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  • Mapping algebras and the Adams spectral sequence
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-10-15
    David Blanc; Surojit Ghosh

    For a suitable ring spectrum, such as $\mathbf{E}=\mathbf{H}\mathbb{F}_p$, the $E_2$-term of the $\mathbf{E}$-based Adams spectral sequence for a spectrum $\mathbf{Y}$ may be described in terms of its cohomology $E^{\ast}\mathbf{Y}$, together with the action of the primary operations $E^{\ast}\mathbf{E}$ on it. We show how the higher terms of the spectral sequence can be similarly described in terms

    更新日期:2020-10-16
  • The trace of the local $\mathbb{A}^1$-degree
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-10-14
    Thomas Brazelton; Robert Burklund; Stephen McKean; Michael Montoro; Morgan Opie

    We prove that the local $\mathbb{A}^1$-degree of a polynomial function at an isolated zero with finite separable residue field is given by the trace of the local $\mathbb{A}^1$-degree over the residue field. This fact was originally suggested by Morel’s work on motivic transfers, and by Kass and Wickelgren’s work on the Scheja–Storch bilinear form. As a corollary, we generalize a result of Kass and

    更新日期:2020-10-16
  • $\mathbb{A}^1$-homotopy equivalences and a theorem of Whitehead
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-10-14
    Eoin Mackall

    We prove analogs of Whitehead’s theorem (from algebraic topology) for both the Chow groups and for the Grothendieck group of coherent sheaves: a morphism between smooth projective varieties whose pushforward is an isomorphism on the Chow groups, or on the Grothendieck group of coherent sheaves, is an isomorphism. As a corollary, we show that there are no nontrivial naive $\mathbb{A}^1$-homotopy equivalences

    更新日期:2020-10-16
  • Vector bundles and cohomotopies of $\operatorname{spin} 5$-manifolds
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-09-02
    Panagiotis Konstantis

    The purpose of this paper is two-fold: On the one side we would like to fill a gap on the classification of vector bundles over $5$‑manifolds. Therefore it will be necessary to study quaternionic line bundles over $5$‑manifolds which are in $\textrm{1-1}$ correspondence to elements in the cohomotopy group $\pi^4(M) = [M,S^4]$ of $M$. From results in [22, 24] this group fits into a short exact sequence

    更新日期:2020-09-03
  • Truncated derived functors and spectral sequences
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-09-02
    Hans-Joachim Baues; David Blanc; Boris Chorny

    The $E_2$-term of the Adams spectral sequence may be identified with certain derived functors, and this also holds for a number of other spectral sequences. Our goal is to show how the higher terms of such spectral sequences are determined by truncations of relative derived functors, defined in terms of certain simplicial functors called mapping algebras.

    更新日期:2020-09-03
  • Flatness and Shipley’s algebraicization theorem
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-09-02
    Jordan Williamson

    We provide an enhancement of Shipley’s algebraicization theorem which behaves better in the context of commutative algebras. This involves defining flat model structures as in Shipley and Pavlov–Scholbach, and showing that the functors still provide Quillen equivalences in this refined context. The use of flat model structures allows one to identify the algebraic counterparts of change of groups functors

    更新日期:2020-09-03
  • The wedge family of the cohomology of the $\mathbb{C}$-motivic Steenrod algebra
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-08-26
    Hieu Thai

    We describe some regular behavior in the motivic wedge, which is an infinite family in the cohomology $\mathrm{Ext}_{\mathbf{A}}(\mathbb{M}_2,\mathbb{M}_2)$ of the $\mathbb{C}$-motivic Steenrod algebra. The key tool is to compare motivic computations to classical computations, to $\mathrm{Ext}_{\mathbf{A}(2)}(\mathbb{M}_2,\mathbb{M}_2)$, or to $h_1$-localization of $\mathrm{Ext}_{\mathbf{A}}(\mathbb{M}_2

    更新日期:2020-08-26
  • Unstable algebras over an operad
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-08-26
    Sacha Ikonicoff

    The aim of this article is to define and study a notion of unstable algebra over an operad that generalises the classical notion of unstable algebra over the Steenrod algebra. For this study we focus on the case of characteristic $2$. We define $\star$-unstable $\mathcal{P}$-algebras, where $\mathcal{P}$ is an operad and $\star$ is a commutative binary operation in $\mathcal{P}$. We then build a functor

    更新日期:2020-08-26
  • Gradient versus proper gradient homotopies
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-08-19
    Piotr Bartłomiejczyk; Piotr Nowak-Przygodzki

    We compare the sets of homotopy classes of gradient and proper gradient vector fields in the plane. Namely, we show that gradient and proper gradient homotopy classifications are essentially different. We provide a complete description of the sets of homotopy classes of gradient maps from $\mathbb{R}^n$ to $\mathbb{R}^n$ and proper gradient maps from $\mathbb{R}^2$ to $\mathbb{R}^2$ with the Brouwer

    更新日期:2020-08-20
  • Biased permutative equivariant categories
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-08-19
    Kayleigh Bangs; Skye Binegar; Young Kim; Kyle Ormsby; Angélica M. Osorno; David Tamas-Parris; Livia Xu

    For a finite group $G$, we introduce the complete suboperad $\mathcal{Q}_G$ of the categorical $G$-Barratt–Eccles operad $\mathcal{P}_G$. We prove that $\mathcal{P}_G$ is not finitely generated, but $\mathcal{Q}_G$ is finitely generated and is a genuine $E_\infty$ $G$-operad (i.e., it is $N_\infty$ and includes all norms). For $G$ cyclic of order $2$ or $3$, we determine presentations of the object

    更新日期:2020-08-20
  • (Co)homology self-closeness numbers of simply-connected spaces
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-08-05
    Pengcheng Li

    The (co)homology self-closeness number of a simply-connected based CW-complex $X$ is the minimal number $k$ such that any self-map $f$ of $X$ inducing an automorphism of the (co)homology groups for dimensions $\leqslant k$ is a self-homotopy equivalence. These two numbers are homotopy invariants and have a close relation with the group of self-homotopy equivalences. In this paper, we compare the (co)homology

    更新日期:2020-08-06
  • Cohomology of the classifying spaces of $U(n)$-gauge groups over the 2-sphere
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-08-05
    Masahiro Takeda

    A gauge group is the topological group of automorphisms of a principal bundle. We compute the integral cohomology ring of the classifying spaces of gauge groups of principal $U(n)$-bundles over the $2$-sphere by generalizing the operation for free loop spaces, called the free double suspension.

    更新日期:2020-08-06
  • The equivariant fundamental groupoid as an orbifold invariant
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-08-05
    Dorette Pronk; Laura Scull

    We construct a $2$-category version of Tom Dieck’s equivariant fundamental groupoid for representable orbifolds and show that the discrete fundamental groupoid is Morita invariant; hence it is an orbifold invariant for representable orbifolds.

    更新日期:2020-08-06
  • A simplicial construction for noncommutative settings
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-08-05
    Samuel Carolus; Jacob Laubacher; Mihai D. Staic

    In this paper we present a general construction that can be used to define the higher order Hochschild homology for a noncommutative algebra. We also discuss other examples where this construction can be used.

    更新日期:2020-08-06
  • Braided categorical groups and strictifying associators
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-05-13
    Oliver Braunling

    A key invariant of a braided categorical group is its quadratic form, introduced by Joyal and Street. We show that the categorical group is braided equivalent to a simultaneously skeletal and strictly associative one if and only if the quadratic form comes from a bilinear form. This generalizes the result of Johnson–Osorno that all Picard groupoids can simultaneously be strictified and skeletalized

    更新日期:2020-07-20
  • Galois theory and the categorical Peiffer commutator
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-05-13
    Alan S. Cigoli; Arnaud Duvieusart; Marino Gran; Sandra Mantovani

    We show that the Peiffer commutator previously defined by Cigoli, Mantovani and Metere can be used to characterize central extensions of precrossed modules with respect to the subcategory of crossed modules in any semi-abelian category satisfying an additional property. We prove that this commutator also characterizes double central extensions, obtaining then some Hopf formulas for the second and third

    更新日期:2020-07-20
  • Crossed modules in the category of Loday QD-Rinehart algebras
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-05-20
    J. M. Casas; S. Çetin; E. Ö. Uslu

    In this paper we introduce the notion of Loday QD-Rinehart algebra as an abstraction of Loday QD-algebroids. Additionally, we study cohomology groups, derivations, abelian extensions and crossed modules of these algebraic structures and analyze the relationships between them.

    更新日期:2020-07-20
  • An elementary computation of the cohomology of the Fomin–Kirillov algebra with $3$ generators
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-05-20
    Estanislao Herscovich

    We give an elementary computation of the algebra structure of the Yoneda algebra of the Fomin–Kirillov algebra FK(3) over a field of characteristic different from $2$ and $3$. The computation is based on a new bootstrap technique we introduce which is built upon the (nonacyclic) Koszul complex of FK(3).

    更新日期:2020-07-20
  • Constructions of self-maps of $\operatorname{SU}(4)$ via Postnikov towers
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-05-20
    Jim Fowler; Chris Kennedy

    Cohomology operations restrict the degree of a self-map of $\operatorname{SU}(4)$ to be either odd or a multiple of $8$; we find self-maps realizing these possible degrees. The notion of the degree of a self-map can be refined to a notion of multidegree which records the effect of the self-map on each of the generators of $H^{\star} (\operatorname{SU}(4))$. We find restrictions on the possible multidegrees

    更新日期:2020-07-20
  • A simple proof of Curtis’ connectivity theorem for Lie powers
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-05-06
    Sergei O. Ivanov; Vladislav Romanovskii; Andrei Semenov

    We give a simple proof of Curtis’ theorem: if $A_{\bullet}$ is a $k$-connected free simplicial abelian group, then $L^n (A_{\bullet})$ is a $k + \lceil \operatorname{log}_2 n \rceil$-connected simplicial abelian group, where $L^n$ is the $n$‑th Lie power functor. In the proof we do not use Curtis’ decomposition of Lie powers. Instead we use the Chevalley–Eilenberg complex for the free Lie algebra.

    更新日期:2020-05-06
  • Euler characteristics of finite homotopy colimits
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-05-06
    John D. Berman

    In this note, we provide a calculation of the Euler characteristic of a finite homotopy colimit of finite cell complexes, which depends only on the Euler characteristics of each space and resembles Mobius inversion. Versions of the result are known when the colimit is indexed by categories with various finiteness conditions, but the behavior is more uniform when we index by a finite quasicategory instead

    更新日期:2020-05-06
  • Verification of the Quillen conjecture in the rank 2 imaginary quadratic case
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-05-06
    Bui Anh Tuan; Alexander D. Rahm

    We confirm a conjecture of Quillen in the case of the $\operatorname{mod} 2$ cohomology of arithmetic groups ${\rm SL}_2({\mathcal{O}}_{\mathbb{Q}(\sqrt{-m}\, )}[\frac{1}{2}])$, where ${\mathcal{O}}_{\mathbb{Q}(\sqrt{-m}\, )}$ is an imaginary quadratic ring of integers. To make explicit the free module structure on the cohomology ring conjectured by Quillen, we compute the $\operatorname{mod} 2$ cohomology

    更新日期:2020-05-06
  • So, what is a derived functor?
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-05-06
    Vladimir Hinich

    We rethink the notion of derived functor in terms of correspondences, that is, functors $\mathcal{E} \to [1]$. While derived functors in our sense, when they exist, are given by Kan extensions, their existence is a strictly stronger property than the existence of Kan extensions. We show, however, that derived functors exist in the cases one expects them to exist. Our definition is especially convenient

    更新日期:2020-05-06
  • Koszul duality and the Hochschild cohomology of Artin–Schelter regular algebras
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-04-29
    Leilei Liu

    We identify two Batalin–Vilkovisky algebra structures, one obtained by Kowalzig and Krahmer on the Hochschild cohomology of an Artin–Schelter regular algebra with semisimple Nakayama automorphism and the other obtained by Lambre, Zhou and Zimmermann on the Hochschild cohomology of a Frobenius algebra also with semisimple Nakayama automorphism, provided that these two algebras are Koszul dual to each

    更新日期:2020-04-29
  • Equivariant Steinberg summands
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-04-29
    Krishanu Sankar

    We construct Steinberg summands of $G$-equivariant spectra with $\operatorname{GL}_n (\mathbb{F}_p)$-action. We prove a lemma about their fixed points when $G$ is a $p$-group, and then use this lemma to compute the fixed points of the Steinberg summand of the equivariant classifying space of $(\mathbb{Z} / p)^n$. These results will be used in a companion paper to study the layers in the $\operatorname{mod}$

    更新日期:2020-04-29
  • Compatible actions in semi-abelian categories
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-04-29
    Davide di Micco; Tim Van der Linden

    The concept of a pair of compatible actions was introduced in the case of groups by Brown and Loday [6] and in the case of Lie algebras by Ellis [14]. In this article we extend it to the context of semi-abelian categories (that satisfy the Smith is Huq condition). We give a new construction of the Peiffer product, which specialises to the definitions known for groups and Lie algebras. We use it to

    更新日期:2020-04-29
  • Crossed modules and symmetric cohomology of groups
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-04-15
    Mariam Pirashvili

    This paper links the third symmetric cohomology (introduced by Staic [10] and Zarelua [12]) to crossed modules with certain properties. The equivalent result in the language of $2$‑groups states that an extension of $2$-groups corresponds to an element of $HS^3$ iff it possesses a section which preserves inverses in the $2$‑categorical sense. This ties in with Staic’s (and Zarelua’s) result regarding

    更新日期:2020-04-15
  • An algebraic representation of globular sets
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-04-15
    Anibal M. Medina-Mardones

    We describe a fully faithful embedding of the category of (reflexive) globular sets into the category of counital cosymmetric $R$-coalgebras when $R$ is an integral domain. This embedding is a lift of the usual functor of $R$-chains and the extra structure consists of a derived form of cup coproduct. Additionally, we construct a functor from group-like counital cosymmetric $R$-coalgebras to $\omega$-categories

    更新日期:2020-04-15
  • A note on a Holstein construction
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-04-15
    Sergey Arkhipov; Daria Poliakova

    We clarify details and fill certain gaps in the construction of a canonical Reedy fibrant resolution for a constant simplicial DG-category due to Holstein.

    更新日期:2020-04-15
  • The homology of principally directed ordered groupoids
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-04-15
    B.O. Bainson; N.D. Gilbert

    We present some homological properties of a relation $\beta$ on ordered groupoids that generalises the minimum group congruence for inverse semigroups. When $\beta$ is a transitive relation on an ordered groupoid $G$, the quotient $G / \beta$ is again an ordered groupoid, and we construct a pair of adjoint functors between the module categories of $G$ and of $G / \beta$. As a consequence, we show that

    更新日期:2020-04-15
  • The non-nil-invariance of periodic topological cyclic homology
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-04-15
    Ryo Horiuchi

    Periodic topological cyclic homology $\operatorname{TP}$ is a topological analogue of periodic cyclic homology $\operatorname{HP}$. It is known that, for R an algebra over a field of characteristic $0$ and $I$ a nilpotent ideal of $R$, the quotient map $R \to R/I$ induces an isomorphism on $\operatorname{HP}$. In this paper, we show that the analogous result for $\operatorname{TP}$ does not hold.

    更新日期:2020-04-15
  • On a conjecture of Mahowald on the cohomology of finite sub-Hopf algebras of the Steenrod algebra
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-03-25
    Paul Shick

    Mahowald’s conjecture arose as part of a program attempting to view chromatic phenomena in stable homotopy theory through the lens of the classical Adams spectral sequence. The conjecture predicts the existence of nonzero classes in the cohomology of the finite sub-Hopf algebras $A(n)$ of the $\operatorname{mod} 2$ Steenrod algebra that correspond to generators in the homotopy rings of certain periodic

    更新日期:2020-03-25
  • Quantifying Quillen’s uniform $\mathcal{F}_p$-isomorphism theorem
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-03-25
    Koenraad van Woerden

    Let $G$ be a finite group with $2$-Sylow subgroup of order less than or equal to $16$. For such a $G$, we prove a quantified version of Quillen’s uniform $\mathcal{F}_p$-isomorphism theorem, which holds uniformly for all $G$-spaces. We do this by bounding from above the exponent of Borel equivariant $\mathbf{F}_2$-cohomology, as introduced by Mathew–Naumann–Noel, with respect to the family of elementary

    更新日期:2020-03-25
  • Algebraic cobordism in mixed characteristic
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-03-25
    Markus Spitzweck

    We compute the geometric part of algebraic cobordism over Dedekind domains of mixed characteristic after inverting the positive residue characteristics and prove cases of a Conjecture of Voevodsky relating this geometric part to the Lazard ring for regular local bases. The method is by analyzing the slice tower of algebraic cobordism, relying on the Hopkins–Morel isomorphism from the quotient of the

    更新日期:2020-03-25
  • A $\operatorname{dg}$ Lie model for relative homotopy automorphisms
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-03-25
    Alexander Berglund; Bashar Saleh

    We construct a $\operatorname{dg}$ Lie algebra model for the universal cover of the classifying space of the grouplike monoid of homotopy automorphisms of a space that fix a given subspace. We derive the model from a known model for based homotopy automorphisms together with general result on rational models for geometric bar constructions.

    更新日期:2020-03-25
  • Topological $K$-theory of equivariant singularity categories
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-02-26
    Michael K. Brown; Tobias Dyckerhoff

    We study the topological $K$-theory spectrum of the $\operatorname{dg}$ singularity category associated to a weighted projective complete intersection. We calculate the topological $K$-theory of the $\operatorname{dg}$ singularity category of a weighted projective hypersurface in terms of its affine Milnor fiber and monodromy operator, and, as an application, we obtain a lift of the Atiyah–Bott–Shapiro

    更新日期:2020-02-26
  • Time-reversal homotopical properties of concurrent systems
    Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-02-26
    Cameron Calk; Eric Goubault; Philippe Malbos

    Directed topology was introduced as a model of concurrent programs, where the flow of time is described by distinguishing certain paths in the topological space representing such a program. Algebraic invariants which reflect this directedness have been introduced to classify directed spaces. In this work we study the properties of such invariants with respect to the reversal of the flow of time in

    更新日期:2020-02-26
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