Recent zbMATH articles in MSC 55Phttps://zbmath.org/atom/cc/55P2023-05-31T16:32:50.898670ZWerkzeugOn virtual cabling and a structure of 4-strand virtual pure braid grouphttps://zbmath.org/1508.200382023-05-31T16:32:50.898670Z"Bardakov, Valeriy G."https://zbmath.org/authors/?q=ai:bardakov.valerii-georgievich"Wu, Jie"https://zbmath.org/authors/?q=ai:wu.jie.2Summary: This paper is dedicated to cabling on virtual braids. This construction gives a new generating set for the virtual pure braid group \(\text{VP}_n\). We define simplicial group \(\text{VP}_\ast\) and its simplicial subgroup \(T_\ast\) which is generated by \(\text{VP}_2\). Consequently, we describe \(\text{VP}_4\) as HNN-extension and find presentation of \(T_2\) and \(T_3\). As an application to classical braids, we find a new presentation of the Artin pure braid group \(P_4\) in terms of the cabled generators.Uniform homotopy invariance of Roe index of the signature operatorhttps://zbmath.org/1508.530372023-05-31T16:32:50.898670Z"Spessato, Stefano"https://zbmath.org/authors/?q=ai:spessato.stefanoSummary: In this paper we study the Roe index of the signature operator of manifolds of bounded geometry. Our main result is the proof of the uniform homotopy invariance of this index. In other words we show that, given an orientation-preserving uniform homotopy equivalence \(f:(M,g) \longrightarrow (N,h)\) between two oriented manifolds of bounded geometry, \(f_{\star} (Ind_{Roe}D_M) = Ind_{Roe}(D_N)\). Moreover we also show that the same result holds if a group \(\Gamma\) acts on \(M\) and \(N\) by isometries and \(f\) is \(\Gamma\)-equivariant. The only assumption on the action of \(\Gamma\) is that the quotients are again manifolds of bounded geometry.String topological robotics. IIhttps://zbmath.org/1508.550072023-05-31T16:32:50.898670Z"Ayoub, Ettaki"https://zbmath.org/authors/?q=ai:ayoub.ettaki"Mamouni, My Ismail"https://zbmath.org/authors/?q=ai:mamouni.my-ismail"Elomary, Mohamed Abdou"https://zbmath.org/authors/?q=ai:elomary.mohamed-abdou\textit{Y. Derfoufi} and \textit{M. I. Mamouni} introduced in [JP J. Geom. Topol. 19, No. 3, 189--208 (2016; Zbl 1364.55010)] a certain version of equivariant motion planning algorithms, similar to the one proposed by \textit{W. Lubawski} and \textit{W. Marzantowicz} [Bull. Lond. Math. Soc. 47, No. 1, 101--117 (2015; Zbl 1311.55004)] but considering loops instead of paths. Namely, taking a transitive action of a compact Lie group \(G\) on a path-connected \(n\)-manifold \(X\), they studied the set \(\mathcal{M}^{LP}(X)\) consisting of all loop motion planning algorithms (LMPA) of \(X\). This is the set of all \(G\times G\)-homotopy sections (not necessarily continuous) of the \(G\times G\)-fibration
\[
ev^{LMP}:LX\times_{X/G}LX\rightarrow X\times X,(\gamma,\tau )\mapsto (\gamma (0),\tau (\tfrac{1}{2})))
\]
where \(LX\) denotes the space of free loops in \(X\) and \(LX\times_{X/G}LX\) is the set of pairs \((\gamma ,\tau )\in LX\times LX\) satisfying \(G\gamma (\frac{1}{2})=G\tau (0)\). Then, considering \textit{F. Laudenbach}'s techniques [Enseign. Math. (2) 57, No. 1--2, 3--21 (2011; Zbl 1229.55004)], which provide a generalization of the Chas-Sullivan approach in string topology, they equipped the shifted homology
\[
\mathbb{H}_*(\mathcal{M}^{LP}(X)):=H_{*+2n}(\mathcal{M}^{LP}(X))
\]
with the structure of an associative and commutative graded algebra.
In the paper under review the authors continue this previous work by showing that, actually, \(\mathbb{H}_*(\mathcal{M}^{LP}(X))\) has the structure of a Batalin-Vilkovisky algebra (and of a Gerstenhaber algebra).
Reviewer: José Calcines (La Laguna)Equivariant sheaves for profinite groupshttps://zbmath.org/1508.550082023-05-31T16:32:50.898670Z"Barnes, David"https://zbmath.org/authors/?q=ai:barnes.david-c|barnes.david|barnes.david-j"Sugrue, Danny"https://zbmath.org/authors/?q=ai:sugrue.danny\textit{J. P. C. Greenlees} (cf. [``Triangulated categories of rational equivariant cohomology theories'', in \textit{R.-O. Buchweitz} (ed.) et al., Oberwolfach Rep. 3, No. 1, 461--509 (2006; Zbl 1109.18301)]) conjectured that for every compact Lie group \(G\) there exists an algebraic model for rational \(G\)-spectra built in terms of sheaves over \(\mathcal{S}G\), the space of subgroups of \(G\). As observed by \textit{L. G. Lewis jun.} et al. [Equivariant stable homotopy theory. With contributions by J. E. McClure. Berlin etc.: Springer-Verlag (1986; Zbl 0611.55001)], the space of orbits, \(\mathcal{S}G/G\), is a profinite space, i.e., a compact Hausdorff totally disconnected topological space. Therefore a good theory of sheaves over \(\mathcal{S}G\) and \(\mathcal{S}G/G\) would provide a better insight into Greenlees's conjecture. Coming from this motivation, the authors develop a theory of \(G\)-equivariant sheaves over profinite spaces for profinite groups \(G\).
Let \(G\) be a profinite group and \(X\) be a profinite space. In particular, \(X\) has a basis \(\mathcal{B}\) consisting of open-closed sets. The category \(\mathcal{O}_G(\mathcal{B})\) is defined by having members of \(\mathcal{B}\) as objects and twisted inclusions \(g:U\subset V\) as maps. Then a \emph{\(G\)-presheaf of sets on \(X\)} is defined as a functor
\[
\mathcal{O}_G(\mathcal{B})^{op}\to Set.
\]
Using this notion of a \(G\)-presheaf, the authors further construct the sheafification functor into the category of \(G\)-sheaves on \(X\) and show that it is idempotent and left adjoint to the forgetful functor. These results are extended to \(G\)-presheaves of modules.
In this setting, the authors show that the category of \(G\)-equivariant sheaves of modules is an abelian category with all small limits and colimits. They also prove that pullback and pushout functors from classical sheaf theory pass to categories of equivariant sheaves of sets/modules and that these functors form an adjoint pair. Further on, they define \textit{equivariant skyscraper sheaves}, which are sheaves with a non-zero stalk over only one orbit and show that forming an equivariant skyscraper sheaf has a left adjoint. The equivariant skyscraper sheaves are used to prove that the category of equivariant sheaves over a profinite space has enough injectives and that they provide a canonical injective resolution, known as the \textit{Godement resolution}.
In [\textit{D. Barnes} and \textit{D. Sugrue}, Math. Proc. Camb. Philos. Soc. 174, No. 2, 345--368 (2023; Zbl 07653000)], the present authors used the theory of equivariant sheaves to construct an equivalence between the category of rational \(G\)-Mackey functors (for profinite \(G\)) and the full subcategory of the category of \(G\)-sheaves of \(\mathbb{Q}\)-modules over the space \(\mathcal{S}G\), called \textit{Weyl-\(G\)-sheaves}. In the current paper, the authors show that the category of Weyl-\(G\)-sheaves is abelian with all small limits and colimits and has enough injectives.
Given that a profinite space is an inverse limit of finite discrete spaces, there arises a natural question on the relation between \(G\)-sheaves over profinite spaces and diagrams of sheaves over spaces defining \(X\). The authors address this question in the final section, providing an equivalence between classes of diagrams of equivariant sheaves and \(G\)-sheaves over profinite spaces.
The work is supported with two Appendices, containing an overview of profinite groups, profinite spaces and discrete modules (App. A) and sheaves and presheaves (App. B).
Reviewer: Igor Sikora (Ankara)The twisted homology of simplicial sethttps://zbmath.org/1508.550122023-05-31T16:32:50.898670Z"Zhang, Meng Meng"https://zbmath.org/authors/?q=ai:zhang.mengmeng"Li, Jing Yan"https://zbmath.org/authors/?q=ai:li.jingyan"Wu, Jie"https://zbmath.org/authors/?q=ai:wu.jie.2The paper gives the \(\Delta\)-twisted homology as a generalization of the \(\delta\)-twisted homology introduced by Jingyan Li, Vladimir Vershinin and Jie Wu [\textit{J. Y. Li} et al., Homology Homotopy Appl. 19, No. 2, 111--130 (2017; Zbl 1384.55016)]. This enriches the theory of \(\delta\)-(co)homology introduced by Alexander Grigoryan, Yuri Muranov, and Shing-Tung Yau [\textit{A. Grigor'yan} et al., J. Homotopy Relat. Struct. 11, No. 2, 209--230 (2016; Zbl 1353.05056)]. The paper proves that the Mayer-Vietoris sequence holds for \(\Delta\)-twisted homology, and the paper introduces the notion of \(\Delta\)-twisted Cartesian product on simplicial sets, which generalizes the classical work of Barratt, Gugenheim and Moore on twisted Cartesian products of simplicial sets [\textit{M. Barratt} et al., Am. J. Math. 81, 639--657 (1959; Zbl 0127.39002)]. Under certain hypotheses, the paper shows that the coordinate projection of \(\Delta\)-twisted Cartesian products admits a fibre bundle structure.
Reviewer: Shiquan Ren (Singapore)A calculus for flow categorieshttps://zbmath.org/1508.570162023-05-31T16:32:50.898670Z"Lobb, Andrew"https://zbmath.org/authors/?q=ai:lobb.andrew"Orson, Patrick"https://zbmath.org/authors/?q=ai:orson.patrick"Schütz, Dirk"https://zbmath.org/authors/?q=ai:schutz.dirk\textit{Framed flow categories} are combinatorial models of topological spectra, first introduced in [\textit{R. L. Cohen} et al., Prog. Math. 133, 297--325 (1995; Zbl 0843.58019)]. One of the succesful applications of these models is the \textit{Khovanov spectrum}, a link invariant with values in spectra discovered by \textit{R. Lipshitz} and \textit{S. Sarkar} [J. Am. Math. Soc. 27, No. 4, 983--1042 (2014; Zbl 1345.57014)]. These link invariants have the particularity of having small cohomological width (their non-trivial cohomology is concentrated in a few consecutive degrees) relative to the number of crossings of their link diagrams. The first purpose of this article is to give a combinatorial viewpoint on the question of whether two framed flow categories yield homotopy equivalent spectra. The authors introduce four ``moves'' between framed flow categories: perturbation, stabilization, handle cancellation and extended Whitney trick. Any two framed flow categories related by any of these moves yield weak equivalent spectra, and they conjecture that the converse holds: Any two weak equivalent framed flow categories are related by a sequence of moves. The main theorem of the article proves this conjecture when the cohomology of the framed flow categories is supported in four consecutive degrees.
On the other hand, homotopy types of topological spectra whose cohomology is supported in four consecutive degrees can be classified using the work of \textit{H.-J. Baues} and \textit{M. Hennes} [Topology 30, No. 3, 373--408 (1991; Zbl 0735.55002)]. The authors give minimal framed flow categories modeling these homotopy types and construct an algorithm that turns any framed flow category of cohomological length four into a minimal one. Remarkably, this algorithm gives an independent proof of the classification of spectra of cohomological length three. If the framed flow category has bigger cohomological length, the algorithm can be still used to give effective computations of the first three Steenrod squares.
This is the fourth paper in a series of articles exploring combinatorial models of weak equivalences (see [\textit{D. Jones} et al., Indiana Univ. Math. J. 66, No. 5, 1603--1657 (2017; Zbl 1394.57012); \textit{A. Lobb} et al., Algebr. Geom. Topol. 18, No. 5, 2821--2858 (2018; Zbl 1418.57023); Exp. Math. 29, No. 4, 475--500 (2020; Zbl 1464.57020)]).
Reviewer: Federico Cantero Morán (Madrid)