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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

    更新日期:2020-07-24
  • 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

    更新日期:2020-07-21
  • 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

    更新日期:2020-07-03
  • 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.

    更新日期:2020-06-23
  • 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.

    更新日期:2020-05-23
  • 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

    更新日期:2020-04-02
  • 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

    更新日期:2020-03-19
  • 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.

    更新日期:2020-03-12
  • 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

    更新日期:2020-03-11
  • 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:

    更新日期:2020-03-06
  • 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

    更新日期:2020-02-12
  • 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

    更新日期:2019-12-30
  • 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

    更新日期:2019-11-21
  • 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)]

    更新日期:2019-11-12
  • 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

    更新日期:2019-11-08
  • 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

    更新日期:2019-10-05
  • 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 \(\mathop {\mathop {\mathsf {Ex}}^\infty }\)

    更新日期:2019-09-24
  • 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.

    更新日期:2019-09-21
  • 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

    更新日期:2019-07-18
  • 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

    更新日期:2019-07-16
  • 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

    更新日期:2019-07-12
  • 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

    更新日期:2019-07-12
  • 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

    更新日期:2019-06-26
  • 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

    更新日期:2019-05-30
  • 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

    更新日期:2019-05-22
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