Recent zbMATH articles in MSC 55https://zbmath.org/atom/cc/552024-07-05T15:31:27.292712ZWerkzeugFrom cubes to twisted cubes via graph morphisms in type theoryhttps://zbmath.org/1535.030642024-07-05T15:31:27.292712Z"Pinyo, Gun"https://zbmath.org/authors/?q=ai:pinyo.gun"Kraus, Nicolai"https://zbmath.org/authors/?q=ai:kraus.nicolaiSummary: Cube categories are used to encode higher-dimensional categorical structures. They have recently gained significant attention in the community of homotopy type theory and univalent foundations, where types carry the structure of higher groupoids. \textit{M. Bezem} et al. [LIPIcs -- Leibniz Int. Proc. Inform. 26, 107--128 (2014; Zbl 1359.03009)] have presented a constructive model of univalence using a specific cube category, which we call the BCH cube category.\par The higher categories encoded with the BCH cube category have the property that all morphisms are invertible, mirroring the fact that equality is symmetric. This might not always be desirable: the field of directed type theory considers a notion of equality that is not necessarily invertible.\par This motivates us to suggest a category of twisted cubes which avoids built-in invertibility. Our strategy is to first develop several alternative (but equivalent) presentations of the BCH cube category using morphisms between suitably defined graphs. Starting from there, a minor modification allows us to define our category of twisted cubes. We prove several first results about this category, and our work suggests that twisted cubes combine properties of cubes with properties of globes and simplices (tetrahedra).
For the entire collection see [Zbl 1445.68007].Simultaneously vanishing higher derived limitshttps://zbmath.org/1535.032432024-07-05T15:31:27.292712Z"Bergfalk, Jeffrey"https://zbmath.org/authors/?q=ai:bergfalk.jeffrey"Lambie-Hanson, Chris"https://zbmath.org/authors/?q=ai:lambie-hanson.chrisSummary: In [Trans. Am. Math. Soc. 307, No. 2, 725--744 (1988; Zbl 0648.55007)], \textit{S. Mardešić} and \textit{A. V. Prasolov} isolated an inverse system \(\boldsymbol{A}\) with the property that the additivity of strong homology on any class of spaces which includes the closed subsets of Euclidean space would entail that \(\lim^n\boldsymbol{A}\) (the \(n\)th derived limit of \(\boldsymbol{A}\)) vanishes for every \(n>0\). Since that time, the question of whether it is consistent with the \(\mathsf{ZFC}\) axioms that \(\lim^n \boldsymbol{A}=0\) for every \(n>0\) has remained open. It remains possible as well that this condition in fact implies that strong homology is additive on the category of metric spaces.
We show that assuming the existence of a weakly compact cardinal, it is indeed consistent with the \(\mathsf{ZFC}\) axioms that \(\lim^n \boldsymbol{A}=0\) for all \(n>0\). We show this via a finite-support iteration of Hechler forcings which is of weakly compact length. More precisely, we show that in any forcing extension by this iteration, a condition equivalent to \(\lim^n\boldsymbol{A}=0\) will hold for each \(n>0\). This condition is of interest in its own right; namely, it is the triviality of every coherent \(n\)-dimensional family of certain specified sorts of partial functions \(\mathbb{N}^2\to \mathbb{Z}\) which are indexed in turn by \(n\)-tuples of functions \(f:\mathbb{N}\to \mathbb{N}\). The triviality and coherence in question here generalise the classical and well-studied case of \(n=1\).One-sided sharp thresholds for homology of random flag complexeshttps://zbmath.org/1535.052402024-07-05T15:31:27.292712Z"Newman, Andrew"https://zbmath.org/authors/?q=ai:newman.andrew-fSummary: We prove that the random flag complex has a probability regime where the probability of nonvanishing homology is asymptotically bounded away from zero and away from one. Related to this main result, we also establish new bounds on a sharp threshold for the fundamental group of a random flag complex to be a free group. In doing so, we show that there is an intermediate probability regime in which the random flag complex has fundamental group that is neither free nor has Kazhdan's property (T).
{\copyright} 2024 The Authors. \textit{Journal of the London Mathematical Society} is copyright {\copyright} London Mathematical Society.Topological combinatorics -- the Kneser conjecturehttps://zbmath.org/1535.052752024-07-05T15:31:27.292712Z"Deo, Satya"https://zbmath.org/authors/?q=ai:deo.satya.2|deo.satya.1Summary: The proof of Kneser conjecture given by L. Lovász is believed to have laid the foundations of topological combinatorics. In this expository paper, we present three distinct proofs of the Kneser conjecture, each of which essentially uses the Bousruk-Ulam theorem of algebraic topology. We also discuss various generalizations of this conjecture along with the methods
employed in their proofs.Polarizations and hook partitionshttps://zbmath.org/1535.130332024-07-05T15:31:27.292712Z"Almousa, Ayah"https://zbmath.org/authors/?q=ai:almousa.ayah"VandeBogert, Keller"https://zbmath.org/authors/?q=ai:vandebogert.kellerSummary: In this paper, we relate combinatorial conditions for polarizations of powers of the graded maximal ideal with rank conditions on submodules generated by collections of Young tableaux. We apply discrete Morse theory to the hypersimplex resolution introduced by Batzies-Welker to show that the \(L\)-complex of Buchsbaum and Eisenbud for powers of the graded maximal ideal is supported on a CW-complex. We then translate the ``spanning tree condition'' of Almousa-Fløystad-Lohne characterizing polarizations of powers of the graded maximal ideal into a condition about which sets of hook tableaux span a certain Schur module. As an application, we give a complete combinatorial characterization of polarizations of so-called ``restricted powers'' of the graded maximal ideal.A glimpse to most of the old and new results on very well-covered graphs from the viewpoint of commutative algebrahttps://zbmath.org/1535.130362024-07-05T15:31:27.292712Z"Kimura, K."https://zbmath.org/authors/?q=ai:kimura.kyouko"Pournaki, M. R."https://zbmath.org/authors/?q=ai:pournaki.mohammad-reza"Seyed Fakhari, S. A."https://zbmath.org/authors/?q=ai:seyed-fakhari.seyed-amin"Terai, N."https://zbmath.org/authors/?q=ai:terai.naoki"Yassemi, S."https://zbmath.org/authors/?q=ai:yassemi.siamakSummary: A very well-covered graph is a well-covered graph without isolated vertices such that the height of its edge ideal is half of the number of vertices. In this survey article, we gather together most of the old and new results on the edge and cover ideals of these graphs.On graded \(\mathbb{E}_\infty\)-rings and projective schemes in spectral algebraic geometryhttps://zbmath.org/1535.140082024-07-05T15:31:27.292712Z"Ohara, Mariko"https://zbmath.org/authors/?q=ai:ohara.mariko"Torii, Takeshi"https://zbmath.org/authors/?q=ai:torii.takeshiSummary: We introduce graded \(\mathbb{E}_\infty\)-rings and graded modules over them, and study their properties. We construct projective schemes associated to connective \(\mathbb{N}\)-graded \(\mathbb{E}_\infty\)-rings in spectral algebraic geometry. Under some finiteness conditions, we show that the \(\infty\)-category of almost perfect quasi-coherent sheaves over a spectral projective scheme \(\text{Proj}\,(A)\) associated to a connective \(\mathbb{N}\)-graded \(\mathbb{E}_\infty\)-ring \(A\) can be described in terms of \(\mathbb{Z}\)-graded \(A\)-modules.Motivic infinite loop spaceshttps://zbmath.org/1535.140572024-07-05T15:31:27.292712Z"Elmanto, Elden"https://zbmath.org/authors/?q=ai:elmanto.elden"Hoyois, Marc"https://zbmath.org/authors/?q=ai:hoyois.marc"Khan, Adeel A."https://zbmath.org/authors/?q=ai:khan.adeel-a"Sosnilo, Vladimir"https://zbmath.org/authors/?q=ai:sosnilo.vladimir-a"Yakerson, Maria"https://zbmath.org/authors/?q=ai:yakerson.mariaSummary: We prove a recognition principle for motivic infinite \(\mathsf{P}^1\)-loop spaces over a perfect field. This is achieved by developing a theory of \textit{framed motivic spaces}, which is a motivic analogue of the theory of \(\mathcal{E}_\infty \)-spaces. A framed motivic space is a motivic space equipped with transfers along finite syntomic morphisms with trivialized cotangent complex in \(K\)-theory. Our main result is that grouplike framed motivic spaces are equivalent to the full subcategory of motivic spectra generated under colimits by suspension spectra. As a consequence, we deduce some representability results for suspension spectra of smooth varieties, and in particular for the motivic sphere spectrum, in terms of Hilbert schemes of points in affine spaces.Triangulated categories of framed bispectra and framed motiveshttps://zbmath.org/1535.140582024-07-05T15:31:27.292712Z"Garkusha, G."https://zbmath.org/authors/?q=ai:garkusha.grigory"Panin, I."https://zbmath.org/authors/?q=ai:panin.ivanThe stable motivic homotopy theory \(SH(k)\) over a field k is obtained from the category of bispectra \(SH_{Nis}(k)\) by a left Bousfield localization on \(\{pr_X:X\times\mathbb{A}^1\to X\,X\in Sm/k\}\). It is a triangulated category and was first introduced by [\textit{F. Morel} and \textit{V. Voevodsky} [Publ. Math., Inst. Hautes Étud. Sci. 90, 45--143 (1999; Zbl 0983.14007); Doc. Math. Extra Vol., 579--604 (1998; Zbl 0907.19002)]. Because of the \(\mathbb{A}^1\)-localization one cannot control stable homotopy types of bispectra \(\mathcal{E}\) by their stable homotopy sheaves \(\pi_{*,*}^{Nis}(\mathcal{E})\) we should rather compute their stable motivic homotopy sheaves \(\pi_{*,*}^{\mathbb{A}^1}(\mathcal{E}):=\pi_{*,*}^{Nis}(L_{\mathbb{A}^1}\mathcal{E})\), where \(L_{\mathbb{A}^1}:SH_{Nis}(k)\to SH_{Nis}(k)\) is an \(\mathbb{A}^1\)-localization functor. In general, it is hard to compute the sheaves \(\pi_{*,*}^{\mathbb{A}^1}(\mathcal{E})\).
In the present paper the authors suggest an alternative approach to \(SH(k)\). This new approach is of local nature and does not involve any motivic equivalences. In particular it is more suitable for computations. More precisely, they introduce the triangulated category of framed bispectra \(SH^{fr}_{Nis}(k)\) based on their paper [\textit{G. Garkusha} and \textit{I. Panin}, J. Am. Math. Soc. 34, No. 1, 261--313 (2021; Zbl 1491.14034)] and show that there is an equivalence of categories \(F:SH^{fr}_{Nis}(k)\to SH(k)\) where \(F\) is the identity on objects. This is Theorem 2.4 of the present paper. In Corollary 2.9 they show that a morphism \(f:\mathcal{E}\to\mathcal{F}\) of framed bispectra is a stable motivic equivalence if and only if the induced morphisms of Nisnevich sheaves of stable homotopy groups \(f_*:\pi_*^{Nis}(\mathcal{E}(q))\to\pi_*^{Nis}(\mathcal{F}(q))\) are isomorphisms in each weight \(q\). Therefore they get rid of motivic equivalences.
In section 3 they introduce the triangulated category of effective framed bispectra \(SH_{Nis}^{fr,eff}(k)\). In [\textit{G. Garkusha} and \textit{I. Panin}, J. Am. Math. Soc. 34, No. 1, 261--313 (2021; Zbl 1491.14034)] they constructed the functor of big framed motives \(\mathcal{M}^b_{fr}\). They prove now in Corollary 3.8 that the big framed motive functor induces a triangle equivalence \(\mathcal{M}^b_{fr}:SH^{eff}(k)\to SH^{fr,eff}_{Nis}(k)\).
Furthermore the authors also defined in [\textit{G. Garkusha} and \textit{I. Panin}, J. Am. Math. Soc. 34, No. 1, 261--313 (2021; Zbl 1491.14034)] the framed motive \(M_{fr}(X)\) of a smooth algebraic variety \(X\in Sm/k\). It is a functor from smooth algebraic varieties to sheaves of framed \(\mathbb{A}^1\)-local \(S^1\)-spectra. In section 6 they show that the construction of \(M_{fr}(-)\) can be extended to all motivic bispectra. Especially, \(M_{fr}(X)\) is recovered as \(M^E_{fr}(X)\) where \(E\) is the motivic sphere bispectrum.
At the end of the paper the authors also introduce the triangulated category of framed motives \(SH^{fr}(k)\). In Theorem 7.2 they prove that \(SH^{eff}(k)\) is naturally triangle equivalent to \(SH^{fr}(k)\).
Reviewer: Xiaowen Dong (Osnabrück)Decomposition of pointwise finite-dimensional \(\mathbb{S}^1\) persistence moduleshttps://zbmath.org/1535.160192024-07-05T15:31:27.292712Z"Hanson, Eric J."https://zbmath.org/authors/?q=ai:hanson.eric-j"Rock, Job Daisie"https://zbmath.org/authors/?q=ai:rock.job-daisieFrom the authors' abstract: Persistent homology is an essential tool in topological data analysis. The authors ``prove that [...] pointwise finite-dimensional persistence modules indexed by \(\mathbb{S}^1\) decompose uniquely, up to isomorphism, into the direct sum of a bar code and finitely-many Jordan cells. In the language of representation theory, this is a direct sum of string modules and band modules. Persistence modules indexed on \(\mathbb{S}^1\) have also been called angle-valued or circular persistence modules''. Either a cyclic order or partial order on \(\mathbb{S}^1\) is allowed. The authors ``also show that a pointwise finite-dimensional \(\mathbb{S}^1\) persistence module is indecomposable if and only if it is a bar or Jordan cell''. Finally, the authors ``classify the isomorphism classes of such indecomposable modules''.
Reviewer: Mee Seong Im (Annapolis)On bialgebras, comodules, descent data and Thom spectra in \(\infty\)-categorieshttps://zbmath.org/1535.160412024-07-05T15:31:27.292712Z"Beardsley, Jonathan"https://zbmath.org/authors/?q=ai:beardsley.jonathanThe paper studies \(\infty\)-categorical coalgebras and bialgebras which are not necessarily commutative nor cocommutative as well as their \(\infty\)-categories of modules and comodules.
The authors establish several results for coalgebraic structure in \(\infty\)-categories, specifically with connections to the spectral noncommutative geometry of cobordism theories. It is proved 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.
The authors give two examples of higher coalgebraic structure. First, they prove that for a map of \(\mathbb{E}_n\)-ring spectra \(\phi: A \to B\), the associated \(\infty\)-category of descent data is equivalent to the \(\infty\)-category of comodules over \(B \otimes_A B\), the so-called descent coring. Secondly, they show that Thom spectra are canonically equipped with a highly structured comodule structure which is equivalent to the \(\infty\)-categorical Thom diagonal of
\textit{M. Ando} et al. [J. Topol. 7, No. 3, 869--893 (2014; Zbl 1312.55011)]. They also show that this highly structured diagonal decomposes the Thom isomorphism for an oriented Thom spectrum in the expected way indicating that Thom spectra are good examples of spectral noncommutative torsors.
Reviewer: Mahender Singh (S.A.S. Nagar)Regularity and stable ranges of \textbf{FI}-moduleshttps://zbmath.org/1535.180012024-07-05T15:31:27.292712Z"Bahran, Cihan"https://zbmath.org/authors/?q=ai:bahran.cihanLet \(FI\) denote the category whose objects are finite sets and whose morphisms are all injective maps. \textit{T. Church} et al. [Duke Math. J. 164, No. 9, 1833--1910 (2015; Zbl 1339.55004)] used the notation \(FI\) for this category as an acronym for \textbf{F}inite sets and \textbf{I}njective maps. The category has appeared in other contexts in algebraic topology, algebraic geometry, and computer science under various names.
The author gives refined bounds for the regularity of \(FI\)-modules and the stable ranges of \(FI\)-modules for various forms of their stabilization studied in the representation stability literature. The first bound, is in terms of the generation and presentation degrees of an \(FI\)-module \(V\). The second bound, is in terms of the local and stable degrees of \(V\) (in the sense of [\textit{T. Church} et al., Adv. Math. 333, 1--40 (2018; Zbl 1392.15030)]) which is also often sharp. Presented results are applied to get explicit stable ranges for diagonal coinvariant algebras and improve those for ordered configuration spaces of manifolds and congruence subgroups of general linear groups.
Reviewer: Marek Golasiński (Olsztyn)The spectrum of derived Mackey functorshttps://zbmath.org/1535.180242024-07-05T15:31:27.292712Z"Patchkoria, Irakli"https://zbmath.org/authors/?q=ai:patchkoria.irakli"Sanders, Beren"https://zbmath.org/authors/?q=ai:sanders.beren"Wimmer, Christian"https://zbmath.org/authors/?q=ai:wimmer.christianSummary: We compute the spectrum of the category of derived Mackey functors (in the sense of Kaledin) for all finite groups. We find that this space captures precisely the top and bottom layers (i.e. the height infinity and height zero parts) of the spectrum of the equivariant stable homotopy category. Due to this truncation of the chromatic information, we are able to obtain a complete description of the spectrum for all finite groups, despite our incomplete knowledge of the topology of the spectrum of the equivariant stable homotopy category. From a different point of view, we show that the spectrum of derived Mackey functors can be understood as the space obtained from the spectrum of the Burnside ring by ``ungluing'' closed points. In order to compute the spectrum, we provide a new description of Kaledin's category, as the derived category of an equivariant ring spectrum, which may be of independent interest. In fact, we clarify the relationship between several different categories, establishing symmetric monoidal equivalences and comparisons between the constructions of Kaledin, the spectral Mackey functors of Barwick, the ordinary derived category of Mackey functors, and categories of modules over certain equivariant ring spectra. We also illustrate an interesting feature of the ordinary derived category of Mackey functors that distinguishes it from other equivariant categories relating to the behavior of its geometric fixed points.Operads, operadic categories and the blob complexhttps://zbmath.org/1535.180322024-07-05T15:31:27.292712Z"Batanin, Michael"https://zbmath.org/authors/?q=ai:batanin.michael-a"Markl, Martin"https://zbmath.org/authors/?q=ai:markl.martinThe blob complex was introduced by \textit{S. Morrison} and \textit{K. Walker} [Geom. Topol. 16, No. 3, 1481--1607 (2012; Zbl 1280.57026)], associating to an -dimensional manifold \(\mathbb{M}\), rigged with a system of fields \(\mathcal{C}\)\ containing an ideal of local relations \(\mathcal{U}\), the \textit{blob complex} \(\mathcal{B}_{\ast}\left( \mathbb{M},\mathcal{C} \right) \), which is a chain complex whose salient feature is the isomorphism
\[
H_{0}\left( \mathcal{B}_{\ast}\left( \mathbb{M},\mathcal{C}\right) \right) \cong A\left( \mathbb{M}\right) :=\mathcal{C}\left( \mathbb{M}\right) /\mathcal{U}\left( \mathbb{M}\right)
\]
where \(A\left( \mathbb{M}\right) \)\ is the skein module associated to \(\mathbb{M}\).
This paper shows in two ways how to interpret the blob complex within operad theory. The first interpretation produces a complex quasi-isomorphic to the Morrison-Walker blob complex, namely, the bar construction of a certain operad of fields over an operadic category of blob configurations in \(\mathbb{M}\). The second interpretation identifies the blob complex with Fresse's bar construction of a traditional colored operad [\textit{B. Fresse}, Contemp. Math. 346, 115--215 (2004; Zbl 1077.18007)], reminiscent of the little discs operad, the colors being blobs in \(\mathbb{M}\)\ with boundaries decorated by fields.
As a by-product, the theory of unary operadic categories is developed while some novel and scintillating phenomena arising in this context are investigated.
Reviewer: Hirokazu Nishimura (Tsukuba)Models for knot spaces and Atiyah dualityhttps://zbmath.org/1535.180332024-07-05T15:31:27.292712Z"Moriya, Syunji"https://zbmath.org/authors/?q=ai:moriya.syunji\textit{D. P. Sinha} [J. Am. Math. Soc. 19, No. 2, 461--486 (2006; Zbl 1112.57004); Am. J. Math. 131, No. 4, 945--980 (2009; Zbl 1184.57023)] constructed cosimplicial models of spaces of knots in a manifold of dimension \(\geq4\), based on Goodwillie-Weise embedding calculus [\textit{T. G. Goodwillie} and \textit{J. R. Klein}, J. Topol. 8, No. 3, 651--674 (2015; Zbl 1329.57029); \textit{T. G. Goodwillie} and \textit{M. Weiss}, Geom. Topol. 3, 103--118 (1999; Zbl 0927.57028); \textit{M. Weiss}, Geom. Topol. 3, 67--101 (1999; Zbl 0927.57027); Geom. Topol. 15, No. 1, 407--409 (2011; Zbl 1208.57013)]. The model was crucially exploited in the affirmative solution to \textit{V. A. Vassiliev}'s conjecture [Transl. Math. Monogr. 185, 155--182 (1998; Zbl 0929.57004)] for the space of long knots in \(\mathbb{R}^{d}\)\ (\(d\geq4\)) for rational coefficients in [\textit{P. Lambrechts} et al., Geom. Topol. 14, No. 4, 2151--2187 (2010; Zbl 1222.57020)]. This paper investigates a version of Sinha's model in stable categories. \(M\)\ denotes a connected closed smooth manidold of dimension \(d\).
The synopsis of the paper goes as follows.
\begin{itemize}
\item[\S 2] introduces basic notions, defining of Sinha's cosimplicial model and showing that its homotopy limit is equivalent to the space \(\mathrm{Emb} \left( S^{1},M\right) \) of smooth embeddings \(S^{1}\rightarrow M\)\ rigged with the \(C^{\infty}\)-topology. The notions of a (co)module and Hochschild complex of a comodule over the associahedral operad are delineated.
\item[\S 3] is the technical heart of the paper, introduccing a version of Cohen's model of Thom spectra (called the \textit{Sinha's cosimplicial model} and denoted by \(\mathcal{C}^{\cdot}\left( M\right) \)) and using it to construct the comodule \(\mathcal{T}_{M}\)\ whose \(n^{\text{th}}\)\ object is a natural model of the Thom spectrum
\[
\Sigma^{-nK}\mathrm{Th}\left( \upsilon^{n}\right) /\mathrm{Th}\left( \upsilon^{n}\mid_{FD_{n}}\right)
\]
where
\begin{itemize}
\item \(\Sigma\) denotes the suspension equivalence and \(\mathrm{Th}\left( -\right) \)\ denotes the associated Thom space,
\item \(FD_{n}\) is the preimage of the fat diagonal by (the product of) the projections \(SM^{n}\rightarrow M^{n}\), and
\item \(\upsilon^{n}\mid_{FD_{n}}\) denotes the restriction of the base to \(FD_{n}\).
\end{itemize}
\item[\S 4] establishes Theorem 1.1 claiming a dual equivalence between the configuration spaces and quotients by a fat diagonal
\[
\left( \mathcal{C}_{M}\right) ^{\vee}\simeq\mathcal{T}_{M}
\]
where \(\left( \mathcal{C}_{M}\right) ^{\vee}\)\ is a comodule whose \(n^{\text{th}}\)\ object is the Spanier-Whitehead dual [\textit{E. H. Spanier} and \textit{J. H. C. Whitehead}, Proc. Natl. Acad. Sci. USA 39, 655--660 (1953; Zbl 0051.14001); Mathematika 2, 56--80 (1955; Zbl 0064.17202); Ann. Math. (2) 67, 203--238 (1958; Zbl 0092.15701)] of the configuration space of \(n\)\ points with a tangent vector in \(M\).
\item[\S 5] defines a chain functor for symmetric spectra and constructs the spectral sequence filtering Hochschild complex of the chains of a resolution of the comodule \(\mathcal{T}_{M}\). It is established that the \(E_{1}\)-page of the Čech spectral sequence is quasi-isomorphic to the total complex of a simplicial differential bigraded algebra. The convergence of the Čech spectral sequence is demonstrated.
\item[\S 6] computes the cohomology ring \(H^{\ast}\left( D_{G}\right) \)\ and maps between them, giving a description of the simplicial algebra in terms of the cohomology ring \(H^{\ast}\left( M\right) \)\ under some assumptions. The computation is standard after Serre spectral sequences.
\item[\S 7] computes examples, establishing under some assumptions that the inclusion
\[
i_{M}:\mathrm{Emb}\left( S^{1},M\right) \rightarrow\mathrm{Imm}\left( S^{1},M\right)
\]
with \(\mathrm{Imm}\left( S^{1},M\right) \)\ the space of smooth immersions \(S^{1}\rightarrow M\)\ with the \(C^{\infty}\)-topology induces an isomorphism on \(\pi_{1}\).
\item[\S 8] resolves Sinha's cosimplicial model, considered as a resolution of \(\mathrm{Emb}\left( S^{1},M\right) \)\ into simpler spaces, further simpler pieces in the category of chain complexes.
\end{itemize}
Reviewer: Hirokazu Nishimura (Tsukuba)From gravity to string topologyhttps://zbmath.org/1535.180342024-07-05T15:31:27.292712Z"Merkulov, Sergei A."https://zbmath.org/authors/?q=ai:merkulov.sergei-aThis article concerns universal operations defined on the \(S^1\)-equivariant homology of the free loop space \(LM\) of \(d\)-manifolds \(M\), denoted \(H^{S^1}(LM)\). The study of such operations, called string topology, goes back to at least Chas-Sullivan [\textit{M. Chas} and \textit{D. Sullivan}, ``String topology'', Preprint, \url{arXiv:math/9911159}]. The operations built by Chas-Sullivan were shown by \textit{C. Westerland} [Math. Ann. 340, No. 1, 97--142 (2007; Zbl 1141.55002)] to produce an action of Getzler's gravity operad \(\mathrm{grav}_{3-d}\), which is built from the shifted homology of moduli spaces of Riemann surfaces of genus \(0\). Moreover, \textit{M. Chas} and \textit{D. Sullivan} [in: The legacy of Niels Henrik Abel. Papers from the Abel bicentennial conference, University of Oslo, Oslo, Norway, June 3--8, 2002. Berlin: Springer. 771--784 (2004; Zbl 1068.55009)] also produced a geometric construction for the involutive \((3-d)\)-Lie bialgebra structure on \(H^{S^1}(LM)\).
For a simply connected closed \(d\)-manifold \(M\), there is an isomorphism between \(\bar{H}^{S^1}(LM)\) and the reduced cyclic Hochschild cohomology \(H(\overline{\mathrm{Cyc}}(\bar{A}^*[-1]))\) of a Poincaré duality model of \(M\). The author built [``Gravity prop and moduli spaces $\mathcal{M}_{g,n}$'', Preprint, \url{arXiv:2108.10644}] a chain-level gravity properad \(\mathrm{ChGrav}_d\), whose cohomology is given the compactly supported cohomology of moduli spaces of Riemann surfaces with marked points divided into inputs and outputs. This properad naturally acts of cyclic Hochschild cohomology of Poincaré duality algebras, which thus produces an action of \(\mathrm{ChGrav}_{3-d}\) on \(\bar{H}^{S^1}(LM)\).
The author proves that elements of \(\mathrm{ChGrav}_{3-d}\) represented by ribbon graphs with at least one \((\ge4)\)-valent black vertex or at least one boundary consisting of black vertex act trivially on \(\bar{H}^{S^1}(LM)\). This thus motivates the introduction of the quotient \(\mathcal{ST}_{3-d}\) (dubbed ``string topology properad'' in this article) of \(\mathrm{ChGrav}_{3-d}\) by the dg-properadic ideal generated by such ribbon graphs. The author eventually proves that the action of \(\mathcal{ST}_{3-d}\) on \(\bar{H}^{S^1}(LM)\) recover, at least conjectually, previously known operations: (1) the shifted Lie bialgebra structure of Chas-Sullivan; (2) an action of \(\mathrm{grav}_{3-d}\) which conjecturally coindices with that of Westerland; (3) a set of four operations described by the author [``Gravity prop and moduli spaces $\mathcal{M}_{g,n}$'', Preprint, \url{arXiv:2108.10644}] that produce an action of the homotopy Lie bialgebra properad.
Reviewer: Najib Idrissi (Paris)Homotopy (co)limits via homotopy (co)ends in general combinatorial model categorieshttps://zbmath.org/1535.180372024-07-05T15:31:27.292712Z"Arkhipov, Sergey"https://zbmath.org/authors/?q=ai:arkhipov.sergey"Ørsted, Sebastian"https://zbmath.org/authors/?q=ai:orsted.sebastianGiven a model category \(\mathcal{C}\) and a (small) category \(\Gamma\), write \(\mathcal{C}^\Gamma\) for the category of functors \(\Gamma\to\mathcal{C}\). The inverse limit functor \(\lim\limits_{\gets} : \mathcal{C}^\Gamma\to\mathcal{C}\), if it exists, is defined as a right adjoint of this trivial functor \(\mathcal{C}\to\mathcal{C}^\Gamma.\) The right derived functor of \(\lim\limits_{\gets}\) is called \textit{the homotopy limit}. Dually, the left derived functor of the direct limit functor \(\lim\limits_{\to}\) is called the \textit{homotopy colimit}.
One of the most popular techniques involves adding a parameter to the limit functor \(\lim\limits_{\gets}\) before deriving it. The result is the end bifunctor \(\int_\Gamma : \Gamma^{op}\times\Gamma\to \mathcal{C}\) which is in general much easier to derive. The classical accounts of this technique is the result by \textit{P. S. Hirschhorn} [Model categories and their localizations. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1017.55001)] who mainly works in the setting of simplicial model categories.
The authors prove and explain several classical formulae for homotopy (co)limits in general (combinatorial) model categories which are not necessarily simplicially enriched. Versions of the Bousfield-Kan formula and the fat totalization formula in this complete generality are discussed. Furthermore, a proof that homotopy-final functors preserve homotopy limits, again in complete generality is presented. At the end, a proof that homotopy-final functors preserve homotopy limits, again in complete generality is presented.
Reviewer: Marek Golasiński (Olsztyn)Composition closed premodel structures and the Kreweras latticehttps://zbmath.org/1535.180382024-07-05T15:31:27.292712Z"Balchin, Scott"https://zbmath.org/authors/?q=ai:balchin.scott"MacBrough, Ethan"https://zbmath.org/authors/?q=ai:macbrough.ethan"Ormsby, Kyle"https://zbmath.org/authors/?q=ai:ormsby.kyle-mA \textit{premodel structure} consists of a category \(\mathcal{C}\) with all finite limits and colimits equipped with two weak factorization systems \((\mathcal{L},\mathcal{R})\) and \((\mathcal{L}',\mathcal{R}')\) such that \(\mathcal{R}\subset \mathcal{R}'\) (equivalently \(\mathcal{L}' \subset \mathcal{L}\)). The weak equivalences of a premodel structure is the class of maps \(\mathcal{W} := \mathcal{R}\circ \mathcal{L}'\). A premodel structure is \textit{composition closed} if \(\mathcal{W}\) is closed under composition. A \textit{Quillen model structure} is a premodel structure such that \(\mathcal{W}\) satisfies the two-out-of-three property. The paper delves into the intricate combinatorial structure of premodel structures on finite lattices.
The authors establish that, on a finite lattice \(L\), there is a refinement of the ordering on weak factorization systems on \(L\) such that the intervals are exactly the composition closed premodel structures. Moreover, the set of weak factorization systems equipped with this ordering is a finite lattice. Then in the specific case of the lattice \([n]\) with \(n\geq 0\), they prove that the composition closed premodel structures are in bijection with intervals of the Kreweras lattice. Finally, the authors prove that there is an explicit bijection between certain tricolored trees and Kreweras pairs under which a tricolored tree represents a model structure if and only if it does not have a red branch descended from any non-red branch.
Reviewer: Philippe Gaucher (Paris)Synthetic fibered \((\infty,1)\)-category theoryhttps://zbmath.org/1535.180392024-07-05T15:31:27.292712Z"Buchholtz, Ulrik"https://zbmath.org/authors/?q=ai:buchholtz.ulrik"Weinberger, Jonathan"https://zbmath.org/authors/?q=ai:weinberger.jonathan-maximilian-lajos|weinberger.jonathanThe paper is a well-written and fairly self-contained development of the theory of (co)cartesian fibrations inside the simplicial type theory of ``synthetic'' \((\infty,1)\)-categories introduced in [\textit{E. Riehl} and \textit{M. Shulman}, High. Struct. 1, No. 1, 147--224 (2017; Zbl 1437.18016)].
Riehl and Shulman define in loc.\ cit.\ a ``simplicial homotopy type theory'' which extends the homotopy type theory of [\textit{The Univalent Foundations Program}, Homotopy type theory. Univalent foundations of mathematics. Princeton, NJ: Institute for Advanced Study; Raleigh, NC: Lulu Press (2013; Zbl 1298.03002)] essentially by an interval type \(\Delta^1\) together with various subdivision operators that can be applied to cartesian powers of \(\Delta^1\). This additional layer of ``tope'' structure enables to discuss simplicial shapes as well as extensions of such inside the type theory, and ultimately to express locality conditions with respect to them. The central objects of this syntax are Rezk types, i.e.\ types which ought to be thought of ``synthetic'' complete Segal spaces (or, more generally, as synthetic complete Segal objects in an \((\infty,1)\)-topos). Indeed, the type theory does come equipped with a formal interpretation in the \(\infty\)-category of complete Segal spaces such that Rezk types do correspond exactly to complete Segal spaces. It is hence possible to formalize various fragments of the practice of \((\infty,1)\)-category theory ``synthetically'' within this type theory. For instance , Riehl and Shulman recover various fundamental results about adjunctions and of left fibrations therein.
The main objective of the paper at hand is to develop a theory of (co)cartesian fibrations inside this type theory. Therefore, it takes the theory of (co)cartesian fibrations inside an \(\infty\)-cosmos of \((\infty,1)\)-categories as developed in Chapter 5 of \textit{E. Riehl} and \textit{D. Verity} [Elements of \(\infty\)-category theory. Cambridge: Cambridge University Press (2022; Zbl 1492.18001)] as a blueprint, and shows that a host of results can be recovered in this type theory when stated accordingly respective (co)cartesian fibrations between (Rezk) types. However, a corresponding semantics is not formally introduced, albeit briefly discussed in Section 2.1.3.
Section 1 is a clearly arranged introduction. Section 2 is a concise exposition of the simplicial type theory of Riehl and Shulman. Among others, it defines Segal types, Rezk types, and covariant type families over Segal types (say, left fibrations of Rezk types). Section 3 defines and studies the concepts of right orthogonal families and LARI-families with respect to a type map \(j\), the latter of which is a weakening of right orthogonality. Section 4 introduces (iso)inner families paralleling (complete) Segal fibrations between (complete) Segal spaces. Therefore, the authors introduce a type that synthesizes the free bi-invertible arrow \(\mathbb{E}\), and hence characterize Rezk types and isoinner families (among all inner families) in terms of a right orthogonality condition respectively the endpoint inclusion into \(\mathbb{E}\). Section 5 contains the definitions of cocartesian arrows, cocartesian families and cocartesian functors between such, closely following Chapter 5 of Riehl and Verity's book. The authors give two equivalent characterizations of cocartesian families over Rezk types -- first, as the isoinner LARI-families with respect to the endpoint inclusion \(\ast\rightarrow\Delta^1\) (Theorem 5.2.6), and second, as the isoinner families with functorial transport (Theorem 5.2.7). The authors furthermore discuss some fundamental examples of cocartesian families, and show that the collection of cocartesian families is closed under various categorical constructions (e.g.\ Proposition 5.3.17). In Section 6, the authors relate the results about cocartesian families from Section 5 to the results about covariant families shown in Riehl and Shulman's paper. Among others, they characterize covariant families over Segal types as exactly the (fiberwise) discrete cocartesian families. In reference to Proposition 5.3.17, the authors then show that the collection of covariant families is closed under various categorical constructions as well (Proposition 6.2.1). Section 7 is primarily concerned with expressions of the Yoneda lemma as stated in Section 5.7 of Riehl and Verity's book, and with recovering the Yoneda lemma for covariant families over Rezk types from Section 9 of Riehl and Shulman's paper as a special case.
Reviewer: Raffael Stenzel (Bonn)Localizations and completions of stable \(\infty\)-categorieshttps://zbmath.org/1535.180402024-07-05T15:31:27.292712Z"Mantovani, Lorenzo"https://zbmath.org/authors/?q=ai:mantovani.lorenzoLet \(\mathcal{M}\)\ be a presentably symmetric monoidal \(\infty\)-category, with monoidal product \(\wedge\)\ and unit \(\boldsymbol{1}\), and let \(E\)\ be an object of \(\mathcal{M}\). One can construct a homology localization of \(\mathcal{M}\)\ by inverting all the maps \(\varphi\)\ in \(\mathcal{M}\)\ such that \(\varphi\wedge E\) is an equivalence, which was first introduced for the topological stable category \(\mathcal{SH}\)\ by \textit{A. K. Bousfield} [Topology 18, 257--281 (1979; Zbl 0417.55007)]. The associated localization functor \(X\mapsto X_{E}\)\ is called \(E\)-homology localization, which has a particularly simple universal property. If \(E\)\ is a commutative algebra object in \(\mathcal{M}\), given any object \(X\in \mathcal{M}\), one can perform a second construction \(X_{\widehat{E}}\), called the nilpotent completion of \(X\)\ at \(E\),\ by setting
\[
X_{\widehat{E}}:=\mathrm{lim}_{\Delta}\mathrm{\,X\wedge E}^{\wedge\ast+1}
\]
\textit{A. K. Bousfield} [Topology 18, 257--281 (1979; Zbl 0417.55007)] has established the following theorem.
Theorem. When \(E\)\ is a \(\left( -1\right) \)-connected commutative algebra in \(\mathcal{SH}\), with \(\pi_{0}\left( E\right) \simeq\mathbb{Z}/n\mathbb{Z}\), then for every bounded-below spectrum \(X\)\ the natural map \(X_{\widehat{E} }\rightarrow X_{\widehat{E}}\)\ is an equivalence. Furthermore, under some assumptions, \(X_{E}\)\ is naturally identified with the derived completion
\[
X_{\widehat{n}}=\lim_{k}\,X/n^{k}
\]
This paper axiomatizes some of the techniques of Bousfield, adapting them to work in a presentably symmetric monoidal stable \(\infty\)-category \(\mathcal{M}\). The author aims to reach a formal analogue for \(\mathcal{M} \)\ of the theorem of Bousfield. The main assumption on \(\mathcal{M}\)\ is that it comes rigged with a \(t\)-structure which has the following two properties.
\begin{itemize}
\item[(1)] the \(t\)-structure is left-complete, i.e.,
\[
X\simeq\mathrm{lim}_{n}P^{n}\left( X\right)
\]
for all \(X\in\mathcal{M}\);
\item[(2)] the \(t\)-structure is multiplicative, i.e.,
\[
\boldsymbol{1}\in\mathcal{M}_{\geq0}
\]
and
\[
\mathcal{M}_{\geq p}\wedge\mathcal{M}_{\geq q}\subseteq\mathcal{M}_{\geq p+q}
\]
\end{itemize}
The main application the author has in mind is motivic homotopy theory. In order to work with an abstract symmetric monoidal \(\infty\)-category \(\mathcal{M}\), the author chooses to axiomatize the elements \(\pi_{0}\left( \mathbb{S}\right) \simeq\mathbb{Z}\) in terms of maps \(L\rightarrow \boldsymbol{1}\), where \(L\)\ is a -invertible object of \(\mathcal{M}\)\ such that \(L\wedge-\)\ respects both \(\mathcal{M}_{\geq0}\)\ and \(\mathcal{M}_{\leq 0}\). Objects satisfying this property are called \textit{tif} objects (tif stands for ``tensor invertible and flat'').
In their recent work [Forum Math. Sigma 10, Paper No. e1, 27 p. (2022; Zbl 1490.14033)], \textit{T. Bachmann} and \textit{P. A. Østvær} have generalized and streamlined the arguments of a previous version of this paper [\textit{L. Mantovani}, ``Localizations and completions in motivic homotopy theory'', Preprint, \url{arXiv:1810.04134}], whose results are phrased for the motivic stable homotopy category \(h\mathcal{SH}\left( K\right) \)\ of a perfect field \(K\). However, the structure and the arguments of this paper are essentially the same as the previous version, so that the present paper and the second section of Bachmann and Østvær's\ present similar results with similar techniques, though Bachmann and Østvær's\ has a more direct and a simpler approach.
The synopsis of the paper goes as follows.
\begin{itemize}
\item[\S 2] recollects some well-known facts about the motivic stable category \(\mathcal{SH}\left( S\right) \) and about the categories of modules \(\mathcal{M}od_{A}\left( S\right) \)\ over a commutative algebra \(A\in\mathcal{SH}\left( S\right) _{\geq0}\).
\item[\S 3 and \S 4] establishes the following theorem
Theorem 4.3.7 and Proposition 3.2.15. In the special case of \(J=\phi\), for every bounded-below object \(X\)\ in \(\mathcal{M}\),\ there is a canonical equivalence
\[
X_{E}\simeq X_{\widehat{f_{1}}}\cdots X_{\widehat{f_{r}}}
\]
compatible with the localization map
\[
\lambda_{E}\left( X\right) :X\rightarrow X_{E}
\]
and the formal completion map
\[
\chi_{\underline{f}}\left( X\right) :X\rightarrow X_{\widehat{f_{1}}}\cdots X_{\widehat{f_{r}}}
\]
under the following assumption on \(E\).
\item[Assumption 4.2.1] Let \(E\)\ be a homotopy commutative algebra in \(\mathcal{M}\). Furthermore, there exists a finite set of tif objects \(\left\{ L_{i}\right\} _{i=1}^{r}\)\ and maps \(f_{i}:L_{i}\rightarrow\boldsymbol{1} \)\ such that the unit \(\boldsymbol{1}\rightarrow E\)\ induces an isomorphism
\[
\tau_{0}\left( \boldsymbol{1}\right) /\left( f_{1},\ldots,f_{r}\right) \simeq\tau_{0}\left( E\right)
\]
\item[\S 5] contains a list of relevant examples and applications in the motivic setting, partially recovering a conservativity result for motives of \textit{T. Bachmann} [Duke Math. J. 167, No. 8, 1525--1571 (2018; Zbl 1390.14064)].
\item[\S 6] gives a construction of \(E\)-nilpotent completions using an axiomatized version of the Adams tower, and a construction of the associated spectral sequence.
\item[\S 7] contains the axiomatization of nilpotent resolutions and a general proof of some of their properties. Using these, the author provides a universal property for the Adams tower as a pro-object. Comparing the pro-objects associated with the Adams tower and with the formal completion, the author obtains the following theorem, though the.result has already appeared in [\textit{P. Hu} et al., J. \(K\)-Theory 7, No. 3, 573--596 (2011; Zbl 1309.14018)].
Theorems 7.3.5 and 7.3.9. Let \(E\)\ be a homotopy commutative algebra in \(\mathcal{M}\)\ abiding by Assumption 4.2.1\ in the special case where either \(J=\phi\)\ or \(I=\phi\). Then for every bounded-below object \(X\)\ in \(\mathcal{M}\)\ the natural map
\[
X_{E}\rightarrow X_{\widehat{E}}
\]
is an equivalence in \(\mathcal{M}\).
\end{itemize}
Reviewer: Hirokazu Nishimura (Tsukuba)Multiplicative equivariant \(K\)-theory and the Barratt-Priddy-Quillen theoremhttps://zbmath.org/1535.190012024-07-05T15:31:27.292712Z"Guillou, Bertrand J."https://zbmath.org/authors/?q=ai:guillou.bertrand-j"May, J. Peter"https://zbmath.org/authors/?q=ai:may.j-peter"Merling, Mona"https://zbmath.org/authors/?q=ai:merling.mona"Osorno, Angélica M."https://zbmath.org/authors/?q=ai:osorno.angelica-mSummary: We prove a multiplicative version of the equivariant Barratt-Priddy-Quillen theorem, starting from the additive version proven in [\textit{B. J. Guillou} and \textit{J. P. May}, Algebr. Geom. Topol. 17, No. 6, 3259--3339 (2017; Zbl 1394.55008)]. The proof uses a multiplicative elaboration of an additive equivariant infinite loop space machine that manufactures orthogonal \(G\)-spectra from symmetric monoidal \(G\)-categories. The new machine produces highly structured associative ring and module \(G\)-spectra from appropriate multiplicative input. It relies on new operadic multicategories that are of considerable independent interest and are defined in a general, not necessarily equivariant or topological, context. Most of our work is focused on constructing and comparing them. We construct a multifunctor from the multicategory of symmetric monoidal \(G\)-categories to the multicategory of orthogonal \(G\)-spectra. With this machinery in place, we prove that the equivariant BPQ theorem can be lifted to a multiplicative equivalence. That is the heart of what is needed for the presheaf reconstruction of the category of \(G\)-spectra in [\textit{B. Guillou} and \textit{J. P. May}, ``Models of G-spectra as presheaves of spectra'', Preprint, \url{arXiv:1110.3571}].Algorithms for twisted conjugacy classes of polycyclic-by-finite groupshttps://zbmath.org/1535.200042024-07-05T15:31:27.292712Z"Dekimpe, Karel"https://zbmath.org/authors/?q=ai:dekimpe.karel"Tertooy, Sam"https://zbmath.org/authors/?q=ai:tertooy.samSummary: We construct two practical algorithms for twisted conjugacy classes of polycyclic groups. The first algorithm determines whether two elements of a group are twisted conjugate for two given endomorphisms, under the condition that their Reidemeister coincidence number is finite. The second algorithm determines representatives of the Reidemeister coincidence classes of two endomorphisms if their Reidemeister coincidence number is finite, or returns ``false'' if this number is infinite. We also discuss a theoretical extension of these algorithms to polycyclic-by-finite groups.Irreducible representations of the symmetric groups from slash homologies of \(p\)-complexeshttps://zbmath.org/1535.200752024-07-05T15:31:27.292712Z"Chan, Aaron"https://zbmath.org/authors/?q=ai:chan.aaron-chun-shing|chan.aaron-c-s"Wong, William"https://zbmath.org/authors/?q=ai:wong.william-p|wong.william-w-l|wong.william-c-mSummary: In the 40s, Mayer introduced a construction of (simplicial) \(p\)-complex by using the unsigned boundary map and taking coefficients of chains modulo \(p\). We look at such a \(p\)-complex associated to an \((n-1)\)-simplex; in which case, this is also a \(p\)-complex of representations of the symmetric group of rank \(n\) -- specifically, of permutation modules associated to two-row compositions. In this article, we calculate the so-called slash homology -- a homology theory introduced by \textit{M. Khovanov} and \textit{Y. Qi} [Quantum Topol. 6, No. 2, 185--311 (2015; Zbl 1352.81038)] -- of such a \(p\)-complex. We show that every non-trivial slash homology group appears as an irreducible representation associated to two-row partitions, and how this calculation leads to a basis of these irreducible representations given by the so-called \(p\)-standard tableaux.Nonrealizability of certain representations in fusion systemshttps://zbmath.org/1535.201072024-07-05T15:31:27.292712Z"Oliver, Bob"https://zbmath.org/authors/?q=ai:oliver.bobLet \(A\) be a finite abelian group. If \(\Gamma \leq \mathrm{Aut}(A)\), then the pair \((\Gamma,A)\) is fusion realizable if there is a saturated fusion system \(\mathcal{F}\) over a finite \(p\)-group \(S \geq A\) such that \(C_{S}(A)=A\), \(\mathrm{Aut}_{\mathcal{F}}(A)=\Gamma\) as subgroups of \(\mathrm{Aut}(A)\), and \(A \not \trianglelefteq \mathcal{F}\).
In the paper under review, the author develops tools to show that certain representations are not fusion realizable in this sense. In particular, if \(p=2\) or \(3\) and \(\Gamma\) is one of the Mathieu groups, he proves that the only \(\mathbb{F}_{p}\Gamma\)-modules that are fusion realizable (up to extensions by trivial modules) are the Todd modules and in some cases, their duals.
Reviewer: Enrico Jabara (Venezia)Characterising actions on trees yielding non-trivial quasimorphismshttps://zbmath.org/1535.201372024-07-05T15:31:27.292712Z"Iozzi, Alessandra"https://zbmath.org/authors/?q=ai:iozzi.alessandra"Pagliantini, Cristina"https://zbmath.org/authors/?q=ai:pagliantini.cristina"Sisto, Alessandro"https://zbmath.org/authors/?q=ai:sisto.alessandroSummary: Using a cocycle defined by \textit{N. Monod} and \textit{Y. Shalom} [J. Differ. Geom. 67, No. 3, 395--455 (2004; Zbl 1127.53035)] we introduce the \textit{median} quasimorphisms for groups acting on trees. Then we characterise actions on trees that give rise to non-trivial median quasimorphisms. Roughly speaking, either the action is highly transitive on geodesics, or it fixes a point in the boundary, or there exists an infinite family of non-trivial median quasimorphisms. In particular, in the last case the second bounded cohomology of the group is infinite dimensional as a vector space. As an application, we show that a cocompact lattice in the automorphism group of a product of trees has only trivial quasimorphisms if and only if the closures of the projections on each of the two factors are locally \(\infty\)-transitive.Green equivalences in equivariant mathematicshttps://zbmath.org/1535.202632024-07-05T15:31:27.292712Z"Balmer, Paul"https://zbmath.org/authors/?q=ai:balmer.paul"Dell'Ambrogio, Ivo"https://zbmath.org/authors/?q=ai:dellambrogio.ivoSummary: We establish Green equivalences for all Mackey 2-functors, without assuming Krull-Schmidt. By running through the examples of Mackey 2-functors, we recover all variants of the Green equivalence and Green correspondence known in representation theory and obtain new ones in several other contexts. Such applications include equivariant stable homotopy theory in topology and equivariant sheaves in geometry.Homotopy patterns in group theoryhttps://zbmath.org/1535.202682024-07-05T15:31:27.292712Z"Mikhailov, Roman"https://zbmath.org/authors/?q=ai:mikhailov.romanSummary: This is a survey. The main subject of this survey is the homotopical or homological nature of certain structures which appear in classical problems about groups, Lie rings and group rings. It is well known that the (generalized) dimension subgroups have complicated combinatorial theories. In this paper we show that, in certain cases, the complexity of these theories is based on homotopy theory. The derived functors of nonadditive functors, homotopy groups of spheres, group homology, etc., appear naturally in problems formulated in purely group-theoretical terms. The variety of structures appearing in the considered context is very rich. In order to illustrate it, we present this survey as a trip passing through examples having a similar nature.
For the entire collection see [Zbl 1532.00038].On the \(\Sigma \)-invariants of Artin groups satisfying the \(K (\pi, 1)\)-conjecturehttps://zbmath.org/1535.202712024-07-05T15:31:27.292712Z"Escartín-Ferrer, Marcos"https://zbmath.org/authors/?q=ai:escartin-ferrer.marcos"Martínez-Perez, Conchita"https://zbmath.org/authors/?q=ai:martinez-perez.conchitaSummary: We consider \(\Sigma \)-invariants of Artin groups that satisfy the \(K (\pi, 1)\)-conjecture. These invariants determine the cohomological finiteness conditions of subgroups that contain the derived subgroup. We extend a known result for even Artin groups of FC-type, giving a sufficient condition for a character \(\chi : A_{\Gamma} \to \mathbb{R}\) to belong to \(\Sigma^{n} (A_{\Gamma}, \mathbb{Z})\). We also prove some partial converses. As applications, we prove that the \(\Sigma^{1}\)-conjecture holds true when there is a prime \(p\) that divides \(l(e) / 2\) for any edge with even label \(l(e) > 2\), we generalize to Artin groups the homological version of the Bestvina-Brady theorem and we compute the \(\Sigma\)-invariants of all irreducible spherical and affine Artin groups and triangle Artin groups, which provide a complete classification of the \(F_{n}\) and \(FP_{n}\) properties of their derived subgroup.
{\copyright} 2024 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.Mod-two cohomology rings of alternating groupshttps://zbmath.org/1535.202722024-07-05T15:31:27.292712Z"Giusti, Chad"https://zbmath.org/authors/?q=ai:giusti.chad"Sinha, Dev"https://zbmath.org/authors/?q=ai:sinha.dev-pSummary: We calculate the direct sum of the mod-two cohomology of all alternating groups, with both cup and transfer product structures, which in particular determines the additive structure and ring structure of the cohomology of individual groups. We show that there are no nilpotent elements in the cohomology rings of individual alternating groups. We calculate the action of the Steenrod algebra and discuss individual component rings. A range of techniques are developed, including an almost Hopf ring structure associated to the embeddings of products of alternating groups and Fox-Neuwirth resolutions, which are new techniques. We also extend understanding of the Gysin sequence relating the cohomology of alternating groups to that of symmetric groups and calculation of restriction to elementary abelian subgroups.Cohomology classes of the \(Qd (p)\) groupshttps://zbmath.org/1535.202742024-07-05T15:31:27.292712Z"Long, Jane H."https://zbmath.org/authors/?q=ai:long.jane-holsappleSummary: The \(Qd (p)\) groups arise in the study of a well-known conjecture by \textit{D. Benson} and \textit{J. Carlson} [Math. Z. 195, 221--238 (1987; Zbl 0593.20062)], stating that a finite group of rank \(n\) acts freely on a finite complex homotopy equivalent to a product of \(n\) spheres. These rank-2 groups are noteworthy in that they lack a particular cohomology class, an effective Euler class, which is used to construct free actions on products of two spheres. The present work describes the modular and integral cohomology of \(Qd (p)\) in terms of the cohomology of its Sylow-\(p\) subgroup, the extra-special \(p\)-group of order \(p^3\) and exponent \(p\), for \(p>3\).Cohomology classes of \(Qd(3)\)https://zbmath.org/1535.202752024-07-05T15:31:27.292712Z"Long, Jane H."https://zbmath.org/authors/?q=ai:long.jane-holsappleSummary: The modular and integral cohomologies of \(Qd(3)\), the special affine group over \(\mathbb{F}_3^2\), are described in terms of the cohomology of its Sylow-3 subgroup, \(P_3\), the extra-special 3-group of order \(3^3\) and exponent 3. \(Qd(3)\) is one of the rank-2 finite groups for which a free action on a finite complex homotopy equivalent to \(S^{m_1} \times S^{m_2}\) has not been constructed. Differences in the cohomology of \(Qd(3)\) versus \(Qd (p)\) for \(p>3\) are briefly discussed.How is algebraic topology algebraic?https://zbmath.org/1535.550012024-07-05T15:31:27.292712Z"Aguadé i Bover, Jaume"https://zbmath.org/authors/?q=ai:aguade-i-bover.jaume(no abstract)Variations on a theme of Borel. An essay on the role of the fundamental group in rigidityhttps://zbmath.org/1535.550022024-07-05T15:31:27.292712Z"Weinberger, Shmuel"https://zbmath.org/authors/?q=ai:weinberger.shmuelThe Borel conjecture asserts that aspherical closed manifolds are topologically rigid, that is, every homotopy equivalence between such manifolds is homotopic to a homeomorphism. The book studies various rigidity aspects of aspherical manifolds by looking at variations of the Borel conjecture. This leads the author to give a concise treatment of different aspects in many different areas including algebraic geometry, controlled topology, algebraic \(K\)-theory and non-commutative geometry.
The 1st chapter is a short introduction to geometric rigidity and the Borel conjecture.
In the 2nd chapter, different constructions of aspherical manifolds are reviewed. This chapter starts with looking at low dimensional examples, in particular 3-manifolds, and then continues with one of the main constructions of (closed) aspherical manifolds. Namely, (uniform) lattices in Lie groups. Here examples of both arithmetic and non-arithmetic constructions are reviewed. The chapter ends with looking at more exotic constructions of aspherical manifolds like Davis's reflection group method and hyperbolization.
The 3rd chapter discusses the failure of rigidity for quotients by non-uniform lattices. This starts with explaining some aspects of \(K\backslash G/ \Gamma\), where \(K\) is the maximal compact subgroup of a connected Lie group \(G\) and \(\Gamma\) is a non-uniform lattice. In particular, the Borel-Serre classification is explained. In Section 3.3, some surgery theory is reviewed to give a result about non-rigidity in the non-compact setting: for \(M=K\backslash G/ \Gamma\) of dimension greater than five and \(\mathbb{Q}-\mathrm{rank}(\Gamma)\geq 3\), there are infinitely many smooth manifolds proper homotopy-equivalent to \(M\) that are not homeomorphic to \(M\) if, for some \(i\), \(H^{4i}(M;\mathbb{Q})\neq 0\). These examples are detected by their rational Pontryagin classes. Motivated by this, the cohomology of lattices is studied to show the existence of non-rigid manifolds. For this different aspects like Property (T) and \(L^2\)-cohomology are discussed.
In the 4th chapter, the Novikov conjecture is discussed. It implies that the rational Pontryagin classes are a homotopy-invariant in the case of closed aspherical manifolds. This chapter starts with Hirzebruch's signature theorem and then discusses the Novikov conjecture. The chapter continues with the discussion of controlled topology and further results in surgery theory. This relates the Borel conjecture as well as the Novikov conjecture to \(L\)-theoretic assembly maps.
The 5th chapter continues the study of the Novikov conjecture this time focussing on index theory. Furthermore, assembly maps for topological and algebraic \(K\)-theory are discussed.
In the 6th chapter, an equivariant version of the Borel conjecture is discussed. At the start of this chapter, \(h\)-cobordisms and Cappell's UNil groups are discussed. In Section 6.5, some nontrivial examples of the equivariant Borel conjecture are discussed and in particular counterexamples are constructed by using the UNil groups. In Section 6.6, an equivariant version of the Novikov conjecture is discussed and in Section 6.7 the Farrell-Jones conjecture is introduced. The chapter ends with a discussion on embedding theory.
In the 7th chapter, the focus is shifted from uniqueness statements as in the Borel conjecture to existence statements. This includes a discussion of the Nielsen realization problem and the Cannon conjecture.
The 8th chapter contains a survey of the main techniques going into the proofs of known cases of the Borel, Farrell-Jones and Baum-Connes conjecture.
Reviewer: Daniel Kasprowski (Southampton)Rational cohomology fixed pointshttps://zbmath.org/1535.550032024-07-05T15:31:27.292712Z"Benkhalifa, Mahmoud"https://zbmath.org/authors/?q=ai:benkhalifa.mahmoudThis work concerns rational homotopy. Let \((\land V, d)\) be the Sullivan minimal model of a simply connected CW-complex, of finite type, \(X\). Denote \({\mathcal{E}_{\sharp}}(\land V,d)\) the group of homotopy self equivalences inducing the identity on \(V\). By definition, the space \(X\) has an \(m\)-rational cohomology fixed point if there exist \([\alpha]\in {\mathcal{E}}_{\sharp}(\land V^{\leq m},d)\) and \(0\neq x\in H^{m+2}(\land V^{\leq m},d)\) such that \(\alpha\not\simeq {\mathrm{id}}\) and \(H^m(\alpha)(x)=x\). The author determines a sufficient condition for having an \((n-1)\)-rational cohomology fixed point, where \(n\) is the greatest integer \(i\) such that the Postnikov invariant \(k_{i}\) is different from 0.
Reviewer: Daniel Tanré (Villeneuve d'Ascq)The Borsuk-Ulam theorem for \(n\)-valued maps between surfaceshttps://zbmath.org/1535.550042024-07-05T15:31:27.292712Z"Laass, Vinicius Casteluber"https://zbmath.org/authors/?q=ai:laass.vinicius-casteluber"de Miranda e Pereiro, Carolina"https://zbmath.org/authors/?q=ai:de-miranda-e-pereiro.carolinaMaking an interesting use of the theory of braid groups, the paper under review proves a type of Borsuk-Ulam theorem for multimaps between surfaces. As an application, the authors determine the conditions for the Borsuk-Ulam theorem to hold for split and non-split multimaps \(\phi : X \to Y\) in the cases when: (i) \(X\) is the 2-sphere equipped with the antipodal involution and \(Y\) is either a closed surface or the Euclidean plane; (ii) \(X\) is a closed surface different from the 2-sphere equipped with a free involution and \(Y\) is the Euclidean plane.
Reviewer: Mahender Singh (S.A.S. Nagar)On the number of fixed points for a mapping of a finite connected graphhttps://zbmath.org/1535.550052024-07-05T15:31:27.292712Z"Llibre, Jaume"https://zbmath.org/authors/?q=ai:llibre.jaume"Reventós, Agustí"https://zbmath.org/authors/?q=ai:reventos.agusti(no abstract)Neighboring mapping points theoremhttps://zbmath.org/1535.550062024-07-05T15:31:27.292712Z"Malyutin, Andrei V."https://zbmath.org/authors/?q=ai:malyutin.andrei-valerevich"Musin, Oleg R."https://zbmath.org/authors/?q=ai:musin.oleg-rThe authors introduce and study a new family of theorems extending the class of Borsuk-Ulam and topological Radon type theorems. The defining idea for this new family is to replace requirements of the form ``the image of a subset that is large in some sense is a singleton'' with requirements of the milder form ``the image of a subset that is large in some sense is a subset that is small in some sense''. This approach covers the case of mappings \(S^m\rightarrow \mathbb{R}^n\) with \(m<n\) and extends to wider classes of spaces.
The main theorems are as follows:
\textbf{Theorem 4}: Let \(f\) be a continuous map of the boundary \(\partial\Delta^n\) of the \(n\)-dimensional complex \(\Delta^n\) to a contractible metric space \(M\). Then a set of spherical \(f\)-neighbors intersects all facets of \(\Delta^n\).
\textbf{Theorem 6}: Let \(C\) be a KKM cover of the \(n\)-sphere \(S^n\), and let \(f:\mathbb{S}^n\rightarrow M\) be a continuous map to a contractible metric space \(M\). Then a set of spherical \(f\)-neighbors intersects all elements of \(C\).
\textbf{Theorem 19}: Let \(X\) be a compact normal space, let \(M\) be a contractible metric space and let \(f:X\rightarrow M\) be a continuous map. Then, for any non-nullhomotopic cover \(C\) of \(X\), a set of spherical \(f\)-neighbors intersects at least \(rk(C)+1\) elements of \(C\). In particular, for any principal cover, a set of spherical \(f\)-neighbors intersects all elements of the cover.
\textbf{Theorem 36}: Let \(A\) be a compact normal space, Let \(C\) be a closed cover of \(A\), and let \([C]\) be the corresponding homotopy class in \([A,N(C)]\), where \(N(C)\) is the nerve. Let \(Z\) be a normal space containing \(A\) as a subspace. If the triple \((Z,A,[C])\) is Eilenberg-Pontryagin, with \(EP\) rank \(rk(Z,A,[C])>0\), then for any metric space \(M\) and any continuous map \(f:A\rightarrow M\) that extends to a continuous map \(Z\rightarrow M\), a set of spherical \(f\)-neighbors intersects at least \(rk(Z,A,[c])+1\) elements of \(C\).
The authors also give an open problem:
\textbf{Question}: Which of the other versions of the topological Helly theorem give sufficient conditions for principal and non-nullhomotopic covers?
Reviewer: Jun Wang (Shijiazhuang)On the realisation problem for mapping degree setshttps://zbmath.org/1535.550072024-07-05T15:31:27.292712Z"Neofytidis, Christoforos"https://zbmath.org/authors/?q=ai:neofytidis.christoforos"Sun, Hongbin"https://zbmath.org/authors/?q=ai:sun.hongbin"Tian, Ye"https://zbmath.org/authors/?q=ai:tian.ye.3|tian.ye.1|tian.ye.2|tian.ye|tian.ye.6|tian.ye.5|tian.ye.4"Wang, Shicheng"https://zbmath.org/authors/?q=ai:wang.shi-cheng"Wang, Zhongzi"https://zbmath.org/authors/?q=ai:wang.zhongziA set \(A \subseteq \mathbb{Z}\) is called multiplicative provided \(a,b \in A\) implies \(ab \in A.\)
The main results of the paper under review are: \((i)\) There exists a multiplicative set \(A\) containing \(\{0,1\}\) which is not the set of degrees \(D(M)\) of a self-mapping of any closed oriented \(n\)-manifold \(M;\) \((ii)\) For each \(n \in \mathbb{N}\) and \(k \geq 3\) every mapping degree set \(D(M,N)\) for closed oriented \(n\)-manifolds \(M\) and \(N\) is equal to \(D(P,Q)\) for some closed oriented \((n+k)\)-manifolds \(P\) and \(Q.\)
As a consequence of \((ii)\) it follows that for each \(n \neq 1,2,4,5\) every finite set of integers containing \(0\) is the mapping degree set for some \(n\)-manifolds.
Reviewer: Valerii V. Obukhovskij (Voronezh)Some smooth circle and cyclic group actions on exotic sphereshttps://zbmath.org/1535.550082024-07-05T15:31:27.292712Z"Quigley, J. D."https://zbmath.org/authors/?q=ai:quigley.james-dOne appealing feature of spheres is their high degree of symmetry. Usually the same does not apply to exotic spheres. The paper has a careful history of existence and nonexistence results of actions on homotopy spheres. \textit{R. Schultz} asked [Contemp. Math. 36, 243--267 (1985; Zbl 0559.57004)] whether every exotic sphere \(\Sigma^n\) of dimension \(n \ge 5\) admits a nontrivial action of the circle group \(\mathbb T\), or nontrivial action of the cyclic groups \(\mathbb Z_p\) for every prime \(p\).
Let \(\Theta_n\) denote the group of \(h\)-cobordism classes of homotopy spheres of dimension \(n\), and \(\Theta_n^{bp}\) the subgroup of those that bound stably parallelizable manifolds.
Theorem. [\textit{R. Schultz}, loc. cit. and \textit{C. N. Lee}, in: Proc. Conf. Transform. Groups, New Orleans 1967, 207 (1968; Zbl 0172.48303)] Every exotic sphere of dimension \(n\) admits a nontrivial smooth \(\mathbb Z_p\) action for each prime \(p\) that does not divide the order of \(\Sigma^n\) in \(\Theta_n\).
The author provides a table with the prime divisiors of the order of \(\Theta_n^{bp}\) and \(\mathrm{coker}\, J_n\) for \(n \le 100\). The table combined with the theorem yields abundant examples of actions on homotopy spheres.
One may use Mahowald invariants to show the existence of nontrivial smooth \(\mathbb T\) and \(\mathbb Z_p\) actions for primes \(p\) on homotopy spheres whose order is divisible by \(p\). Examples are indicated in the table.
Reviewer: Karl Heinz Dovermann (Honolulu)Note on homeomorphic spaces into open subsets of compact polyhedrahttps://zbmath.org/1535.550092024-07-05T15:31:27.292712Z"Gómez, F."https://zbmath.org/authors/?q=ai:gomez.federico|gomez.fernando-palacios|gomez.felix-pedro-quispe|gomez.fabio|gomez.francisco-j|gomez.florian(no abstract)\(K\)-theory of real Grassmann manifoldshttps://zbmath.org/1535.550102024-07-05T15:31:27.292712Z"Podder, Sudeep"https://zbmath.org/authors/?q=ai:podder.sudeep"Sankaran, Parameswaran"https://zbmath.org/authors/?q=ai:sankaran.parameswaranSummary: 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 \(n\equiv 0\pmod 4\), \(k\equiv 1\pmod 2\), we use the Hodgkin spectral sequence to determine the \(K\)-ring completely.A unified view on the functorial nerve theorem and its variationshttps://zbmath.org/1535.550112024-07-05T15:31:27.292712Z"Bauer, Ulrich"https://zbmath.org/authors/?q=ai:bauer.ulrich"Kerber, Michael"https://zbmath.org/authors/?q=ai:kerber.michael"Roll, Fabian"https://zbmath.org/authors/?q=ai:roll.fabian"Rolle, Alexander"https://zbmath.org/authors/?q=ai:rolle.alexanderIn this paper, the authors present together several different functional nerve theorems under various assumptions. The nerve theorem is a fundamental part of the algebraic topology tool box. It plays a very important part in computational and applied algebraic topology. In particular, in topological data analysis, practitioners regularly require an appropriate functorial nerve theorem, and often such a theorem in proven in a specific context. This paper presents several versions of the nerve theorem, and unifies them into a theorem which implies several of these variants. Due to the functorial nature of these theorems, the authors present the necessary category theory to prove the theorems, as well as the unified version.
Beginning with `nerve theorems for covers by closed convex sets in Euclidean spaces', which they prove using fundamental techniques. After this, they prove nerve theorems for `covers of simplicial complexes by subcomplexes', before finally presenting a `unified nerve theorem', which absorbs many versions of the nerve theorem. To prove this unified nerve theorem, they use techniques from abstract homotopy theory, in particular model categories.
Reviewer: Yossi Bokor Bleile (Aalborg)Topological invariance of torsion-sensitive intersection homologyhttps://zbmath.org/1535.550122024-07-05T15:31:27.292712Z"Friedman, Greg"https://zbmath.org/authors/?q=ai:friedman.gregCategories of torsion-sensitive perverse sheaves were introduced in [\textit{G. Friedman}, Mich. Math. J. 68, No. 4, 675--726 (2019; Zbl 1439.55008)] in order to unify several intersection homology duality theorems, see [\textit{M. Goresky} and \textit{R. MacPherson}, Topology 19, 135--165 (1980; Zbl 0448.55004), \textit{M. Goresky} and \textit{R. MacPherson}, Invent. Math. 72, 77--129 (1983; Zbl 0529.55007), \textit{M. Goresky} and \textit{P. Siegel}, Comment. Math. Helv. 58, 96--110 (1983; Zbl 0529.55008), \textit{S. E. Cappell} and \textit{J. L. Shaneson}, Ann. Math. (2) 134, No. 2, 325--374 (1991; Zbl 0759.55002)], as special cases of a more general duality theorem in which torsion phenomena are encoded into the sheaf complexes.
The torsion-sensitive-Deligne sheaves are a generalisation of Deligne sheaves on stratified pseudomaniflods, [\textit{A. Beilinson} et al., Faisceaux pervers. Actes du colloque ``Analyse et Topologie sur les Espaces Singuliers''. Partie I. 2nd edition. Paris: Société Mathématique de France (SMF) (2018; Zbl 1390.14055)], since they are the intermediate extensions of torsion-sensitive coefficient systems and they can be characterised by a set of axioms which generalise the ones by Goresky and MacPherson for ordinary Deligne sheaves.
This paper is devoted to the study of the topological invariance (up to quasi-isomorphism) of the torsion-sensitive-Deligne sheaves. For the classical Deligne sheaves this problem was considered in [Goresky and MacPherson, Zbl 0529.55007, \textit{H. C. King}, Topology Appl. 20, 149--160 (1985; Zbl 0568.55003)]. The topological invariance in the torsion sensitive category is achieved in Theorem 3.6; this theorem recovers (with a different proof) the original Goresky-MacPherson theorem [Goresky and MacPherson, loc. cit.], as well as results in [\textit{D. Chataur} et al., Ill. J. Math. 63, No. 1, 127--163 (2019; Zbl 1451.55002), \textit{N. Habegger} and \textit{L. Saper}, Invent. Math. 105, No. 2, 247--272 (1991; Zbl 0759.55003)].
Reviewer: Alessio Cipriani (Verona)A short note on simplicial stratificationshttps://zbmath.org/1535.550132024-07-05T15:31:27.292712Z"Wrazidlo, Dominik J."https://zbmath.org/authors/?q=ai:wrazidlo.dominik-jThe classical approach to generalize Poincaré duality and related structures from manifolds to manifolds with singularities is intersection homology and cohomology. A recent new idea to achieve the same goal is to assign an ``intersection space'' to the singular manifold and consider the ordinary homology and cohomology of it. This spatial version of intersection homology is not the same as the classical version but is related to it.
The first -- somewhat restrictive -- construction of intersection spaces, due to \textit{M. Banagl} [Electron. Res. Announc. Math. Sci. 16, 63--73 (2009; Zbl 1215.55003) and Intersection spaces, spatial homology truncation, and string theory. Dordrecht: Springer (2010; Zbl 1219.55001)], was generalized by \textit{M. Agustín Vicente} and \textit{J. Fernandez de Bobadilla} [Doc. Math. 25, 1653--1725 (2020; Zbl 1457.32082)] to a wide variety of spaces satisfying extra assumptions. The paper under review proves that these assumptions hold in great generality: they hold for triangulated PL pseudomanifolds, for example for complex algebraic varieties.
Reviewer: Richárd Rimányi (Chapel Hill)Weighted (co)homology and weighted Laplacianhttps://zbmath.org/1535.550142024-07-05T15:31:27.292712Z"Wu, Chengyuan"https://zbmath.org/authors/?q=ai:wu.chengyuan"Ren, Shiquan"https://zbmath.org/authors/?q=ai:ren.shiquan"Wu, Jie"https://zbmath.org/authors/?q=ai:wu.jie.2"Xia, Kelin"https://zbmath.org/authors/?q=ai:xia.kelinIn this paper, the authors extend the combinatorial Laplace operator introduced in [\textit{D. Horak} and \textit{J. Jost}, Adv. Math. 244, 303--336 (2013; Zbl 1290.05103)] to weighted simplicial complexes, in turn providing new examples and computations.
For a simplicial complex \(K\), denote by \(d_i\) the \(i\)-th face map. Consider an \(R\)-module \(G\) over a commutative unital ring \(R\). Then, Definition 2.1 in the paper at hand defines a weight function \(\phi\) as a map
\[
\phi\colon K\times K\to R
\]
such that \(\phi(d_i\sigma, d_j d_i \sigma)\phi(\sigma,d_i\sigma)g= \phi(d_j\sigma, d_j d_i \sigma)\phi(\sigma,d_j\sigma)g\) for all \(\sigma\) in \(K\), \(g\in G\), and \(j<i\). This definition is shown to encompass both the definition of the classical differential, and of the weighted boundary of [\textit{R. J. MacG. Dawson}, Cah. Topologie Géom. Différ. Catégoriques 31, No. 3, 229--243 (1990; Zbl 0735.18011)]. The weight function also yields \(\phi\)-weighted cohomology groups. After analyzing \(\phi\)-weighted cohomology theories, the \(\phi\)-weighted Laplacian is introduced in Section 5. The main generalization of Horak and Jost's results on the calculation of the multiplicity of the eigenvalue zero in the Laplacian spectrum is given in Section 6. The theory is then applied in Section 7 to concrete examples as weighted polygons and digraphs.
Reviewer: Luigi Caputi (Torino)External Spanier-Whitehead duality and homology representation theorems for diagram spaceshttps://zbmath.org/1535.550152024-07-05T15:31:27.292712Z"Lackmann, Malte"https://zbmath.org/authors/?q=ai:lackmann.malteRecall that Brown's representability theorem states that any abelian presheaf \(\mathrm{Ho}(\mathbf{CW}_*)^{\mathrm{op}} \to \mathbf{Ab}\) satisfying certain exactness properties is represented by an abelian group object in \(\mathrm{Ho}(\mathbf{CW}_*)\) (see [\textit{E. H. Brown jun.}, Ann. Math. (2) 75, 467--484 (1962; Zbl 0101.40603)]). This result may be used to show that cohomology theories are representable by spectra. To show that homology theories are representable by spectra one requires a strengthening of Brown's result due to \textit{J. F. Adams} [Topology 10, 185--198 (1971; Zbl 0197.19604)], namely that an abelian presheaf \(\mathrm{Ho}(\mathbf{CW}_*)^{\mathrm{op}} \to \mathbf{Ab}\) as above is determined by its restriction to the subcategory of \(\mathrm{Ho}(\mathbf{CW}_*)\) spanned by finite pointed CW-complexes. Then one may use Spanier-Whitehead duality to reduce the question of the representability of homology theories to one about the representability of cohomology theories.
The present article carries out a similar strategy as the one outlined above to prove a representability theorem for homology theories on the category of presheaves \(C^{\mathrm{op}} \to \mathbf{Top}_*\), where \(C\) is a small category. In this more general setting, the author must overcome two hurdles:
\begin{itemize}
\item[1.] The category of \(C\)-indexed spectra is no longer canonically self-dual for general \(C\).
\item[2.] Adams' representability theorem is no longer true for arbitrary \(C\).
\end{itemize}
To deal with the first hurdle the author observes that \([C^{\mathrm{op}}, \mathbf{Spectra}]\) is canonically dual to \([C, \mathbf{Spectra}]\) in the monoidal \(2\)-category with objects small categories, and morphism categories \(\mathrm{Hom}(A,B) = \mathrm{Ho} [A^{\mathrm{op}} \times B, \mathbf{Spectra}]\), and the fact that \([C^{\mathrm{op}}, \mathbf{Spectra}]\) and \([C, \mathbf{Spectra}]\) are generated under colimits by \(C\) and \(C^{\mathrm{op}}\), respectively. Thus, the author is faced with the second hurdle, which is surmounted using a result of Neeman that Adams' representability theorem still goes through for \(C\) countable (see [\textit{A. Neeman}, Topology 36, No. 3, 619--645 (1997; Zbl 0869.55008)]).
Finally, the author specialises to the case of homology theories on \([C^{\mathrm{op}}, \mathbf{Top}_*]\) valued in \(\mathbf{Q}\)-vector spaces, and studies when such homology theories split into simpler pieces.
A large part of the present article is dedicated to proving model categorical results in order to make the above precise.
Reviewer: Adrian Clough (Abu Dhabi)Cofibration-fibration extensionshttps://zbmath.org/1535.550162024-07-05T15:31:27.292712Z"Hernández Paricio, Luis Javier"https://zbmath.org/authors/?q=ai:hernandez-paricio.luis-javier(no abstract)Homotopy type through homology groupshttps://zbmath.org/1535.550172024-07-05T15:31:27.292712Z"Bravo, Andrés Carnero"https://zbmath.org/authors/?q=ai:carnero-bravo.andres"Antolín Camarena, Omar"https://zbmath.org/authors/?q=ai:antolin-camarena.omarIn this paper the authors consider the homotopy type of a simply connected CW-complex \(X\) satisfying the following condition \((*)\):
\[
\tilde{H}_q(X;\mathbb{Z})\cong \begin{cases} \mathbb{Z}^a & \text{ for }q=d \\ \mathbb{Z}^b & \text{ for }q=d+k \\ 0 & \text{ for }q\not=d,d+k, \end{cases} \tag{\(*\)}
\]
where \(a,b,d,k\) are positive integers with \(d>k\). In particular, they prove that there is a homotopy equivalence
\[
X\simeq \big(\bigvee^aS^d\big)\vee \big(\bigvee^bS^{d+k}\big)
\]
when \(k\in \{1,5,6,13,62\}\).
Their proof is based on the computation of the Serre spectral sequence and the Blakers-Massey theorem.
Reviewer: Kohhei Yamaguchi (Tōkyō)Manifold topology: a preludehttps://zbmath.org/1535.550182024-07-05T15:31:27.292712Z"Roushon, S. K."https://zbmath.org/authors/?q=ai:roushon.sayed-kSummary: In this expository article we describe some of the fundamental
questions in manifold topology and the obstruction groups which give answers to the questions.On the space of free loops of an odd spherehttps://zbmath.org/1535.550192024-07-05T15:31:27.292712Z"Aguadé i Bover, Jaume"https://zbmath.org/authors/?q=ai:aguade-i-bover.jaume(no abstract)Periodic self maps and thick ideals in the stable motivic homotopy category over \(\mathbb{C}\) at odd primeshttps://zbmath.org/1535.550202024-07-05T15:31:27.292712Z"Stahn, Sven-Torben"https://zbmath.org/authors/?q=ai:stahn.sven-torbenIn this paper, the author studies thick subcategories of the stable motivic homotopy category \(\mathcal{SH}(\mathbb{C})\) over the base field \(\mathbb{C}\), building upon work of Ruth Joachimi [\textit{R. Joachimi}, Springer Proc. Math. Stat. 309, 109--219 (2020; Zbl 1475.55023)] and in relation to the thick subcategory theorem [\textit{M. J. Hopkins} and \textit{J. H. Smith}, Ann. Math. (2) 148, No. 1, 1--49 (1998; Zbl 0924.55010)] of classical stable homotopy theory. More precisely, he works with quasi-finite cellular spectra, locally with respect to an odd prime \(\ell\) or after completion.
Betti realization yields \(R : \mathcal{SH}(\mathbb{C})^{qfin}_{(\ell)} \rightarrow \mathcal{SH}_{(\ell)}^{fin}\) to the localization of the classical stable homotopy category of finite spectra and one also has the `constant sheaf' functor \(c : \mathcal{SH}_{(\ell)}^{fin} \rightarrow \mathcal{SH}(\mathbb{C})^{qfin}_{(\ell)}\). This gives, for \(\mathcal{C}_n \subset \mathcal{SH}_{(\ell)}^{fin}\) the thick subcategory of type \(n\) spectra, the tensor ideals
\[
\mathrm{thickid} (c \mathcal{C}_n) \subseteq R^{-1} (\mathcal{C}_n).
\]
Joachimi pointed out that there are more thick tensor ideals in the motivic setting. Moreover, nilpotence is more delicate: for example, the motivic \(\eta\) is not nilpotent. She defined the thick ideal \(\mathcal{C}_{AK(n)}\) as the spectra for which \(AK(n)_{**}(-)\) vanishes, for \(AK(n)\) the \(n\)th algebraic Morava \(K\)-theory, and showed that \[\mathcal{C}_{AK(n+1)} \subseteq \mathcal{C}_{AK(n)} \subseteq R^{-1} (\mathcal{C}_{n+1}).\]
The author first considers the full subcategory of \(\mathcal{SH}(\mathbb{C})^{qfin}_{(\ell)}\) (or its \(\ell\)-complete variant) admitting a motivic \(v_n\)-self map. Assuming that a motivic nilpotence conjecture holds (this only involves certain bidegrees), he proves that this subcategory is thick.
To construct examples of motivic \(v_n\)-self maps (in the \(\ell\)-complete setting), he considers Joachimi's motivic spectrum \(\mathbb{X}_n\), a motivic counterpart of Smith's type \(n\) spectrum. Joachimi showed that \(AK(s)_{**}(\mathbb{X}_n)\) vanishes for \(s<n\) and is non-zero for \(s=n\). The author shows that \(AK(n)_{**}(\mathbb{X}_n)\) is free over \(AK(n)_{**}\), which allows the calculation of \(AK(n)_{**}(D \mathbb{X}_n \wedge \mathbb{X}_n)\) by Künneth. A motivic Adams spectral sequence adaptation of the classical argument then yields a motivic \(v_n\)-self map.
To deal with non-nilpotence of motivic \(\eta\), it is natural to smash with the cone \(C_\eta\), thus considering the thick ideals \(\mathrm{thickid}(C_\eta) \cap \mathcal{C}_{AK(n)}\). The author proves that \(C_\eta \wedge \mathbb{X}_{n+1}\) lies in \(\mathcal{C}_{AK(n)}\) but not in \(\mathcal{C}_{AK(n+1)}\), showing that these thick ideals are nonzero and distinct.
Finally he shows that the inclusion \(\mathcal{C}_{AK(1)} \subsetneq R^{-1}(\mathcal{C}_2)\) is proper and that \(\mathrm{thickid} (c \mathcal{C}_2)\not \subset \mathcal{C}_{AK(1)}\) (this corrects an assertion of Joachimi's). He indicates that the argument also applies for higher chromatic \(n\).
Reviewer: Geoffrey Powell (Angers)Realizability of localized groups and spaceshttps://zbmath.org/1535.550212024-07-05T15:31:27.292712Z"Aguadé i Bover, Jaume"https://zbmath.org/authors/?q=ai:aguade-i-bover.jaume(no abstract)Formality and finiteness in rational homotopy theoryhttps://zbmath.org/1535.550222024-07-05T15:31:27.292712Z"Suciu, Alexander I."https://zbmath.org/authors/?q=ai:suciu.alexander-iThe present work is a very welcomed survey on formality and finiteness in the context of rational homotopy theory.
First, we recall these two central notions. A nilpotent finite \(\mathbb Q\)-type space \(X\) is \textit{formal} if the Sullivan algebra of piecewise linear polynomial rational forms \(A_{\mathrm{PL}}(X)\) is connected to the rational cohomology algebra \(H^*(X;\mathbb Q)\) via a zig-zag of differential graded commutative algebras (cdga's, henceforth)
\[
A_{\mathrm{PL}}(X) \xleftarrow{\simeq} \bullet \xrightarrow{\simeq} H^*(X;\mathbb Q).
\]
Here, the cohomology algebra carries the trivial differential. In other words: \(X\) is formal if the algebras \(A_{\mathrm{PL}}(X)\) and \(H^*(X;\mathbb Q)\) represent the same homotopy type in the corresponding homotopy category of cdga's. Formal spaces form one of the most celebrated classes of spaces since the notion was introduced, due to their relative simplicity with respect to arbitrary homotopy types.
Finiteness refers to the possibility of approximating an object of interest by another object of the same type with convenient finiteness hypotheses. For example, a \(q\)-finite space is a path-connected topological space \(X\) homotopy equivalent to a finite CW-complex whose skeleton stabilizes at stage \(q\). Similarly, a cdga \((A,d)\) is \(q\)-finite if it is connected (i.e., \(A^0=k\) is the ground field) and each degree \(n\) component \(A^n\) is a finite dimensional \(k\)-vector space for all \(n\leq q\).
This survey collects many of the foundational results on these topics, as well as many of the results of the prolific author and his collaborators on them. In any case, the theory has developed too broadly for a survey, or even a book, to contain most of what is known. Along the text, the reader will find abundant examples to illustrate the results or to show that some of the statements are sharp. The text also poses some conjectures and open questions on the topics.
The paper is divided into five main sections that we briefly summarize.
Part I presents the basics on cdga's and their homotopy theory, formality and its variants (such as intrinsic and partial formality), and the first basic results on formality. These include the behavior of formality under field extension, formality of cdga maps, the relationship to Massey products, the descent property, the theory of positive weight decompositions, minimal models, and Poincaré duality.
Part II deals with several of the Lie algebras involved in this theory, discussing some of their properties and interconnections. The reader should not confuse this section to be the study of formality of differential graded Lie algebras (i.e., the Quillen side of rational homotopy theory), the defining characteristic of \textit{coformal} spaces. This section includes graded and filtered, Malcev, and holonomy Lie algebras. Some of these Lie algebras arise as the associated Lie algebra to a group, whose graded pieces are the successive quotients of the lower central series of the group tensored with the ground field, and whose Lie bracket is induced from the commutators of the group. In some other instances, these Lie algebras are related to group cohomology, or are constructed from some special cdga's (that might arise as a model of a topological space).
Part III contains the basics of rational homotopy theory of spaces, such as completions, rationalizations, and algebraic models for spaces and groups. The main focus is on the formality and finiteness properties of such models. Important results and constructions are given for polyhedral products, configuration spaces, right-angled Artin groups, among many others.
The remaining two parts are slightly more specialized, and touch upon topics mostly on algebraic geometry.
Part IV deals with Alexander invariants and the cohomology jump loci of spaces and suitable algebraic models. It further connects the characteristic and resonance varieties to various formality and finiteness properties previously established.
Part V applies the theory presented to the study of Kähler manifolds and smooth, quasi-projective varieties, compact Lie group actions on manifolds, and closed, orientable 3-manifolds.
Reviewer: José Manuel Moreno Fernández (Málaga)A kind of characterization of homeomorphism and homeomorphic spaces by core fundamental groupoid a good invarianthttps://zbmath.org/1535.550232024-07-05T15:31:27.292712Z"Badiger, Chidanand"https://zbmath.org/authors/?q=ai:badiger.chidanand"Venkatesh, T."https://zbmath.org/authors/?q=ai:venkatesh.t-a|venkatesh.t-n|venkatesh.t-gIn this paper the \textit{core fundamental groupoid} of a topological space \(M\), which is a wide subgroupoid of the fundamental groupoid \(\pi_1(M)\) is defined to be the disjoint union of the fundamental groups. This definition is also called bundle of groups. In this paper, using the core groupoid, a kind of characterization of homeomorphisms is given. The organization of the paper is as follows:
Following an introduction, in Section 2 some basic notions about groupoids are introduced as in the literature. In the third section a full account of the construction of core groupoid is given. Some of these properties can be obtained from the fundamental groupoid. The notion of fundamental groupoid defines a functor from topological spaces to groupoids and therefore a characterisation of homeomorphism or homeomorphic spaces can be obtained. In this paper these relations are considered in terms of core fundamental groupoids. In Section 4 the topological structure of the core fundamental groupoid is studied
Reviewer: Osman Mucuk (Kayseri)Projection spaces and twisted Lie algebrashttps://zbmath.org/1535.550242024-07-05T15:31:27.292712Z"Knudsen, Ben"https://zbmath.org/authors/?q=ai:knudsen.benThis article introduces the notion of \emph{projection space}, which axiomatizes the following behavior of configuration spaces of topological spaces. Given a (discrete) finite set \(I\), let \(\mathrm{Conf}_I(X)\) be the space of injections \(I \hookrightarrow X\). Any bijection \(f : I \sqcup J \xrightarrow{\cong} K\) defines a natural projection map \(f^* : \mathrm{Conf}_K(X) \twoheadrightarrow \mathrm{Conf}_I(X)\), the Fadell-Neuwirth fibration that forgets some points in a configuration. Moreover, this map is invariant under endomorphisms of \(J\).
To model this situation, the author defines the \emph{projection category} \(\mathrm{Pr}(\mathcal{C})\) of a monoidal category \(\mathcal{C}\). Its objects are the same as \(\mathcal{C}\). Morphisms from \(C_1\) to \(C_2\) are equivalence classes of morphisms of the form \(C_1 \otimes D \to C_2\), with an equivalence relation generated by \((C_1 \otimes D \to C_2) \sim (C_1 \otimes D' \to C_2)\) if there exists a morphism \(D \to D'\) making the obvious triangle commute. A projection space is then a presheaf on \(\mathrm{Pr}(\mathcal{C})\). The example of classical configuration spaces above is a projection space for \(\mathcal{C}\) the category of finite sets and bijections. The author recovers many other kinds of configuration spaces with this framework, such as orbit configuration spaces (for \(\mathcal{C}\) the category of \(G\)-sets), graphical configuration spaces (for \(\mathcal{C}\) the category of graphs), simplicial configuration spaces, generalized configuration spaces, Stiefel configuration spaces, etc.
In the spirit of the author's previous results, this article defines a functor \(L\) from the category of topological presheaves over \(\mathrm{Pr}(\mathcal{C})\) to the category of Lie algebras in dg-\(\mathbb{Q}\)-presheaves over \(\mathcal{C}\). Philosophically, a projection space is a collection of spaces equipped with a cocommutative comultiplication; its ``Koszul dual'' should thus be a Lie algebra, of which the functor \(L\) is a representative.
The article's main theorem is that, for a reduced projection space \(X\) over a combinatorial symmetric monoidal category \(\mathcal{C}\), the homology of \(X\) is isomorphic to the Chevalley-Eilenberg homology of \(L(X)\) as a \(\mathcal{C}\)-twisted cocommutative coalgebra. Using this theorem, the author deduces some results on representation stability. Given an \(\mathrm{FI}^{\mathrm{op}}\)-space \(X\), that is, a presheaf over the category of finite sets and injections -- or equivalently, a projection space over the category of finite sets and bijections -- the author proves that if (i) \(H_i(L(X)_k)\) is finite dimensional; (ii) \(H_0(L(X)_k) = 0\) for \(k > 0\); (iii) \(H_i(L(X)_k) = 0\) for \(k \gg i\); then \(H^*(X; \mathbb{Q})\) is representation stable, that is, the decomposition as a representation of the symmetric group of \(H^i(X_k; \mathbb{Q})\) stabilizes when \(k \gg i\).
For the entire collection see [Zbl 1522.14005].
Reviewer: Najib Idrissi (Paris)A note on the hit problem for the polynomial algebra in the case of odd primes and its applicationhttps://zbmath.org/1535.550252024-07-05T15:31:27.292712Z"Phúc, Đặng Võ"https://zbmath.org/authors/?q=ai:dang-vo-phuc.Let \(p\) be a prime and let \(P_h:=\mathbb{F}_p[t_1,t_2,\dots,t_h]\) denote the graded polynomial algebra over the prime field of \(p\) elements and let \(GL (h, {\mathbb F}_{p}) \) denote the general linear group of rank \(h\) over \({\mathbb F}_p.\) The paper under review is concerned with the \textit{hit problem} set up by F. Peterson, of finding a minimal generating set for the polynomial algebra \(P_h\) as a module over the mod \(p\) Steenrod algebra \(\mathcal{A}_p,\) or, equivalently, of determining a monomial basis of the \({\mathbb F}_{p}\)-vector space \({\mathbb F}_p \otimes_{\mathcal{A}_p} { P}_{h}.\) Denote by \(( {\mathbb F}_p \otimes_{\mathcal{A}_p} { P}_{h})_n\) the homogeneous component of degree \(n\) in \({\mathbb F}_p \otimes_{\mathcal{A}_p} { P}_{h}.\)
The hit problem has been utilized to investigate the Singer algebraic transfer [\textit{W. M. Singer}, Math. Z. 202, No. 4, 493--523 (1989; Zbl 0687.55014)], which is a homomorphism
\[
Tr_h^{\mathcal{A}_p} : ({\mathbb F}_p \otimes_{GL (h, {\mathbb F}_{p})}\mathrm{Ann}_{\overline{\mathcal{A}_p}}H_*(V;{\mathbb F}_{p}))_n \rightarrow \mathrm{Ext}_{\mathcal{A}_p}^{h,h+n}({\mathbb F}_p, {\mathbb F}_p).
\]
Here \(V\) denotes the rank \(h\) elementary abelian \(p\)-group, \(\prod_{i=1}^{h}({\mathbb Z}/p{\mathbb Z}).\) In this paper the author investigates the behavior of \(Tr_h^{\mathcal{A}_p}\) for the cases \(p>2\) and \(h=3\) in some \textit{generic} degrees. He shows that the transfer homomorphism \(Tr_h^{\mathcal{A}_p}\) is one-to-one for \(h=3\) and any odd prime \(p.\)
Reviewer: Mbakiso Fix Mothebe (Gaborone)Coherence for monoidal groupoids in HoTThttps://zbmath.org/1535.550262024-07-05T15:31:27.292712Z"Piceghello, Stefano"https://zbmath.org/authors/?q=ai:piceghello.stefanoSummary: We present a proof of coherence for monoidal groupoids in homotopy type theory. An important role in the formulation and in the proof of coherence is played by groupoids with a free monoidal structure; these can be represented by 1-truncated higher inductive types, with constructors freely generating their defining objects, natural isomorphisms and commutative diagrams. All results included in this paper have been formalised in the proof assistant Coq.
For the entire collection see [Zbl 1445.68007].Correction and addendum to: ``The solution on the geography-problem of non-formal (almost) contact manifolds''https://zbmath.org/1535.570412024-07-05T15:31:27.292712Z"Bock, Christoph"https://zbmath.org/authors/?q=ai:bock.christophSummary: Proposition 2.1 and Theorem 2.2 in the author's paper [ibid. 317, Article ID 108186, 7 p. (2022; Zbl 1503.57029)], can easily be generalised: Any odd-dimensional solvmanifold is contact.Linking in tree-manifoldshttps://zbmath.org/1535.570432024-07-05T15:31:27.292712Z"Wang, Xueqi"https://zbmath.org/authors/?q=ai:wang.xueqi|wang.xueqi.1Let \(T\) be a tree whose vertices are \(2m\)-dimensional oriented vector fields over the sphere \(S^{2m}\). Plumbing, as described in [\textit{W. Browder}, Surgery on simply-connected manifolds. Berlin-Heidelberg-New York: Springer-Verlag (1972; Zbl 0239.57016)], gives an oriented manifold \(W(T)\) whose boundary \(M(T)\) is called a tree manifold. The fibers of the associated sphere bundles to the vertices of \(T\) provide embedded \((2m-1)\)-dimensional spheres in \(M(T)\). In the case when \(M(T)\) is a rational homology sphere, the author calculates the linking number of these spheres. This paper corrects earlier work of \textit{T. tom Dieck} [Geom. Dedicata 34, No. 1, 57--65 (1990; Zbl 0759.57016)].
Reviewer: James Hebda (St. Louis)Birman-Hilden bundles. IIhttps://zbmath.org/1535.570442024-07-05T15:31:27.292712Z"Malyutin, A. V."https://zbmath.org/authors/?q=ai:malyutin.andrey-v|malyutin.andrei-valerevichSummary: We study the structure of self-homeomorphism groups of fibered manifolds. A fibered topological space is a Birman-Hilden space whenever in each isotopic pair of its fiber-preserving (taking each fiber to a fiber) self-homeomorphisms the homeomorphisms are also fiber-isotopic (isotopic through fiber-preserving homeomorphisms). We prove in particular that the Birman-Hilden class contains all compact connected locally trivial surface bundles over the circle, including nonorientable ones and those with nonempty boundary, as well as all closed orientable Haken 3-manifold bundles over the circle, including nonorientable ones.
For Part I, see [the author, Sib. Math. J. 65, No. 1, 106--117 (2024; Zbl 1533.57064); translation from Sib. Mat. Zh. 65, No. 1, 125--139 (2024)].Diffeomorphism groups of prime 3-manifoldshttps://zbmath.org/1535.570552024-07-05T15:31:27.292712Z"Bamler, Richard H."https://zbmath.org/authors/?q=ai:bamler.richard-h"Kleiner, Bruce"https://zbmath.org/authors/?q=ai:kleiner.bruceFor a compact (non-Haken) 3-manifold modeled on the Thurston geometry Nil (the nilpotent Heisenberg group), the main result of the paper states that the inclusion of the isometry group of \(X\) into its diffeomorphism group \(\mathrm{Diff}(X)\) is a homotopy equivalence (basically, these manifolds are Seifert fiber spaces over Euclidean 2-orbifolds with non-zero Euler number). As the authors note, combining this with earlier work by many authors, this completes the determination of the homotopy type of \(\mathrm{Diff}(X)\) for any compact, orientable, prime 3-manifold \(X\). The proof of the main theorem is based on Ricci flow and follows closely the lines of a previous paper by the present authors ([J. Am. Math. Soc. 36, No. 2, 563--589 (2023; Zbl 1518.53072)]).
``Together with previous work, we have shown that singular Ricci flow offers a uniform approach to studying diffeomorphism groups of large classes of prime manifolds: spherical space forms, hyperbolic manifolds, \(S^2 \times S^1\) and non-Haken manifolds modeled on Nil. We expect that the methods from this paper can be readily adapted to also cover the non-Haken case modeled on \(\mathbb H^2 \times \mathbb R\) and \(\tilde{\mathrm{SL}}(2, \mathbb R)\) and the Haken case modeled on Solv, which were covered by McCullough and Soma and Hatcher).''
Reviewer: Bruno Zimmermann (Trieste)Universal complexes in toric topologyhttps://zbmath.org/1535.570562024-07-05T15:31:27.292712Z"Baralić, Đorđe"https://zbmath.org/authors/?q=ai:baralic.djordje"Vavpetič, Aleš"https://zbmath.org/authors/?q=ai:vavpetic.ales"Vučić, Aleksandar"https://zbmath.org/authors/?q=ai:vucic.aleksandarThe paper is devoted to the so called universal complexes \(X(\mathbb F^n_p)\) and \(K(\mathbb F^n_p)\), that play an important role in mod \(p\) toric topology and its applications. The faces of those simplicial complexes are certain unimodular subsets of \(\mathbb F^n_p\). The authors compute \(f\)-vectors of those complexes as well as their bigraded Betti numbers. Furthermore, they show that the complexes are shellable. Turning from combinatorial commutative algebra to homotopy theory, the authors prove that the Lusternik-Schnirelmann category of the moment-angle-complex of the simplicial complex \(X(\mathbb F^n_p)\) equals \(n\), provided \(p\) is an odd prime, whereas the Lusternik-Schnirelmann category of the moment-angle-complex of the simplicial complex \(K(\mathbb F^n_p)\) equals \([n/2]\). Finally, they introduce the Buchstaber invariant \(s_p\) for a prime number \(p\) based on the theory of universal complexes.
Reviewer: Ivan Limonchenko (Toronto)Sculptures in concurrencyhttps://zbmath.org/1535.681602024-07-05T15:31:27.292712Z"Fahrenberg, Uli"https://zbmath.org/authors/?q=ai:fahrenberg.uli"Johansen, Christian"https://zbmath.org/authors/?q=ai:johansen.christian"Trotter, Christopher A."https://zbmath.org/authors/?q=ai:trotter.christopher-a"Ziemiański, Krzysztof"https://zbmath.org/authors/?q=ai:ziemianski.krzysztofSummary: We give a formalization of Pratt's intuitive sculpting process for higher-dimensional automata (HDA). Intuitively, an HDA is a sculpture if it can be embedded in (i.e., sculpted from) a single higher dimensional cell (hypercube). A first important result of this paper is that not all HDA can be sculpted, exemplified through several natural acyclic HDA, one being the famous ``broken box'' example of van Glabbeek. Moreover, we show that even the natural operation of unfolding is completely unrelated to sculpting, e.g., there are sculptures whose unfoldings cannot be sculpted. We investigate the expressiveness of sculptures, as a proper subclass of HDA, by showing them to be equivalent to regular ST-structures (an event-based counterpart of HDA) and to (regular) Chu spaces over 3 (in their concurrent interpretation given by Pratt). We believe that our results shed new light on the intuitions behind sculpting as a method of modeling concurrent behavior, showing the precise reaches of its expressiveness. Besides expressiveness, we also develop an algorithm to decide whether an HDA can be sculpted. More importantly, we show that sculptures are equivalent to Euclidean cubical complexes (being the geometrical counterpart of our combinatorial definition), which include the popular PV models used for deadlock detection. This exposes a close connection between geometric and combinatorial models for concurrency which may be of use for both areas.Continuous Kendall shape variational autoencodershttps://zbmath.org/1535.683012024-07-05T15:31:27.292712Z"Vadgama, Sharvaree"https://zbmath.org/authors/?q=ai:vadgama.sharvaree"Tomczak, Jakub M."https://zbmath.org/authors/?q=ai:tomczak.jakub-m"Bekkers, Erik"https://zbmath.org/authors/?q=ai:bekkers.erik-jSummary: We present an approach for unsupervised learning of geometrically meaningful representations via \textit{equivariant} variational autoencoders (VAEs) with \textit{hyperspherical} latent representations. The equivariant encoder/decoder ensures that these latents are geometrically meaningful and grounded in the input space. Mapping these geometry- grounded latents to hyperspheres allows us to interpret them as points in a Kendall shape space. This paper extends the recent \textit{Kendall-shape VAE} paradigm by Vadgama et al. by providing a general definition of Kendall shapes in terms of group representations to allow for more flexible modeling of KS-VAEs. We show that learning with generalized Kendall shapes, instead of landmark-based shapes, improves representation capacity.
For the entire collection see [Zbl 1528.94003].On the emergent behavior of the swarming models on the complex spherehttps://zbmath.org/1535.810742024-07-05T15:31:27.292712Z"Kim, Dohyun"https://zbmath.org/authors/?q=ai:kim.dohyun.2"Kim, Jeongho"https://zbmath.org/authors/?q=ai:kim.jeonghoSummary: We study swarming models on the complex unit sphere in the Hilbert space, which are generalizations of the well-known swarming model on the real unit sphere. We introduce two possible models: The first one is induced from the standard swarming model on the real sphere, and the second one is obtained by taking the conjugate to the inner product in the first one, which seems analogous to the projection in quantum mechanics. We provide various asymptotic behaviors of the two models and clarified their differences from the perspective of emergent dynamics. Finally, we conduct several numerical simulations that support the analytical results and provide further qualitative insight beyond our theoretical results.
{{\copyright} 2021 Wiley Periodicals LLC}Semiclassical theory and the Koopman-van Hove equationhttps://zbmath.org/1535.811182024-07-05T15:31:27.292712Z"Joseph, Ilon"https://zbmath.org/authors/?q=ai:joseph.ilonSummary: The phase space Koopman-van Hove (KvH) equation can be derived from the asymptotic semiclassical analysis of partial differential equations. Semiclassical theory yields the Hamilton-Jacobi equation for the complex phase factor and the transport equation for the amplitude. These two equations can be combined to form a nonlinear semiclassical version of the KvH equation in configuration space. There is a natural injection of configuration space solutions into phase space and a natural projection of phase space solutions onto configuration space. Hence, every solution of the configuration space KvH equation satisfies both the semiclassical phase space KvH equation and the Hamilton-Jacobi constraint. For configuration space solutions, this constraint resolves the paradox that there are two different conserved densities in phase space. For integrable systems, the KvH spectrum is the Cartesian product of a classical and a semiclassical spectrum. If the classical spectrum is eliminated, then, with the correct choice of Jeffreys-Wentzel-Kramers-Brillouin (JWKB) matching conditions, the semiclassical spectrum satisfies the Einstein-Brillouin-Keller quantization conditions which include the correction due to the Maslov index. However, semiclassical analysis uses different choices for boundary conditions, continuity requirements, and the domain of definition. For example, use of the complex JWKB method allows for the treatment of tunneling through the complexification of phase space. Finally, although KvH wavefunctions include the possibility of interference effects, interference is not observable when all observables are approximated as local operators on phase space. Observing interference effects requires consideration of nonlocal operations, e.g. through higher orders in the asymptotic theory.
{{\copyright} 2023 The Author(s). Published by IOP Publishing Ltd}Higher structures in algebraic quantum field theoryhttps://zbmath.org/1535.811692024-07-05T15:31:27.292712Z"Benini, Marco"https://zbmath.org/authors/?q=ai:benini.marco.1"Schenkel, Alexander"https://zbmath.org/authors/?q=ai:schenkel.alexanderSummary: A brief overview of the recent developments of operadic and higher categorical techniques in algebraic quantum field theory is given. The relevance of such mathematical structures for the description of gauge theories is discussed.
{\copyright} 2019 WILEY-VCH Verlag GmbH \& Co. KGaA, WeinheimThe rational higher structure of M-theoryhttps://zbmath.org/1535.811962024-07-05T15:31:27.292712Z"Fiorenza, Domenico"https://zbmath.org/authors/?q=ai:fiorenza.domenico"Sati, Hisham"https://zbmath.org/authors/?q=ai:sati.hisham"Schreiber, Urs"https://zbmath.org/authors/?q=ai:schreiber.ursSummary: We review how core structures of string/M-theory emerge as \textit{higher structures} in super homotopy theory; namely from systematic analysis of the \textit{brane bouquet} of universal invariant higher central extensions growing out of the superpoint. Since super homotopy theory is immensely rich, to start with we consider this in the rational/infinitesimal approximation which ignores torsion-subgroups in brane charges and focuses on tangent spaces of super space-time. Already at this level, super homotopy theory discovers all super \(p\)-brane species, their intersection laws, their M/IIA-, T- and S-duality relations, their black brane avatars at ADE-singularities, including their instanton contributions, and, last not least, Dirac charge quantization: for the D-branes it recovers twisted K-theory, rationally, but for the M-branes it gives \textit{cohomotopy cohomology theory}. We close with an outlook on the lift of these results beyond the rational/infinitesimal approximation to a candidate formalization of microscopic M-theory in super homotopy theory.
{\copyright} 2019 WILEY-VCH Verlag GmbH \& Co. KGaA, WeinheimHow space-times emerge from the superpointhttps://zbmath.org/1535.811992024-07-05T15:31:27.292712Z"Huerta, John"https://zbmath.org/authors/?q=ai:huerta.johnSummary: We describe how the super Minkowski space-times relevant to string theory and M-theory, complete with their Lorentz metrics and spin structures, emerge from a much more elementary object: the superpoint. In the sense of higher structures, this comes from treating the superpoint as an object in a flavor of rational homotopy theory, and repeatedly constructing central extensions. We will fit this story into the larger picture of the brane bouquet of \textit{D. Fiorenza} et al. [Int. J. Geom. Methods Mod. Phys. 12, No. 2, Article ID 1550018, 35 p. (2015; Zbl 1309.81216)]: string theories and membrane theories emerge from super Minkowski space-times in precisely the same way as the super Minkowski space-times themselves emerge from the superpoint. This note is adapted from a talk I gave at the Durham symposium \textit{Higher Structures in M-Theory}.
{\copyright} 2019 WILEY-VCH Verlag GmbH \& Co. KGaA, Weinheim