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Unstable algebras over an operad II Homol. Homotopy Appl. (IF 0.5) Pub Date : 2024-02-21 Sacha Ikonicoff
$\def\P\{\mathcal{P}}$We work over the finite field $\mathbb{F}_q$. We introduce a notion of unstable $\P$-algebra over an operad $\P$. We show that the unstable $\P$-algebra freely generated by an unstable module is itself a free $\P$-algebra under suitable conditions. We introduce a family of ‘$q$-level’ operads which allows us to identify unstable modules studied by Brown–Gitler, Miller and Carlsson
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Homotopy theory of spectral sequences Homol. Homotopy Appl. (IF 0.5) Pub Date : 2024-02-21 Muriel Livernet, Sarah Whitehouse
Let $R$ be a commutative ring with unit. We consider the homotopy theory of the category of spectral sequences of $R$-modules with the class of weak equivalences given by those morphisms inducing a quasi-isomorphism at a certain fixed page. We show that this admits a structure close to that of a category of fibrant objects in the sense of Brown and in particular the structure of a partial Brown category
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On strict polynomial functors with bounded domain Homol. Homotopy Appl. (IF 0.5) Pub Date : 2024-02-21 Marcin Chałupnik, Patryk Jaśniewski
$\def\Pdn\{\mathcal{P}_{d,n}}$We introduce a new functor category: the category $\Pdn$ of strict polynomial functors of degree $d$ with domain of dimension bounded by $n$. It is equivalent to the category of finite dimensional modules over the Schur algebra $S(n,d)$, hence it allows one to apply the tools available in functor categories to representations of the algebraic group $\mathrm{GL}_n$. We
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Magnitude, homology, and the Whitney twist Homol. Homotopy Appl. (IF 0.5) Pub Date : 2024-02-21 Emily Roff
Magnitude is a numerical invariant of metric spaces and graphs, analogous, in a precise sense, to Euler characteristic. Magnitude homology is an algebraic invariant constructed to categorify magnitude. Among the important features of the magnitude of graphs is its behaviour with respect to an operation known as the Whitney twist.We give a homological account of magnitude’s invariance under Whitney
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An elementary proof of the chromatic Smith fixed point theorem Homol. Homotopy Appl. (IF 0.5) Pub Date : 2024-02-21 William Balderrama, Nicholas J. Kuhn
A recent theorem by T. Barthel, M. Hausmann, N. Naumann, T. Nikolaus, J. Noel, and N. Stapleton says that if $A$ is a finite abelian $p$-group of rank $r$, then any finite $A$-space $X$ which is acyclic in the $n$th Morava $K$-theory with $n \geqslant r$ will have its subspace $X^A$ of fixed points acyclic in the $(n-r)$th Morava Ktheory. This is a chromatic homotopy version of P. A. Smith’s classical
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Polynomial generators of $\mathbf{MSU}^\ast [1/2]$ related to classifying maps of certain formal group laws Homol. Homotopy Appl. (IF 0.5) Pub Date : 2024-01-24 Malkhaz Bakuradze
This paper presents a commutative complex oriented cohomology theory that realizes the Buchstaber formal group law $F_B$ localized away from $2$. It is shown that the restriction of the classifying map of $F_B$ on the special unitary cobordism ring localized away from $2$ defines a four parameter genus, studied by Hoehn and Totaro.
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Independence complexes of $(n \times 6)$-grid graphs Homol. Homotopy Appl. (IF 0.5) Pub Date : 2024-01-24 Takahiro Matsushita, Shun Wakatsuki
We determine the homotopy types of the independence complexes of the $(n \times 6)$-square grid graphs. In fact, we show that these complexes are homotopy equivalent to wedges of spheres.
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On finite domination and Poincaré duality Homol. Homotopy Appl. (IF 0.5) Pub Date : 2024-01-24 John R. Klein
The object of this paper is to show that non-homotopy finite Poincaré duality spaces are plentiful. Let $π$ be a finitely presented group. Assuming that the reduced Grothendieck group $\widetilde{K}_0 (\mathbb{Z} [\pi])$ has a non-trivial $2$-divisible element, we construct a finitely dominated Poincaré space $X$ with fundamental group $π$ such that $X$ is not homotopy finite. The dimension of $X$
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Lifespan functors and natural dualities in persistent homology Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-11-22 Ulrich Bauer, Maximilian Schmahl
We introduce lifespan functors, which are endofunctors on the category of persistence modules that filter out intervals from barcodes according to their boundedness properties. They can be used to classify injective and projective objects in the category of barcodes and the category of pointwise finite-dimensional persistence modules. They also naturally appear in duality results for absolute and relative
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Sharpness of saturated fusion systems on a Sylow $p$-subgroup of $\mathrm{G}_2 (p)$ Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-11-22 Valentina Grazian, Ettore Marmo
We prove that the Díaz–Park sharpness conjecture holds for saturated fusion systems defined on a Sylow $p$-subgroup of the group $\mathrm{G}_2 (p)$, for $p \geqslant 5$.
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The cohomology of free loop spaces of rank $2$ flag manifolds Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-11-22 Matthew Burfitt, Jelena Grbić
By applying Gröbner basis theory to spectral sequences algebras, we develop a new computational methodology applicable to any Leray–Serre spectral sequence for which the cohomology of the base space is the quotient of a finitely generated polynomial algebra. We demonstrate the procedure by deducing the cohomology of the free loop space of flag manifolds, presenting a significant extension over previous
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Classifying space via homotopy coherent nerve Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-11-22 Kensuke Arakawa
We prove that the classifying space of a simplicial group is modeled by its homotopy coherent nerve. We will also show that the claim remains valid for simplicial groupoids.
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$K$-theory of real Grassmann manifolds Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-11-22 Sudeep Podder, Parameswaran Sankaran
Let $G_{n,k}$ denote the real Grassmann manifold of $k$-dimensional vector subspaces of $\mathbb{R}^n$. We compute the complex $K$-ring of $G_{n,k}\:$, up to a small indeterminacy, for all values of $n,k$ where $2 \leqslant k \leqslant n - 2$. When $n \equiv 0 (\operatorname{mod} 4), k \equiv 1 (\operatorname{mod} 2)$, we use the Hodgkin spectral sequence to determine the $K$-ring completely.
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Erratum to “From loop groups to 2-groups” Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-11-22 John C. Baez, Alissa S. Crans, Urs Schreiber, Danny Stevenson
There were a number of sign errors in our paper “From loop groups to 2-groups” $\href{https://dx.doi.org/10.4310/HHA.2007.v9.n2.a4 }{[\textit{Homology Homotopy Appl.}\;\textbf{9}\;\textrm{(2007), 101–135}]}$. Here we explain how to correct those errors.
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When Bousfield localizations and homotopy idempotent functors meet again Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-11-01 Victor Carmona
We generalize Bousfield–Friedlander’s Theorem and Hirschhorn’s Localization Theorem to settings where the hypotheses are not satisfied at the expense of obtaining semi-model categories instead of model categories. We use such results to answer, in the world of semi-model categories, an open problem posed by May–Ponto about the existence of Bousfield localizations for Hurewicz and mixed type model structures
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On bialgebras, comodules, descent data and Thom spectra in $\infty$-categories Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-11-01 Jonathan Beardsley
This paper establishes several results for coalgebraic structure in $\infty$-categories, specifically with connections to the spectral noncommutative geometry of cobordism theories. We prove that the categories of comodules and modules over a bialgebra always admit suitably structured monoidal structures in which the tensor product is taken in the ambient category (as opposed to a relative (co)tensor
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Zig-zag modules: cosheaves and $k$-theory Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-11-01 Ryan Grady, Anna Schenfisch
Persistence modules have a natural home in the setting of stratified spaces and constructible cosheaves. In this article, we first give explicit constructible cosheaves for common data-motivated persistence modules, namely, for modules that arise from zig‑zag filtrations (including monotone filtrations), and for augmented persistence modules (which encode the data of instantaneous events). We then
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The structuring effect of a Gottlieb element on the Sullivan model of a space Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-11-01 Gregory Lupton, Samuel Bruce Smith
We show a Gottlieb element in the rational homotopy of a simply connected space $X$ implies a structural result for the Sullivan minimal model, with different results depending on parity. In the even-degree case, we prove a rational Gottlieb element is a terminal homotopy element. This fact allows us to complete an argument of Dupont to prove an even-degree Gottlieb element gives a free factor in the
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Homology transfer products on free loop spaces: orientation reversal on spheres Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-10-11 Philippe Kupper
We consider the space $\Lambda M := H^1 (S^1, M)$ of loops of Sobolev class $H^1$ of a compact smooth manifold $M$, the so-called free loop space of $M$. We take quotients $\Lambda M / G$ where $G$ is a finite subgroup of $O(2)$ acting by linear reparametrization of $S^1$. We use the existence of transfer maps $\operatorname{tr} : H_\ast (\Lambda M / G) \to H_\ast (\Lambda M)$ to define a homology
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The homology of connective Morava $E$-theory with coefficients in $\mathbb{F}_p$ Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-10-11 Lukas Kattän, Sean Tilson
Let $e_n$ be the connective cover of the Morava $E$-theory spectrum $E_n$ of height $n$. In this paper we compute its homology $H_\ast (e_n; \mathbb{F}_p)$ for any prime $p$ and $n \leqslant 4$ up to possible multiplicative extensions. In order to accomplish this we show that the Künneth spectral sequence based on an $\mathbb{E}_3$-algebra $R$ is multiplicative when the $R$-modules in question are
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Haefliger’s approach for spherical knots modulo immersions Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-10-04 Neeti Gauniyal
$\def\Emb{\overline{Emb}}$We show that for the spaces of spherical embeddings modulo immersions $\Emb (S^n, S^{n+q})$ and long embeddings modulo immersions $\Emb_\partial (D^n, D^{n+q})$, the set of connected components is isomorphic to $\pi_{n+1} (SG, SG_q)$ for $q \geqslant 3$. As a consequence, we show that all the terms of the long exact sequence of the triad $(SG; SO, SG_q)$ have a geometric meaning
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The homotopy solvability of compact Lie groups and homogenous topological spaces Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-10-04 Marek Golasiński
$\def\F{\mathbb{F}} \def\O{\mathbb{O}} \def\R{\mathbb{R}} \def\C{\mathbb{C}} \def\H{\mathbb{H}}$We analyse the homotopy solvability of the classical Lie groups $O(n)$, $U(n)$, $Sp(n)$ and derive its heredity by closed subgroups. In particular, the homotopy solvability of compact Lie groups is shown. Then, we study the homotopy solvability of the loop spaces $\Omega (G_{n,m} (\F))$, $\Omega (V_{n,m}
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The homotopy-invariance of constructible sheaves Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-10-04 Peter J. Haine, Mauro Porta, Jean-Baptiste Teyssier
In this paper we show that the functor sending a stratified topological space $S$ to the $\infty$-category of constructible (hyper)sheaves on $S$ with coefficients in a large class of presentable $\infty$-categories is homotopy-invariant. To do this, we first establish a number of results for locally constant (hyper)sheaves. For example, if $X$ is a locally weakly contractible topological space and
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Two theorems on cohomological pairings Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-09-27 Ambrus Pál, Tomer M. Schlank
We give a new, elegant description of the Tate duality pairing as a Brauer–Manin pairing for associated embedding problems and prove a new theorem on the cup product.
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Self-closeness numbers of non-simply-connected spaces Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-09-27 Yichen Tong
The self-closeness number of a connected CW complex is the least integer $n$ such that any of its self-maps inducing an isomorphism in $\pi_\ast$ for $\ast \leqslant n$ is a homotopy equivalence. We prove that under a mild condition, the self-closeness number of a non-simply-connected finite complex coincides with that of its universal cover whenever the universal cover is a finite $\mathrm{H}_0$-space
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The classifying space of the 1+1 dimensional $G$-cobordism category Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-09-27 Carlos Segovia
For a finite group $G$, we define the $G$-cobordism category in dimension two. We show there is a one-to-one correspondence between the connected components of its classifying space and the abelianization of $G$. Also, we find an isomorphism of its fundamental group onto the direct sum $\mathbb{Z} \oplus \Omega^{SO}_{2} (BG)$, where $\Omega^{SO}_{2} (BG)$ is the free oriented $G$-bordism group in dimension
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The category of Silva spaces is not integral Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-05-02 Marianne Lawson, Sven-Ake Wegner
We establish that the category of Silva spaces, aka $\mathrm{LS}$-spaces, formed by countable inductive limits of Banach spaces with compact linking maps as objects and linear and continuous maps as morphisms, is not an integral category. The result carries over to the category of $\mathrm{PLS}$-spaces, i.e., countable projective limits of $\mathrm{LS}$-spaces—which contains prominent spaces of analysis
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Homotopy type of the space of finite propagation unitary operators on $\mathbb{Z}$ Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-05-02 Tsuyoshi Kato, Daisuke Kishimoto, Mitsunobu Tsutaya
The index theory for the space of finite propagation unitary operators was developed by Gross, Nesme, Vogts and Werner from the viewpoint of quantum walks in mathematical physics. In particular, they proved that $\pi_0$ of the space is determined by the index. However, nothing is known about the higher homotopy groups. In this article, we describe the homotopy type of the space of finite propagation
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Cyclic $A_\infty$-algebras and cyclic homology Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-04-26 Estanislao Herscovich
We provide a new description of the complex computing the Hochschild homology of an $H$-unitary $A_\infty$-algebra $A$ as a derived tensor product $A \oplus^\infty_{A^\epsilon}$ such that: (1) there is a canonical morphism from it to the complex computing the cyclic homology of $A$ that was introduced by Kontsevich and Soibelman, (2) this morphism induces the map $I$ in the well-known SBI sequence
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Complex orientations and $\mathrm{TP}$ of complete DVRs Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-04-26 Gabriel Angelini-Knoll
Let $L$ be a finite extension of $\mathbb{Q}_p$ with ring of integers $\mathcal{O}_L$. We show that periodic topological cyclic homology of $\mathcal{O}_L$, over the base $\mathbb{E}_\infty$-ring $\mathbb{S}_{W(\mathbb{F}_q)} [z]$ carries a $p$-height one formal group law $\operatorname{mod}(p)$ that depends on an Eisenstein polynomial of $L$ over $\mathbb{Q}_p$ for a choice of uniformizer $\varpi
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Magnitude homology of graphs and discrete Morse theory on Asao–Izumihara complexes Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-04-26 Yu Tajima, Masahiko Yoshinaga
Recently, Asao and Izumihara introduced CW-complexes whose homology groups are isomorphic to direct summands of the graph magnitude homology group. In this paper, we study the homotopy type of the CW-complexes in connection with the diagonality of magnitude homology groups. We prove that the Asao–Izumihara complex is homotopy equivalent to a wedge of spheres for pawful graphs introduced by Y. Gu. The
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A degree formula for equivariant cohomology rings Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-04-26 Mark Blumstein, Jeanne Duflot
This paper generalizes a result of Lynn on the “degree” of an equivariant cohomology ring H^\ast_G (X)$. The degree of a graded module is a certain coefficient of its Poincaré series, and is closely related to multiplicity. In the present paper, we study these commutative algebraic invariants for equivariant cohomology rings. The main theorem is an additivity formula for degree:\[\deg (H^\ast_G (X))
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The homotopy types of $Sp(n)$-gauge groups over $\mathbb{C}P^2$ Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-04-12 Sajjad Mohammadi
Let $n \gt 2$ and $\mathcal{G}_k (\mathbb{C}P^2)$ be the gauge groups of the principal $Sp(n)$-bundles over $\mathbb{C}P^2$. In this article we partially classify the homotopy types of $\mathcal{G}_k (\mathbb{C}P^2)$ by showing that if there is a homotopy equivalence $\mathcal{G}_k (\mathbb{C}P^2) \simeq \mathcal{G}_{k^\prime} (\mathbb{C}P^2)$ then $(k, 4n(2n + 1)) = (k^\prime , 4n(2n + 1))$.
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Coformality around fibrations and cofibrations Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-04-12 Ruizhi Huang
We show that in a fibration the coformality of the base space implies the coformality of the total space under reasonable conditions, and these conditions can not be weakened. The result is partially dual to the classical work of Lupton on the formality within a fibration. Our result has two applications. Firstly, we show that for certain homotopy cofibrations, the coformality of the cofiber implies
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Self-closeness numbers of product spaces Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-04-12 Pengcheng Li
The self-closeness number of a CW-complex is a homotopy invariant defined by the minimal number $n$ such that every selfmap of $X$ which induces automorphisms on the first $n$ homotopy groups of $X$ is a homotopy equivalence. In this article we study the self-closeness numbers of finite Cartesian products, and prove that under certain conditions (called reducibility), the self-closeness number of product
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A(nother) model for the framed little disks operad Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-04-12 Erik Lindell, Thomas Willwacher
We describe new graphical models of the framed little disks operad which exhibit large symmetry $\mathrm{dg}$ Lie algebras.
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On invertible $2$-dimensional framed and $r$-spin topological field theories Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-03-22 Lóránt Szegedy
We classify invertible $2$-dimensional framed and $r$-spin topological field theories by computing the homotopy groups and the $k$-invariant of the corresponding bordism categories. The zeroth homotopy group of a bordism category is the usual Thom bordism group, the first homotopy group can be identified with a Reinhart vector field bordism group, or the so called SKK group as observed by Ebert, Bökstedt–Svane
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Extra structure on the cohomology of configuration spaces of closed orientable surfaces Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-03-22 Roberto Pagaria
The rational homology of unordered configuration spaces of points on any surface was studied by Drummond–Cole and Knudsen. We compute the rational cohomology of configuration spaces on a closed orientable surface, keeping track of the mixed Hodge numbers and the action of the symplectic group on the cohomology. We find a series with coefficients in the Grothendieck ring of $\mathfrak{sp}(2g)$ that
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Multicategories model all connective spectra Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-03-22 Niles Johnson, Donald Yau
There is a free construction from multicategories to permutative categories, left adjoint to the endomorphism multicategory construction. The main result shows that these functors induce an equivalence of homotopy theories. This result extends a similar result of Thomason, that permutative categories model all connective spectra.
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On the topological $K$-theory of twisted equivariant perfect complexes Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-03-22 Michael K. Brown, Tasos Moulinos
We construct a comparison map from the topological $K$-theory of the dg-category of twisted perfect complexes on certain global quotient stacks to twisted equivariant $K$-theory, generalizing constructions of Halpern-Leistner–Pomerleano [HLP15] and Moulinos [Mou19]. We prove that this map is an equivalence if a version of the projective bundle theorem holds for twisted equivariant $K$-theory. Along
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Derived universal Massey products Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-03-22 Fernando Muro
We define an obstruction to the formality of a differential graded algebra over a graded operad defined over a commutative ground ring. This obstruction lives in the derived operadic cohomology of the algebra. Moreover, it determines all operadic Massey products induced on the homology algebra, hence the name of derived universal Massey product.
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E-infinity structure in hyperoctahedral homology Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-03-01 Daniel Graves
Hyperoctahedral homology for involutive algebras is the homology theory associated to the hyperoctahedral crossed simplicial group. It is related to equivariant stable homotopy theory via the homology of equivariant infinite loop spaces. In this paper we show that there is an E-infinity algebra structure on the simplicial module that computes hyperoctahedral homology. We deduce that hyperoctahedral
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Fiber integration of gerbes and Deligne line bundles Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-03-01 Ettore Aldrovandi, Niranjan Ramachandran
Let $\pi : X \to S$ be a family of smooth projective curves, and let $L$ and $M$ be a pair of line bundles on $X$. We show that Deligne’s line bundle $\langle L,M \rangle$ can be obtained from the $\mathcal{K}_2$-gerbe $G_{L,M}$ constructed in [AR16] via an integration along the fiber map for gerbes that categorifies the well known one arising from the Leray spectral sequence of $\pi$. Our construction
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Koszul duality in higher topoi Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-03-01 Jonathan Beardsley, Maximilien Péroux
We show that there is an equivalence in any $n$-topos $\mathscr{X}$ between the pointed and $k$-connective objects of $\mathscr{X}$ and the $\mathbb{E}_k$-group objects of the $(n-k-1)$-truncation of $\mathscr{X}$. This recovers, up to equivalence of $\infty$-categories, some classical results regarding algebraic models for $k$-connective, $(n-1)$-coconnective homotopy types. Further, it extends those
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Constructing coproducts in locally Cartesian closed $\infty$-categories Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-03-01 Jonas Frey, Nima Rasekh
We prove that every locally Cartesian closed $\infty$-category with a subobject classifier has a strict initial object and disjoint and universal binary coproducts.
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Stable equivariant complex cobordism of the symmetric group on three elements Homol. Homotopy Appl. (IF 0.5) Pub Date : 2023-03-01 Po Hu, Igor Kriz, Yunze Lu
In this paper, we calculate the coefficient ring of equivariant Thom complex cobordism for the symmetric group on three elements. We also make some remarks on general methods of calculating certain pullbacks of rings which typically occur in calculations of equivariant cobordism.
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Persistent homology with non-contractible preimages Homol. Homotopy Appl. (IF 0.5) Pub Date : 2022-09-14 Konstantin Mischaikow, Charles Weibel
For a fixed $N$, we analyze the space of all sequences $z=(z_1,\dotsc,z_N)$, approximating a continuous function on the circle, with a given persistence diagram $P$, and show that the typical components of this space are homotopy equivalent to $S^1$. We also consider the space of functions on $Y$-shaped (resp., starshaped) trees with a $2$-point persistence diagram, and show that this space is homotopy
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On the string topology coproduct for Lie groups Homol. Homotopy Appl. (IF 0.5) Pub Date : 2022-09-14 Maximilian Stegemeyer
The free loop space of a Lie group is homeomorphic to the product of the Lie group itself and its based loop space.We show that the coproduct on the homology of the free loop space that was introduced by Goresky and Hingston splits into the diagonal map on the group and a based coproduct on the homology of the based loop space. This result implies that the coproduct is trivial for even-dimensional
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Path homology of directed hypergraphs Homol. Homotopy Appl. (IF 0.5) Pub Date : 2022-09-14 Yuri Muranov, Anna Szczepkowska, Vladimir Vershinin
We describe various path homology theories constructed for a directed hypergraph. We introduce the category of directed hypergraphs and the notion of a homotopy in this category. Also, we investigate the functoriality and the homotopy invariance of the introduced path homology groups.We provide examples of computation of these homology groups.
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Magnitude meets persistence: homology theories for filtered simplicial sets Homol. Homotopy Appl. (IF 0.5) Pub Date : 2022-09-14 Nina Otter
The Euler characteristic is an invariant of a topological space that in a precise sense captures its canonical notion of size, akin to the cardinality of a set. The Euler characteristic is closely related to the homology of a space, as it can be expressed as the alternating sum of its Betti numbers, whenever the sum is well-defined. Thus, one says that homology categorifies the Euler characteristic
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On the convergence of the orthogonal spectral sequence Homol. Homotopy Appl. (IF 0.5) Pub Date : 2022-09-14 Cesar Galindo, Pablo Pelaez
We show that the orthogonal spectral sequence introduced by the second author is strongly convergent in Voevodsky’s triangulated category of motives $DM$ over a field $k$. In the context of the Morel–Voevodsky $\mathbb{A}^1$‑stable homotopy category we provide concrete examples where the spectral sequence is not strongly convergent, and give a criterion under which the strong convergence still holds
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Graphs associated to fold maps from closed surfaces to the projective plane Homol. Homotopy Appl. (IF 0.5) Pub Date : 2022-08-24 Catarina Mendes de Jesus Sánchez, Maria del Carmen Romero Fuster
We describe in this paper how to attach a weighted graph to any fold map (i.e., a stable map without cusps) with planar apparent contour from a closed surface to the projective plane. Also we study necessary and sufficient conditions for a weighted graph with non negatively weighted vertices to be the graph of a fold map from a closed surface to the projective plane.
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Spectral sequences of a Morse shelling Homol. Homotopy Appl. (IF 0.5) Pub Date : 2022-08-24 Jean-Yves Welschinger
We recently introduced a notion of tilings of geometric realizations of finite relative simplicial complexes and related those tilings to the discrete Morse theory of R. Forman, especially when they have the property of being shellable, a property shared by the classical shellable complexes. We now observe that every such tiling supports a quiver which is acyclic precisely when the tiling is shellable
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Nondegenerate homotopy and geometric flows Homol. Homotopy Appl. (IF 0.5) Pub Date : 2022-08-24 Jiří Minarčík, Michal Beneš
Formulating geometric flows of space curves using quantities derived from the Frenet frame restricts the motion to one connected component of the space of locally convex curves. A new invariant quantity called tangent turning sign is proposed to determine the nondegenerate homotopy type of the initial curve and identify its possible shapes during the geometric flow.
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On generalized projective product spaces and Dold manifolds Homol. Homotopy Appl. (IF 0.5) Pub Date : 2022-08-24 Soumen Sarkar, Peter Zvengrowski
D. Davis introduced projective product spaces in 2010 as a generalization of real projective spaces and discussed some of their topological properties. On the other hand, Dold manifolds were introduced by A. Dold in 1956 to study the generators of the non-oriented cobordism ring. Recently, in 2019, A. Nath and P. Sankaran made a modest generalization of Dold manifolds. In this paper we simultaneously
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Hyperplane restrictions of indecomposable $n$-dimensional persistence modules Homol. Homotopy Appl. (IF 0.5) Pub Date : 2022-08-24 Samantha Moore
Understanding the structure of indecomposable $n$‑dimensional persistence modules is a difficult problem, yet is foundational for studying multipersistence. To this end, Buchet and Escolar showed that any finitely presented rectangular $(n-1)$‑dimensional persistence module with finite support is a hyperplane restriction of an indecomposable $n$‑dimensional persistence module. We extend this result
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Exponentials of non-singular simplicial sets Homol. Homotopy Appl. (IF 0.5) Pub Date : 2022-08-24 Vegard Fjellbo, John Rognes
A simplicial set is non-singular if the representing map of each non-degenerate simplex is degreewise injective. The simplicial mapping set $X^K$ has $n$‑simplices given by the simplicial maps $\Delta [n] \times K \to X$. We prove that $X^K$ is non-singular whenever $X$ is non-singular. It follows that non-singular simplicial sets form a cartesian closed category with all limits and colimits, but it
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Poincaré/Koszul duality for general operads Homol. Homotopy Appl. (IF 0.5) Pub Date : 2022-08-10 Araminta Amabel
We record a result concerning the Koszul dual of the arity filtration on an operad. This result is then used to give conditions under which, for a general operad, the Poincaré/Koszul duality arrow of Ayala and Francis is an equivalence, using a proof similar to theirs. We discuss how the Poincaré/Koszul duality arrow for the little disks operad $\mathcal{E}_n$ relates to the work of Ayala and Francis
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A Wells type exact sequence for non-degenerate unitary solutions of the Yang–Baxter equation Homol. Homotopy Appl. (IF 0.5) Pub Date : 2022-08-10 Valeriy Bardakov, Mahender Singh
Cycle sets are known to give non-degenerate unitary solutions of the Yang–Baxter equation and linear cycle sets are enriched versions of these algebraic systems. The paper explores the recently developed cohomology and extension theory for linear cycle sets. We derive a four term exact sequence relating 1-cocycles, second cohomology and certain groups of automorphisms arising from central extensions
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$1$-smooth pro-$p$ groups and Bloch–Kato pro-$p$ groups Homol. Homotopy Appl. (IF 0.5) Pub Date : 2022-08-10 Claudio Quadrelli
Let $p$ be a prime. A pro‑$p$ group $G$ is said to be $1$-smooth if it can be endowed with a homomorphism of pro‑$p$ groups of the form $G \to 1 + p \mathbb{Z}_p$ satisfying a formal version of Hilbert 90. By Kummer theory, maximal pro‑$p$ Galois groups of fields containing a root of $1$ of order $p$, together with the cyclotomic character, are $1$-smooth. We prove that a finitely generated padic analytic