• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-05-06

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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• Homol. Homotopy Appl. (IF 0.493) Pub Date : 2020-03-25

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
• 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
• 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
• 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
• 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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