spectra, which are products of Eilenberg–Mac
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Algebraic Hopf invariants and rational models for mapping spaces J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2019-01-03 Felix Wierstra
The main goal of this paper is to define an invariant \(mc_{\infty }(f)\) of homotopy classes of maps \(f:X \rightarrow Y_{\mathbb {Q}}\), from a finite CW-complex X to a rational space \(Y_{\mathbb {Q}}\). We prove that this invariant is complete, i.e. \(mc_{\infty }(f)=mc_{\infty }(g)\) if and only if f and g are homotopic. To construct this invariant we also construct a homotopy Lie algebra structure
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Computations of orbits for the Lubin–Tate ring J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-12-18 Agnès Beaudry; Naiche Downey; Connor McCranie; Luke Meszar; Andy Riddle; Peter Rock
We take a direct approach to computing the orbits for the action of the automorphism group \(\mathbb {G}_2\) of the Honda formal group law of height 2 on the associated Lubin–Tate rings \(R_2\). We prove that \((R_2/p)_{\mathbb {G}_2} \cong \mathbb {F}_p\). The result is new for \(p=2\) and \(p=3\). For primes \(p\ge 5\), the result is a consequence of computations of Shimomura and Yabe and has been
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A multiplicative K -theoretic model of Voevodsky’s motivic K -theory spectrum J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-11-28 Youngsoo Kim
Voevodsky defined a motivic spectrum representing algebraic K-theory, and Panin, Pimenov, and Röndigs described its ring structure up to homotopy. We construct a motivic symmetric spectrum with a strict ring structure. Then we show that these spectra are stably equivalent and that their ring structures are compatible up to homotopy.
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A comonadic interpretation of Baues–Ellis homology of crossed modules J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-11-27 Guram Donadze; Tim Van der Linden
We introduce and study a homology theory of crossed modules with coefficients in an abelian crossed module. We discuss the basic properties of these new homology groups and give some applications. We then restrict our attention to the case of integral coefficients. In this case we regain the homology of crossed modules originally defined by Baues and further developed by Ellis. We show that it is an
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Equivariant chromatic localizations and commutativity J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-11-27 Michael A. Hill
In this paper, we study the extent to which Bousfield and finite localizations relative to a thick subcategory of equivariant finite spectra preserve various kinds of highly structured multiplications. Along the way, we describe some basic, useful results for analyzing categories of acyclics in equivariant spectra, and we show that Bousfield localization with respect to an ordinary spectrum (viewed
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Yoga of commutators in DSER elementary orthogonal group J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-11-15 A. A. Ambily
In this article, we consider the Dickson-Siegel-Eichler-Roy (DSER) elementary orthogonal subgroup of the orthogonal group of a non-degenerate quadratic space with a hyperbolic summand over a commutative ring, introduced by Roy. We prove a set of commutator relations among the elementary generators of the DSER elementary orthogonal group. As an application, we prove that this group is perfect and an
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A theorem on multiplicative cell attachments with an application to Ravenel’s X ( n ) spectra J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-11-15 Jonathan Beardsley
We show that the homotopy groups of a connective \(\mathbb {E}_k\)-ring spectrum with an \(\mathbb {E}_k\)-cell attached along a class \(\alpha \) in degree n are isomorphic to the homotopy groups of the cofiber of the self-map associated to \(\alpha \) through degree 2n. Using this, we prove that the \(2n-1\)st homotopy groups of Ravenel’s X(n) spectra are cyclic for all n. This further implies that
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The Dold–Thom theorem via factorization homology J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-11-10 Lauren Bandklayder
We give a new proof of the classical Dold–Thom theorem using factorization homology. Our method is direct and conceptual, avoiding the Eilenberg–Steenrod axioms entirely in favor of a more general geometric argument.
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Isotropic reductive groups over discrete Hodge algebras J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-11-10 Anastasia Stavrova
Let G be a reductive group over a commutative ring R. We say that G has isotropic rank \(\ge n\), if every normal semisimple reductive R-subgroup of G contains \(({{\mathrm{{{\mathbf {G}}}_m}}}_{,R})^n\). We prove that if G has isotropic rank \(\ge 1\) and R is a regular domain containing an infinite field k, then for any discrete Hodge algebra \(A=R[x_1,\ldots ,x_n]/I\) over R, the map \(H^1_{\mathrm
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Stabilization of derivators revisited J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-11-10 Ian Coley
We revisit and improve Alex Heller’s results on the stabilization of derivators in Heller (J Pure Appl Algebra 115(2):113–130, 1997), recovering his results entirely. Along the way we give some details of the localization theory of derivators and prove some new results in that vein.
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Koszuality of the $$\mathcal V^{(d)}$$ V ( d ) dioperad J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-10-30 Kate Poirier; Thomas Tradler
Define a \(\mathcal V^{(d)}\)-algebra as an associative algebra with a symmetric and invariant co-inner product of degree d. Here, we consider \(\mathcal V^{(d)}\) as a dioperad which includes operations with zero inputs. We show that the quadratic dual of \(\mathcal V^{(d)}\) is \((\mathcal V^{(d)})^!=\mathcal V^{(-d)}\) and prove that \(\mathcal V^{(d)}\) is Koszul. We also show that the corresponding
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The Koszul–Tate type resolution for Gerstenhaber–Batalin–Vilkovisky algebras J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-10-25 Jeehoon Park; Donggeon Yhee
Tate provided an explicit way to kill a nontrivial homology class of a commutative differential graded algebra over a commutative noetherian ring R in Tate (Ill J Math 1:14–27, 1957). The goal of this article is to generalize his result to the case of GBV (Gerstenhaber–Batalin–Vilkovisky) algebras and, more generally, the descendant \(L_\infty \)-algebras. More precisely, for a given GBV algebra \((\mathcal
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Delooping derived mapping spaces of bimodules over an operad J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-10-17 Julien Ducoulombier
To any topological operad O, we introduce a cofibrant replacement in the category of bimodules over itself such that for every map \(\eta :O\rightarrow O'\) of operads, the corresponding model \({\textit{Bimod}}_{O}^{h}(O\,;\,O')\) of derived mapping space of bimodules is an algebra over the one dimensional little cubes operad \(\mathcal {C}_{1}\). We also build an explicit weak equivalence of \(\mathcal
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A combinatorial model for the path fibration J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-09-29 Manuel Rivera; Samson Saneblidze
We introduce the abstract notion of a necklical set in order to describe a functorial combinatorial model of the path fibration over the geometric realization of a path connected simplicial set. In particular, to any path connected simplicial set X we associate a necklical set \({\widehat{{\varvec{\Omega }}}}X\) such that its geometric realization \(|{\widehat{{\varvec{\Omega }}}}X|\), a space built
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Unstable splittings in Hodge filtered Brown–Peterson cohomology J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-09-27 Gereon Quick
We construct Hodge filtered function spaces associated to infinite loop spaces. For Brown–Peterson cohomology, we show that the corresponding Hodge filtered spaces satisfy an analog of Wilson’s unstable splitting. As a consequence, we obtain an analog of Quillen’s theorem for Hodge filtered Brown–Peterson cohomology for complex manifolds.
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Topology of scrambled simplices J. Homotopy Relat. Struct. (IF 0.537) Pub Date : 2018-09-27 Dmitry N. Kozlov
In this paper we define a family of topological spaces, which contains and vastly generalizes the higher-dimensional Dunce hats. Our definition is purely combinatorial, and is phrased in terms of identifications of boundary simplices of a standard d-simplex. By virtue of the construction, the obtained spaces may be indexed by words, and they automatically carry the structure of a \(\Delta \)-complex