Recent zbMATH articles in MSC 55https://zbmath.org/atom/cc/552024-03-13T18:33:02.981707ZWerkzeugCategorical structures for type theory in univalent foundationshttps://zbmath.org/1528.031002024-03-13T18:33:02.981707Z"Ahrens, Benedikt"https://zbmath.org/authors/?q=ai:ahrens.benedikt"Lumsdaine, Peter LeFanu"https://zbmath.org/authors/?q=ai:lumsdaine.peter-lefanu"Voevodsky, Vladimir"https://zbmath.org/authors/?q=ai:voevodsky.vladimir-aleksandrovichSummary: In this paper, we analyze and compare three of the many algebraic structures that have been used for modeling dependent type theories: categories with families, split type-categories, and representable maps of presheaves. We study these in the setting of univalent foundations, where the relationships between them can be stated more transparently. Specifically, we construct maps between the different structures and show that these maps are equivalences under suitable assumptions. \par We then analyze how these structures transfer along (weak and strong) equivalences of categories, and, in particular, show how they descend from a category (not assumed univalent/saturated) to its Rezk completion. To this end, we introduce relative universes, generalizing the preceding notions, and study the transfer of such relative universes along suitable structure. \par We work throughout in (intensional) dependent type theory; some results, but not all, assume the univalence axiom. All the material of this paper has been formalized in Coq, over the UniMath library.
For the entire collection see [Zbl 1372.68009].Univalent foundations and the equivalence principlehttps://zbmath.org/1528.031012024-03-13T18:33:02.981707Z"Ahrens, Benedikt"https://zbmath.org/authors/?q=ai:ahrens.benedikt"North, Paige Randall"https://zbmath.org/authors/?q=ai:north.paige-randallSummary: In this paper, we explore the `equivalence principle' (EP): roughly, statements about mathematical objects should be invariant under an appropriate notion of equivalence for the kinds of objects under consideration. In set theoretic foundations, EP may not always hold: for instance, `\(1 \in \mathbb{N} \)' under isomorphism of sets. In univalent foundations, on the other hand, EP has been proven for many mathematical structures. We first give an overview of earlier attempts at designing foundations that satisfy EP. We then describe how univalent foundations validates EP.
For the entire collection see [Zbl 1502.03003].Univalent foundations and the unimath libraryhttps://zbmath.org/1528.031042024-03-13T18:33:02.981707Z"Bordg, Anthony"https://zbmath.org/authors/?q=ai:bordg.anthonySummary: We give a concise presentation of the Univalent Foundations of mathematics outlining the main ideas, followed by a discussion of the UniMath library of formalized mathematics implementing the ideas of the Univalent Foundations (Sect. 8.1), and the challenges one faces in attempting to design a large-scale library of formalized mathematics (Sect. 8.2). This leads us to a general discussion about the links between architecture and mathematics where a meeting of minds is revealed between architects and mathematicians (Sect. 8.3). On the way our odyssey from the foundations to the ``horizon'' of mathematics will lead us to meet the mathematicians David Hilbert and Nicolas Bourbaki as well as the architect Christopher Alexander.
For the entire collection see [Zbl 1502.03003].Higher structures in homotopy type theoryhttps://zbmath.org/1528.031052024-03-13T18:33:02.981707Z"Buchholtz, Ulrik"https://zbmath.org/authors/?q=ai:buchholtz.ulrikSummary: The intended model of the homotopy type theories used in Univalent Foundations is the \(\infty \)-category of homotopy types, also known as \(\infty \)-groupoids. The problem of \textit{higher structures} is that of constructing the homotopy types needed for mathematics, especially those that aren't sets. The current repertoire of constructions, including the usual type formers and higher inductive types, suffice for many but not all of these. We discuss the problematic cases, typically those involving an infinite hierarchy of coherence data such as semi-simplicial types, as well as the problem of developing the meta-theory of homotopy type theories in Univalent Foundations. We also discuss some proposed solutions.
For the entire collection see [Zbl 1502.03003].Homotopy canonicity for cubical type theoryhttps://zbmath.org/1528.031062024-03-13T18:33:02.981707Z"Coquand, Thierry"https://zbmath.org/authors/?q=ai:coquand.thierry"Huber, Simon"https://zbmath.org/authors/?q=ai:huber.simon"Sattler, Christian"https://zbmath.org/authors/?q=ai:sattler.christianSummary: Cubical type theory provides a constructive justification of homotopy type theory and satisfies canonicity: every natural number is convertible to a numeral. A crucial ingredient of cubical type theory is a path lifting operation which is explained computationally by induction on the type involving several non-canonical choices. In this paper we show by a sconing argument that if we remove these equations for the path lifting operation from the system, we still retain homotopy canonicity: every natural number is path equal to a numeral.
For the entire collection see [Zbl 1414.68005].Decomposing the univalence axiomhttps://zbmath.org/1528.031102024-03-13T18:33:02.981707Z"Orton, Ian"https://zbmath.org/authors/?q=ai:orton.ian"Pitts, Andrew M."https://zbmath.org/authors/?q=ai:pitts.andrew-mSummary: This paper investigates Voevodsky's univalence axiom in intensional Martin-Löf type theory. In particular, it looks at how univalence can be derived from simpler axioms. We first present some existing work, collected together from various published and unpublished sources; we then present a new decomposition of the univalence axiom into simpler axioms. We argue that these axioms are easier to verify in certain potential models of univalent type theory, particularly those models based on cubical sets. Finally we show how this decomposition is relevant to an open problem in type theory.
For the entire collection see [Zbl 1407.68018].Models of HoTT and the constructive view of theorieshttps://zbmath.org/1528.031122024-03-13T18:33:02.981707Z"Rodin, Andrei"https://zbmath.org/authors/?q=ai:rodin.andrei-sSummary: Homotopy Type theory and its Model theory provide a novel formal semantic framework for representing scientific theories. This framework supports a constructive view of theories according to which a theory is essentially characterised by its methods. The constructive view of theories was earlier defended by Ernest Nagel and a number of other philosophers of the past but available logical means did not allow these people to build formal representational frameworks that implement this view.
For the entire collection see [Zbl 1502.03003].Cubical assemblies, a univalent and impredicative universe and a failure of propositional resizinghttps://zbmath.org/1528.031142024-03-13T18:33:02.981707Z"Uemura, Taichi"https://zbmath.org/authors/?q=ai:uemura.taichiSummary: We construct a model of cubical type theory with a univalent and impredicative universe in a category of cubical assemblies. We show that this impredicative universe in the cubical assembly model does not satisfy a form of propositional resizing.
For the entire collection see [Zbl 1433.68014].Additive posets, CW-complexes, and graphshttps://zbmath.org/1528.060082024-03-13T18:33:02.981707Z"Turaev, Vladimir"https://zbmath.org/authors/?q=ai:turaev.vladimir-gSummary: We introduce and study additive posets. We show that the top homology group (with coefficients in \(\mathbb{Z}/2\mathbb{Z}\)) of a finite dimensional CW-complex carries a structure of an additive poset invariant under subdivisions. Applications to CW-complexes and graphs are discussed.Denominators of special values of \(\zeta\)-functions count \(KU\)-local homotopy groups of mod \(p\) Moore spectrahttps://zbmath.org/1528.110772024-03-13T18:33:02.981707Z"Salch, Andrew"https://zbmath.org/authors/?q=ai:salch.andrewSummary: For each odd prime \(p\), we show that the orders of the \(KU\)-local homotopy groups of the \(\operatorname{mod}\,p\) Moore spectrum are equal to denominators of special values of certain quotients of Dedekind zeta-functions of totally real number fields. With this observation in hand, we give a cute topological proof of the Leopoldt conjecture for those number fields, by showing that it is a consequence of periodicity properties of \(KU\)-local stable homotopy groups.Bad representations and homotopy of character varietieshttps://zbmath.org/1528.140062024-03-13T18:33:02.981707Z"Guérin, Clément"https://zbmath.org/authors/?q=ai:guerin.clement"Lawton, Sean"https://zbmath.org/authors/?q=ai:lawton.sean"Ramras, Daniel"https://zbmath.org/authors/?q=ai:ramras.daniel-aGiven a connected reductive complex affine algebraic group \(G\) and a finitely generated group \(\Gamma\), the \(G\)-character variety of \(\Gamma\) is defined as the GIT quotient
\[
\mathcal{X}_{\Gamma}(G)=\Hom(\Gamma,G) /\!\!/ G,
\]
where \(G\) acts by conjugation. Character varieties play an important role in several areas, such as Representation Theory, Nonabelian Hodge Theory or Geometric Topology. In this paper, the authors focus on the case where \(\Gamma=F_r\), the free group of rank r, and compute the higher homotopy groups \(\pi_k(\mathcal{X}_{F_r}(G))\), for \(0\leq k \leq 4\), extending previous results [\textit{C. Florentino} et al., Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 17, No. 1, 143--185 (2017; Zbl 1403.14011)]. They also prove that
\[
\pi_k(\mathcal{X}_{F_r}(G))\cong \pi_k(G)^r \times \pi_{k-1}(PG),
\]
in a certain range, where \(PG\) is the quotient of \(G\) by its center \(Z(G)\), for both classical \(\left( \text{type } A_n,B_n,C_n,D_n \right)\) and exceptional groups \(G\) \(\left(G_2,F_4, E_6,E_7,E_8 \right)\).
Results are based on a detailed analysis of the singular locus of the character variety, which leads to the study of bad and ugly representations, which are irreducible representations whose \(G\)-stabilizer is larger than \(Z(G)\) and representations that produce topological singularities of \(\mathcal{X}_{F_r}(G)\), respectively. Proofs rely on previous results of Richardson on the algebraic singularities of \(G^r\) [\textit{R. W. Richardson}, Duke Math. J. 57, No. 1, 1--35 (1988; Zbl 0685.20035)] and codimension bounds of the singular locus, built upon results of \textit{C. Guérin} [J. Group Theory 21, No. 5, 789--816 (2018; Zbl 1437.20027); Geom. Dedicata 195, 23--55 (2018; Zbl 1418.20005)]. Their analysis leads to the study of Borel-de Siebenthal subalgebras of the Lie algebra \(\mathfrak{g}\) of \(G\), which are also classified in the article and used to describe bad representations.
Reviewer: Javier Martínez-Martínez (Madrid)\(\mathbb{Z}\)-local system cohomology of hyperplane arrangements and a Cohen-Dimca-Orlik type theoremhttps://zbmath.org/1528.140672024-03-13T18:33:02.981707Z"Sugawara, Sakumi"https://zbmath.org/authors/?q=ai:sugawara.sakumiThis article generalizes a theorem by \textit{D. C. Cohen} et al. [Ann. Inst. Fourier 53, No. 6, 1883--1896 (2003; Zbl 1054.32016)] for certain generic \(\mathbb{C}\)-local systems to \(\mathbb{Z}\)-local systems.
Consider a complexified arrangement \(\mathcal{A}= \{H_1, \dots, H_n\}\) of hyperplanes in \(\mathbb{C}^n\) (with real equations) and a \(\mathbb{Z}\)-local system \(\mathcal{L}\) on the complement \(M(\mathcal{A}) = \mathbb{C}^n \setminus \bigcup_{i=1}^n H_i\). Assume that the local system \(\mathcal{L}\) satisfies a certain CDO condition, defined in [loc. cit.].
{Theorem}. In this setting
\[
H^k(M(\mathcal{A}); \mathcal{L}) =
\begin{cases}
\mathbb{Z}^{\beta_l} \oplus \mathbb{Z}_2^{\beta_{l-1}} & k=l \\
\mathbb{Z}_2^{\beta_{k-1}} & 1 \leq k \leq l-1 \\
0 & \text{otherwise}
\end{cases}
\]
where \(\beta_k= \lvert \sum_{i=0}^k (-1)^i b_i(M(\mathcal{A})) \rvert\).
The proof is based on the minimality of the complement \(M(\mathcal{A})\) and on a careful algebraic-combinatorial inspection of an explicit complex.
Reviewer: Roberto Pagaria (Bologna)\(S\)-colocalization and Adams cocompletionhttps://zbmath.org/1528.180042024-03-13T18:33:02.981707Z"Choudhury, Snigdha Bharati"https://zbmath.org/authors/?q=ai:choudhury.snigdha-bharati"Behera, A."https://zbmath.org/authors/?q=ai:behera.abhisek-k|behera.adikanda|behera.abhishek|behera.ajay-kumar|behera.amit|behera.amiya-kumar|behera.akrur\textit{A. K. Bousfield} [Topology 14, 133--150 (1975; Zbl 0309.55013)] introduced the idea of localization in categories, explaining the determination of an \(S\)-localization\ functor \(E:\mathcal{C} \rightarrow\mathcal{C}\) by a class of morphisms \(S\) in a category \(\mathcal{C}\). \textit{A. Deleanu} et al. [Cah. Topologie Géom. Différ. Catégoriques 15, 61--82 (1974; Zbl 0291.18003)] introduced the idea of Adams completion [\textit{J. F. Adams}, Proc. int. Conf. Manifolds relat. Top. Topol. 1973, 247--253 (1975; Zbl 0313.55011); Stable homotopy and generalised homology. Chicago - London: The University of Chicago Press (1974; Zbl 0309.55016); Localisation and completion. Chicago, Illinois: Department of Mathematics (1975; Zbl 0527.55016)], suggesting its dual notion known as the Adams cocompletion.
The principal objective in this paper is to deduce an isomorphism between the notions of \(S\)-colocalization [\textit{W. G. Dwyer}, NATO Sci. Ser. II, Math. Phys. Chem. 131, 3--28 (2004; Zbl 1062.55009)], the dual of \(S\)-localization\ of an object, and the Adams cocompletion of the same object in a complete small \(\mathcal{U}\)-category together with a specific set of morphisms \(S\) and a fixed Grothendieck universe \(\mathcal{U}\) [\textit{H. Schubert}, Categories. Translated from the German by Eva Gray. Berlin-Heidelberg-New York: Springer-Verlag (1972; Zbl 0253.18002)].
Reviewer: Hirokazu Nishimura (Tsukuba)On the coverings of Hantzsche-Wendt manifoldhttps://zbmath.org/1528.200922024-03-13T18:33:02.981707Z"Chelnokov, Grigory"https://zbmath.org/authors/?q=ai:chelnokov.grigory-r"Mednykh, Alexander"https://zbmath.org/authors/?q=ai:mednykh.alexander-dThe authors' paper is a part of a series of papers devoted to the enumeration of finite-sheeted coverings over closed Euclidean 3-manifolds, which are also known as flat 3-dimensional manifolds or Euclidean 3-forms.
In their previous papers, the authors also described isomorphism types of finite index subgroups \(H\) in the fundamental group \(G\) of manifolds \(\mathcal{B}_1\) and \(\mathcal{B}_2\) and calculated the respective numbers \(s_{H,G}(n)\) and \(c_{H,G}(n)\) for each isomorphism type \(H\). Similar questions also were solved by the authors for manifolds \(\mathcal{G}_2\), \(\mathcal{G}_3\), \(\mathcal{G}_4\), \(\mathcal{G}_5\), \(\mathcal{B}_3\) and \(\mathcal{B}_4\) too.
In the reviewed paper, the authors solve the same questions for the Hantzsche-Wendt manifold \(\mathcal{G}_6,\) undoubtedly the most weird among Euclidean \(3\)-manifolds. This is the unique Euclidean \(3\)-form with finite first homology group \(H_1(\mathcal{G}_6) = \mathbb{Z}^2_4.\) The authors also classify finite index subgroups in the fundamental group \(\pi_1(\mathcal{G}_{6})\) up to isomorphism. Given index \(n\), the authors calculate the numbers of subgroups and the numbers of conjugacy classes of subgroups for each isomorphism type and provide the Dirichlet generating series for the above sequences too.
Reviewer: Stepan Moskaliuk (Wien)Signature cocycles on the mapping class group and symplectic groupshttps://zbmath.org/1528.200972024-03-13T18:33:02.981707Z"Benson, Dave"https://zbmath.org/authors/?q=ai:benson.david-john"Campagnolo, Caterina"https://zbmath.org/authors/?q=ai:campagnolo.caterina"Ranicki, Andrew"https://zbmath.org/authors/?q=ai:ranicki.andrew-a"Rovi, Carmen"https://zbmath.org/authors/?q=ai:rovi.carmenSummary: \textit{W. Meyer} [Math. Ann. 201, 239--264 (1973; Zbl 0241.55019)] constructed a cocycle in \(H^2(\mathsf{Sp}(2g, \mathbb{Z}); \mathbb{Z})\) which computes the signature of a closed oriented surface bundle over a surface. By studying properties of this cocycle, he also showed that the signature of such a surface bundle is a multiple of 4. In this paper, we study signature cocycles both from the geometric and algebraic points of view. We present geometric constructions which are relevant to the signature cocycle and provide an alternative to Meyer's decomposition of a surface bundle. Furthermore, we discuss the precise relation between the Meyer and Wall-Maslov index. The main theorem of the paper, Theorem 6.6, provides the necessary group cohomology results to analyze the signature of a surface bundle modulo any integer \(N\). Using these results, we are able to give a complete answer for \(N = 2, 4, \text{ and } 8\), and based on a theorem of \textit{P. Deligne} [C. R. Acad. Sci., Paris, Sér. A 287, 203--208 (1978; Zbl 0416.20042)], we show that this is the best we can hope for using this method.The homotopy-invariance of constructible sheaveshttps://zbmath.org/1528.320462024-03-13T18:33:02.981707Z"Haine, Peter J."https://zbmath.org/authors/?q=ai:haine.peter-j"Porta, Mauro"https://zbmath.org/authors/?q=ai:porta.mauro"Teyssier, Jean-Baptiste"https://zbmath.org/authors/?q=ai:teyssier.jean-baptisteSummary: In this paper we show that the functor sending a stratified topological space \(S\) to the \(\infty\)-category of constructible (hyper)sheaves on \(S\) with coefficients in a large class of presentable \(\infty\)-categories is homotopy-invariant. To do this, we first establish a number of results for locally constant (hyper)sheaves. For example, if \(X\) is a locally weakly contractible topological space and \(\mathcal{E}\) is a presentable \(\infty\)-category, then we give a concrete formula for the constant hypersheaf functor \(\mathcal{E}\to\mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})\), implying that the constant hypersheaf functor is a right adjoint, and is fully faithful if \(X\) is also weakly contractible. It also lets us prove a general monodromy equivalence and categorical Künneth formula for locally constant hypersheaves.Quasicircles and quasiperiodic surfaces in pseudo-hyperbolic spaceshttps://zbmath.org/1528.530422024-03-13T18:33:02.981707Z"Labourie, François"https://zbmath.org/authors/?q=ai:labourie.francois"Toulisse, Jérémy"https://zbmath.org/authors/?q=ai:toulisse.jeremyThe pseudo-hyperbolic space \(\mathbb{H}^{2,n}\) can be defined as the space of unit time-like vectors in \(\mathbb{R}^{n+3}\) endowed with a bilinear form of signature \((2, n+1)\). It is a pseudo-Riemannian manifold of constant sectional curvature \(-1\). Its boundary at infinity \(\partial_{\infty}\mathbb{H}^{2,n}\) is called Einstein Universe.
The paper studies pointed complete maximal surfaces in \(\mathbb{H}^{2,n}\). The group \(\mathrm{SO}_{0}(2, n+1)\) acts cocompactly on the space \(\mathcal{M}(n)\) of complete pointed maximal surfaces in \(\mathbb{H}^{2,n}\). A pointed maximal surface \((x,\Sigma)\) is called quasiperiodic if its orbit closure in \(\mathcal{M}(n)\) does not contain the Barbot surface, which is the unique flat complete maximal surface in \(\mathbb{H}^{2,n}\). The main result of the paper is the following:
Theorem. Let \(\Sigma\) be a complete maximal surfaces in \(\mathbb{H}^{2,n}\). Then the following are equivalent:
\begin{itemize}
\item[1.] \(\Sigma\) is quasiperiodic;
\item[2.] The induced metric on \(\Sigma\) has curvature bounded above by some negative constant;
\item[3.] The induced metric on \(\Sigma\) is Gromov hyperbolic;
\item[4.] \(\Sigma\) is of conformal hyperbolic type and any uniformization is bi-Lipschitz;
\item[5.] The limit curve \(\partial_{\infty}\Sigma\) is the image of a quasisymmetric map \(S^{1} \rightarrow \partial_{\infty}\mathbb{H}^{2,n}\).
\end{itemize}
As a consequence, the authors deduce that any \(C^{1}\) space-like curve in \(\partial_{\infty}\mathbb{H}^{2,n}\) is quasisymmetric and bounds a complete maximal surface, with curvature tending to \(-1\) at infinity.
Reviewer: Andrea Tamburelli (Houston)Iterated \(S^3\) Sasaki joins and Bott orbifoldshttps://zbmath.org/1528.530452024-03-13T18:33:02.981707Z"Boyer, Charles P."https://zbmath.org/authors/?q=ai:boyer.charles-p"Tønnesen-Friedman, Christina W."https://zbmath.org/authors/?q=ai:tonnesen-friedman.christina-wiisSummary: We present a categorical relationship between iterated \(S^3\) Sasaki-joins and Bott orbifolds. Then we show how to construct smooth Sasaki-Einstein (SE) structures on the iterated joins. These become increasingly complicated as dimension grows. We give an explicit nontrivial construction of (infinitely many) smooth SE structures up through dimension eleven, and conjecture the existence of smooth SE structures in all odd dimensions.Dominant energy condition and spinors on Lorentzian manifoldshttps://zbmath.org/1528.530632024-03-13T18:33:02.981707Z"Ammann, Bernd"https://zbmath.org/authors/?q=ai:ammann.bernd-eberhard"Glöckle, Jonathan"https://zbmath.org/authors/?q=ai:glockle.jonathanThis article is devoted to the study of the topology of initial data sets \(\mathcal{I}^{\ge}(M)\), in particular their homotopy groups, satisfying the dominant energy condition. The authors use index theory and a Lorentzian version of the index difference in order to detect homotopy groups of \(\mathcal{I}^{\ge}(M)\), and construct non-trivial elements in \(\pi_{k}(\mathcal{I}^{\ge}(M))\) combining known methods for constructing non-trivial homotopy groups in the space of positive scalar curvature metrics together with a given suspension mechanism. The index theory used by the authors revolves around the Dirac-Witten operator, introduced by \textit{E. Witten} [Commun. Math. Phys. 80, 381--402 (1981; Zbl 1051.83532)] and formalized by \textit{T. Parker} and \textit{C. H. Taubes} [Commun. Math. Phys. 84, 223--238 (1982; Zbl 0528.58040)] in their proof of the positive mass theorem in general relativity, which is obtained from the standard Dirac operator via a modification depending on a symmetric two-form. For initial data satisfying the strict dominant energy condition, the Dirac-Witten operator is an invertible, self-adjoint, Fredholm operator. This allowed the second author to study the topology of initial data sets satisfying the strict dominant energy condition in a prior publication via a Lorentzian analog of the \(\alpha\)-invariant introduced by \textit{N. J. Hitchin} [Adv. Math. 14, 1--55 (1974; Zbl 0284.58016)]. In this regard, the present article focuses on the differences and difficulties that arise when the strictness in the dominant energy condition is relaxed. In this case, the Diract-Witten operator may be non-invertible and the authors study its kernel, proving that spinors in this kernel define solutions to the initial data problem determined by a parallel spinor. Given this connection, the authors characterize compact manifolds admitting such initial data, proving that their fundamental group is virtually solvable of derived length at most two.
For the entire collection see [Zbl 1517.53004].
Reviewer: Carlos Shabazi (Hamburg)Proper equivariant stable homotopy theoryhttps://zbmath.org/1528.550012024-03-13T18:33:02.981707Z"Degrijse, Dieter"https://zbmath.org/authors/?q=ai:degrijse.dieter"Hausmann, Markus"https://zbmath.org/authors/?q=ai:hausmann.markus"Lück, Wolfgang"https://zbmath.org/authors/?q=ai:luck.wolfgang"Patchkoria, Irakli"https://zbmath.org/authors/?q=ai:patchkoria.irakli"Schwede, Stefan"https://zbmath.org/authors/?q=ai:schwede.stefanThis monograph establishes an analogue of genuine equivariant homotopy theory for compact Lie groups for arbitrary Lie groups \(G\) and studies its fundamental properties. The main idea to retain all the structure implicit in the term ``genuine'' (e.g.\ transfers) is to restrict attention to fixed point information coming from compact subgroups. The authors choose to describe this homotopy theory through a model structure on orthogonal \(G\)-spectra whose weak equivalences are those maps which induce a \(\pi_*^H\)-isomorphism for every compact subgroup \(H\) of \(G\). The underlying stable \(\infty\)-category, which the authors describe in terms of the triangulated category \(\mathrm{Ho}(\mathrm{Sp}_G)\), is compactly generated, comes equipped with a symmetric monoidal structure and carries a canonical t-structure. In particular, Brown representability holds for \(\mathrm{Ho}(\mathrm{Sp}_G)\).
After introducing the equivariant homotopy groups of an orthogonal \(G\)-spectrum and discussing (non-)compactness of the equivariant sphere spectrum, the authors turn their attention to the case of a discrete group \(G\). There, the equivariant homotopy groups of an orthogonal \(G\)-spectrum can be refined to a \(G\)-Mackey functor (in the sense of an additive functor on the category of spans of cofinite \(G\)-sets with finite stabilisers). It turns out that the heart of the t-structure on \(\mathrm{Ho}(\mathrm{Sp}_G)\) is equivalent to the category of \(G\)-Mackey functors. After rationalisation, \(\mathrm{Ho}(\mathrm{Sp}_G)\) becomes equivalent to the derived category of rational \(G\)-Mackey functors.
The final section is dedicated to the study of cohomology theories on finite proper \(G\)-CW-complexes. It is shown that every orthogonal \(G\)-spectrum represents such a cohomology theory. In the discrete case, the authors also provide an alternative description of these cohomology theories in terms of parametrised homotopy theory.
The discussion also includes a description of change-of-groups functors. In particular, the authors show that a Lie group homomorphism \(H \to G\) which is also a homotopy equivalence on underlying spaces induces an equivalence \(\mathrm{Ho}(\mathrm{Sp}_H) \simeq \mathrm{Ho}(\mathrm{Sp}_G)\) of tensor triangulated categories.
The treatment also includes a number of key examples. This includes stable cohomotopy, Borel cohomology and equivariant topological K-theory.
Reviewer: Christoph Winges (Regensburg)Homology of the \(MSU\) spectrumhttps://zbmath.org/1528.550022024-03-13T18:33:02.981707Z"Abramyan, Semyon A."https://zbmath.org/authors/?q=ai:abramyan.semyon-aThe paper is devoted to the description of the Thom spectrum \(MSU\) of the SU-bordism theory and can serve as a review on this classical subject of algebraic topology. The theorem by S. P. Novikov asserts that the SU-bordism ring, after the number 2 is reversed, is isomorphic to the polynomial algebra with infinitely many generators: \(\mathbb{Z}[\frac{1}{2}][y_2,y_3,\ldots]\), where \(\deg(y_n)=2n\) for all \(n\geq 2\). The author provides a complete proof of this well-known result based on the use of the Adams spectral sequence and a description of the comodule structure of \(H_\ast(MSU;F_p)\) over the dual mod \(p\) Steenrod algebra, where \(p\) is an odd prime. The aim of the author is to fill in the gap in the literature dedicated to these classical results.
Reviewer: Ivan Limonchenko (Toronto)Tate blueshift and vanishing for real oriented cohomology theorieshttps://zbmath.org/1528.550032024-03-13T18:33:02.981707Z"Li, Guchuan"https://zbmath.org/authors/?q=ai:li.guchuan"Lorman, Vitaly"https://zbmath.org/authors/?q=ai:lorman.vitaly"Quigley, J. D."https://zbmath.org/authors/?q=ai:quigley.james-dIn 1842, Christian Doppler described the phenomenon by which electromagnetic radiation emitted by objects moving away from an observer has increased wavelength. In the visible light spectrum, this increase corresponds to a shift towards red light, hence the term redshift. The opposite phenomenon in which wavelength is decreased when objects approach an observer is called blueshift.
The stable homotopy theory of \(p\)-local finite complexes is filtered by an infinite discrete set of `wavelengths' \(0\le n\le \infty\) referred to as chromatic heights. If an operation raises chromatic height, we say it exhibits \textit{redshift}, while an operation lowering chromatic height exhibits \textit{blueshift}. If we are willing to strain the analogy past the point of usefulness, we may take ourselves to be fixed observers of stable homotopical phenomena and envision a blueshift operator hurtling objects toward us.
In the work under review, the authors prove that the Tate construction relative to a trivial \(C_2\)-action on Real Johnson-Wilson spectra exhibits blueshift. In particular, the height \(n\) spectrum \(ER(n)\) has a \(C_2\)-Tate construction that splits as a wedge of suspensions of \(ER(n-1)\)'s. Furthermore, a similar Tate construction on real Morava \(K\)-theory \(KR(n)\) creates a nullhomotopic spectrum, a phenomenon referred to as \textit{Tate vanishing}.
The Tate construction in question is familiar to equivariant homotopy theorists, and arises as the homotopy cofiber of the canonical map from homotopy orbits to homotopy fixed points. Real Johnson-Wilson spectra and real Morava \(K\)-theory spectra are defined as the categorical fixed points of \(C_2\)-equivariant lifts of Johnson-Wilson spectra and Morava \(K\)-theories which are real oriented (\textit{i.e.}, admit a Thom class for real vector bundles in the sense of Atiyah).
The results here are of special interest since \(ER(n)\) and \(KR(n)\) are \textit{not} complex oriented. Rather than simply mimicking classical proofs, the authors must effect some labor intensive calculations involving the parametrized Tate construction (in genuine \(C_2\)-equivariant spectra) to prove their theorems, and their techniques should prove useful in an array of computational stable homotopy theory settings.
Reviewer: Kyle Ormsby (Portland)A directed persistent homology theory for dissimilarity functionshttps://zbmath.org/1528.550042024-03-13T18:33:02.981707Z"Méndez, David"https://zbmath.org/authors/?q=ai:mendez.david.1|mendez.david"Sánchez-García, Rubén J."https://zbmath.org/authors/?q=ai:sanchez-garcia.ruben-jThe authors develop a theory of persistent homology for directed simplicial complexes which detects persistent directed cycles in odd dimensions, unlike the traditional approach of persistent homology. They relate directed persistent homology to classical persistent homology, prove some stability results, and discuss the computational challenges of this approach. The directed persistent homology theory is motivated by homology with semiring coefficients: by explicitly removing additive inverses, it is possible to detect directed cycles algebraically. The motivation to develop directed persistent homology is the applications to real-world data sets where asymmetry or directionality, in particular the presence of directed cycles, plays a fundamental role. Examples include biological neural networks (directed synaptic connections), time series data (directed temporal connections) or biological molecular networks such as protein-protein interaction networks (inhibitory and excitatory connections).
Reviewer: Philippe Gaucher (Paris)Recovering the homology of immersed manifoldshttps://zbmath.org/1528.550052024-03-13T18:33:02.981707Z"Tinarrage, Raphaël"https://zbmath.org/authors/?q=ai:tinarrage.raphaelIn the paper under review, the author describes a new method to recover the homology of an abstract manifold from a sample of this manifold immersed in some Euclidean space. In order to estimate the homology of a manifold from an immersion of it, the author proposes to estimate its tangent bundle, seen as a subset of another Euclidean space. But, in the process of estimating this tangent bundle, there are some errors, which will result in anomalous points. This issue can be solved by using the distance-to-measure filtrations, which require to use a measure theoretical framework. It is shown that the method of estimating tangent bundles is stable in Wasserstein distance. Using distance-to-measure filtrations the author can define a filtration of the space \(\mathbb R^n\times M(\mathbb R^n)\) whose persistence module contains information about the homology of the original abstract manifold. The method given to estimate the tangent bundles of manifolds may be used to estimate other topological invariants like characteristic classes.
Reviewer: Semra Pamuk (Ankara)The toral rank conjecture and variants of equivariant formalityhttps://zbmath.org/1528.550062024-03-13T18:33:02.981707Z"Amann, Manuel"https://zbmath.org/authors/?q=ai:amann.manuel"Zoller, Leopold"https://zbmath.org/authors/?q=ai:zoller.leopoldThis work concerns the restrictions on the topology of a space, \(M\), inherited from the existence of an almost free, non-trivial, action of a Lie group \(G\). (Recall that an action is said almost free if all isotropy groups are finite.) As a typical result in this direction, \textit{W. Y. Hsiang} proved [Cohomology theory of topological transformation groups. Berlin-Heidelberg-New York: Springer-Verlag (1975; Zbl 0429.57011)] that, if \(G\) is compact and \(M\) is a compact \(G\)-manifold, then the action is almost free if, and only if, the rational equivariant cohomology of \(M\) is finite dimensional.
More precisely, the main goal of this paper is the study of a long standing conjecture of S. Halperin, called the toral rank conjecture or TRC, and which is stated as follows: if a (reasonable) space is equipped with an almost free action of a torus of dimension \(n\), then the sum of all its Betti numbers is greater than or equal to \(2^n\). This conjecture appears in [\textit{S. Halperin}, Lond. Math. Soc. Lect. Note Ser. 93, 293--306 (1985; Zbl 0562.57015)] where the author proves that the TRC is true for homogeneous spaces, \(G/H\), with \(G\) connected and \(H\) closed and connected. Next, this conjecture has been the subject of numerous publications. For instance it has been verified for Kähler manifolds, some nilmanifolds, or with dimensional bounds linear in \(n\), but it is still open in general.
Here, the authors extend previous works in two directions when \(X\) is a compact Hausdorff \(T^n\)-space. -- In the first one, they prove that the TRC is true if the associated Borel construction is formal in the sense of rational homotopy theory. -- For the second one, their hypothesis is on the equivariant cohomology. Classical equivariant formality of an action means that the Serre spectral sequence of the Borel fibration degenerates at the \(E_{2}\)-term. Here, this notion is replaced by that of hyperformality. Let \(\varphi\colon H^*(BT^n;\mathbb Q)\to H^*_{T^n}(X;\mathbb Q)\) be the map induced by the Borel fibration \(X_{T^n}\to BT^n\). The action is said hyperformal if the kernel of \(\varphi\) is generated by a homogeneous regular sequence. M. Amann and L. Zoller prove that the TRC is true for hyperformal actions.
Apart from these two cases, they also expand the known hypotheses restricting the dimension of the spaces on which the torus operates and under which the TRC is true. The paper contains many concrete examples and a careful study of variations on the definition of hyperformality. Two appendices on minimal models for differential graded modules over a differential commutative graded algebra, or over an \(A_{\infty}\)-algebra, are also included.
Reviewer: Daniel Tanré (Villeneuve d'Ascq)Cup product on relative bounded cohomologyhttps://zbmath.org/1528.550072024-03-13T18:33:02.981707Z"Park, HeeSook"https://zbmath.org/authors/?q=ai:park.hee-sookSummary: In this paper, we define cup product on relative bounded cohomology, and study its basic properties. Then, by extending it to a more generalized formula, we prove that all cup products of bounded cohomology classes of an amalgamated free product \(G_1 \ast_A G_2\) are zero for every positive degree, assuming that free factors \(G_i\) are amenable and amalgamated subgroup \(A\) is normal in both of them. As its consequences, we show that all cup products of bounded cohomology classes of the groups \(\mathbb{Z} \ast \mathbb{Z}\) and \(\mathbb{Z}_n \ast_{\mathbb{Z}_d}\mathbb{Z}_m\), where \(d\) is the greatest common divisor of \(n\) and \(m\), are zero for every positive degree.Self-closeness numbers of non-simply-connected spaceshttps://zbmath.org/1528.550082024-03-13T18:33:02.981707Z"Tong, Yichen"https://zbmath.org/authors/?q=ai:tong.yichenFor a pointed connected CW complex $X$, let $[X,X]$ denote the monoid consisting of all pointed homotopy classes of self-maps $f:X\to X$ whose multiplication is induced from the composition of maps. Let $\mathcal{E}(X)\subset [X,X]$ denote the group of self-homotopy equivalences of $X$. For each integer $n\geq 0$, let $\mathcal{A}^n_{\#}(X)\subset [X,X]$ denote the subset consisting of all $f\in [X,X]$ inducing isomorphisms on the homotopy groups $\pi_k(\#)$ for any $k\leq n$. Note that $\mathcal{A}^n_{\#}(X)$ is a monoid such that there is the following filtration of submonoids
\[
\mathcal{A}^1_{\#}(X)\supset\mathcal{A}^2_{\#}(X)\supset\cdots\supset \mathcal{A}^n_{\#}(X)\supset\mathcal{A}^{n+1}_{\#}(X)\supset\cdots\supset \mathcal{A}^{\infty}_{\#}(X)=\mathcal{E}(X).
\]
Recall that $\mathcal{A}^n_{\#}(X)=\mathcal{E}(X)$ if $\dim X=n$. Then the self-closeness number $\mathrm{N}\mathcal{E}(X)$ is defined as $\mathrm{N}\mathcal{E}(X)=\min\{n:\mathcal{A}^n_{\#}(X)=\mathcal{E}(X)\}. $
Recall that a simply connected space $X$ is called an $\mathrm{H}_0$-space if its rationalization is an $\mbox{H}$-space. For a graded algebra $A$, let $\mbox{d}(A)$ denote the maximal degree of generators of $A$. Note that the cohomological dimension of $X$ is the number defined by \[
\mbox{cd}(X)=\sup\{n:H^n(X,M)\not= 0\mbox{ for some }\pi_1(X)\text{-module } M\}.
\]
In this paper, the author studies the self-closeness number $\mathrm{N}\mathcal{E}(X)$ when $X$ is a non-simply connected CW complex. In particular, he proves the following two results:
Theorem. Let $X$ be a finite non-simply connected complex, let $\tilde{X}$ denote its universal cover, and assume that $\pi_1(X)$ acts trivially on $H^*(\tilde{X};\mathbb{Q}).$ \begin{itemize}
\item[1.] If $\tilde{X}$ is a finite $\mathrm{H}_0$-space and $\mathrm{d}(H^*(\tilde{X};\mathbb{Z}))=\mathrm{d}(H^*(\tilde{X};\mathbb{Q}))$, $\mathrm{N}\mathcal{E}(X)=\mathrm{N}\mathcal{E}(\tilde{X})=\mathrm{d}(H^*(\tilde{X};\mathbb{Q})).$
\item[2.] If $\tilde{X}$ is a finite co-$\mathrm{H}_0$-space and $\mathrm{cd}(X)=\mathrm{d}(H^*(\tilde{X};\mathbb{Q}))$, $\mathrm{N}\mathcal{E}(X)=\mathrm{N}\mathcal{E}(\tilde{X})=\mathrm{d}(H^*(\tilde{X};\mathbb{Q})).$
\end{itemize}
As an application, he computes the self closeness number $\mathrm{N}\mathcal{E}(X)$ for several interesting cases. For example, he computes it for
\begin{align*}
X&=\mbox{the topological spherical space for of dimension }2n-1, \\
X&=(S^{2m+1}\times S^n)/H,\mbox{ or }X=K/G,
\end{align*}
where $H$ is the symmetric group $\Sigma_3$ for order $3$ or $D_{2q}$ (the dihedral group), $K$ is a compact simply connected Lie group with no-torsion in homology, and $G$ is a finite subgroup of $G$.
Reviewer: Kohhei Yamaguchi (Tōkyō)Homology transfer products on free loop spaces: orientation reversal on sphereshttps://zbmath.org/1528.550092024-03-13T18:33:02.981707Z"Kupper, Philippe"https://zbmath.org/authors/?q=ai:kupper.philippeLet \(\Lambda M\) be the free loop space on a compact smooth oriented manifold \(M\). The author studies the topology of quotients \(\land M/G\) for finite subgroups \(G\) of the group \(O(2)\) acting by linear reparametrization of \(S^1\). He uses the existence of a transfer \( H_i(\Lambda M/G; R) \longrightarrow H_i(\Lambda M;R)\) for singular homology with coefficients in a ring \(R\) to lift the Chas-Sullivan product in what he calls the transfer product,
\[
H_i(\Lambda M/G;R) \otimes H_j(\Lambda M/G;R)\longrightarrow H_{i+j-1}(\Lambda M/G;R)\,.
\]
The main result is: The quotient map \(\Lambda M \longrightarrow \Lambda M/G\) induces an algebra isomorphism up to scaling between the rational Chas-Sullivan algebra restricted to classes that are invariant under the action of \(G\) and \(H_\ast(\Lambda M/G;\mathbb Q)\) equipped with the transfer product. In the last section, he computes the transfer product associated to orientation reversal of loops on spheres with rational coefficients. In particular he shows that there are nonnilpotent classes for the transfer product on spheres.
Reviewer: Jean-Claude Thomas (Angers)Stable homotopy groups of spheres: from dimension 0 to 90https://zbmath.org/1528.550102024-03-13T18:33:02.981707Z"Isaksen, Daniel C."https://zbmath.org/authors/?q=ai:isaksen.daniel-c"Wang, Guozhen"https://zbmath.org/authors/?q=ai:wang.guozhen"Xu, Zhouli"https://zbmath.org/authors/?q=ai:xu.zhouliThis paper provides \(2\)-primary information on the stable homotopy groups of spheres up to dimension \(90\), with only a few undetermined cases. Whilst this continues on from the first author's monograph [\textit{D. C. Isaksen}, Stable stems. Providence, RI: American Mathematical Society (AMS) (2019; Zbl 1454.55001)], the authors exploit crucial new ingredients here. This is an impressive body of work, representing a snapshot of substantial progress achieved using modern methods.
A key ingredient is the interplay between classical and \(\mathbb{C}\)-motivic (cellular) homotopy theory. Recall that motivic homotopy theory provides a second grading (the weight) and one has the motivic homotopy class \(\tau\) (working \(2\)-completely); inverting \(\tau\) allows passage to classical homotopy theory. (As the authors note, the usage of \(\mathbb{C}\)-motivic cellular stable homotopy theory can now be circumvented by using synthetic homotopy theory [\textit{P. Pstragowski}, Invent. Math. 232, No. 2, 553--681 (2023; Zbl 07676257)], but essentially applying the same approach to the calculations.)
Here, these methods are made much more powerful and precise by using the results of [\textit{B. Gheorghe} et al., Acta Math. 226, No. 2, 319--407 (2021; Zbl 1478.55006)] which show that \(\mathbb{C}\)-motivic cellular stable homotopy theory is a `deformation' of classical stable homotopy theory: the `generic fibre' is classical stable homotopy theory whereas the `special fibre' is algebraic, identifying with the stable derived category of \(BP_*BP\)-comodules [\textit{M. Hovey}, Contemp. Math. 346, 261--304 (2004; Zbl 1067.18012)]. This is intimately related to the role of the cofibre \(C \tau\) of \(\tau\) in the \(\mathbb{C}\)-motivic stable homotopy category. In particular, the \(\mathbb{C}\)-motivic Adams spectral sequence for \(C \tau\) is isomorphic to the algebraic Novikov spectral sequence that computes the \(E_2\)-page of the \(BP\) Adams-Novikov spectral sequence. The work of the second author [\textit{G. Wang}, Chin. Ann. Math., Ser. B 42, No. 4, 551--560 (2021; Zbl 1471.55001)] has made the latter accessible to machine computation.
A further new ingredient is the usage of the motivic modular forms spectrum \(mmf\). The \(\mathbb{C}\)-motivic Adams spectral sequence of \(mmf\) is understood [\textit{D. C. Isaksen}, Homology Homotopy Appl. 11, No. 2, 251--274 (2009; Zbl 1193.55009); \textit{B. Gheorghe} et al., J. Eur. Math. Soc. (JEMS) 24, No. 10, 3597--3628 (2022; Zbl 1498.14050)]; via the unit map this provides information for the motivic sphere spectrum.
With these tools in hand, the authors proceed as follows. They first compute by machine the cohomology of the \(\mathbb{C}\)-motivic Steenrod algebra and the algebraic Novikov spectral sequence. Using the cell structure of \(C\tau\), Adams differentials for \(C\tau\) can be pushed forward or pulled back to yield Adams differentials for the motivic sphere spectrum.
Further information on Adams differentials is given by standard arguments (such as Toda bracket shuffles) and comparison with the case of \(mmf\). Thereafter, hidden \(\tau\)-extensions are treated.
Finally, inverting \(\tau\) yields the required classical information.
For some background material, the reader is referred to the first author's monograph; some details are provided on the finer points of Adams spectral sequence calculations, such as hidden extensions, Massey products, Toda brackets and their relations.
Of necessity, the paper is not self-contained: the charts and data sets (available on Zenobo, as per references) are essential companions. Indeed, one of the main results is stated as follows: \textit{The classical Adams spectral sequence for the sphere spectrum is displayed in the charts in [\textit{D. C. Isaksen} et al., ``Classical and \(\mathbf{C} \)-motivic Adams charts'', Zenobo (2022; \url{doi:10.5281/zenodo.6987156})], up to the \(90\)-stem}.
Most of the results covered in the text are provided in tabular form in the final section of the paper (containing 25 tables over 31 pages). These are mostly for the motivic setting, using the trigrading \((s,f,w)\), where \(s\) is the stem, \(f\) is the Adams filtration, and \(w\) is the motivic weight. The body of the paper serves to explain how these computations are obtained and to offer a guide to the techniques used. The text is not necessarily intended for linear reading; the user will truly appreciate its value when delving deep to understand particular calculations and phenomena.
The authors document carefully the remaining unresolved uncertainties that start in dimension \(84\). There are some differentials that are undetermined and also some unresolved hidden \(2\)-extensions. This means that the additive structure of a few stable homotopy groups in this range is not determined here.
Reviewer: Geoffrey Powell (Angers)Abelian cycles in the homology of the Torelli grouphttps://zbmath.org/1528.550112024-03-13T18:33:02.981707Z"Lindell, Erik"https://zbmath.org/authors/?q=ai:lindell.erikLet \(S_{g,r}^{s}\) be a connected, orientable surface, of genus \(g\), \(r\) marked points and \(s\) boundary components. Let \(\Gamma_{g,r}^{s}\) be its mapping class group and let \(H=H_{1}(S_{g},\mathbb{Q})\) and \(H_{\mathbb{Z}}=H_{1}(S_{g},\mathbb{Z})\). The natural action of \(\Gamma_{g,r}^{s}\) on \(S_{g}\) preserves the symplectic form and gives a homomorphism \(\Gamma_{g,r}^{s}\to Sp(H)\). The Torelli group of \(S_{g,r}^{s}\), denoted by \(\mathcal{I}_{g,r}^{s}\), is the kernel of this homomorphism. Since the image of the above homomorphism is \(Sp(H_{\mathbb{Z}})\), the natural action of this group on \(H_{n}(\mathcal{I}_{g,r}^{s})\) transforms this into an \(Sp(H_{\mathbb{Z}})\)-representation for all \(n\geq 0\). On the other hand D. Johnson constructed a homomorphism \(\tau: H_{1}(\mathcal{I}_{g,r}^{s})\to \bigwedge^{3}H_{\mathbb{Z}}\) that naturally extends to homomorphisms, for \(n\geq1\),
\[
\psi_{n}:=(\tau_{J})_{*}: H_{n}(\mathcal{I}_{g,r}^{s})\to \bigwedge^{n}(\bigwedge^{3}H).
\]
The author uses these \(\psi_{n}\) to explore properties of \(H_{n}(\mathcal{I}_{g,r}^{s})\) such as finite dimensionality, or when they are algebraic \(S_{p}(H_{\mathbb{Z}})\)-representations, and considers the injectivity of \(\psi_{n}\).
The first result is that the image \(Im(\psi_{n})\), of \(\psi_{n}\), contains all irreducible subrepresentations of weight \(3n\) for \(n\geq 1\) and \(g\geq 3n\). The second result is that \(Im(\psi_{n})\) contains many more subrepresentation defined by certain partitions in a stable range, that is for \(g\geq n+2k+l\) for certain \(k\) and \(l\). The author also proves that for \(n\geq 2\) and \(g>>0\) one has that \(im(\psi_{n})\) is a proper subgroup of \(H_{n}(\mathcal{I}_{g,r}^{s})\). The main tool is to construct certain \textit{abelian cycles} in \(H_{n}(\mathcal{I}_{g,r}^{s})\) to obtain elements in these groups and detect elements in \(Im(\psi_{n})\).
Reviewer: Daniel Juan Pineda (Michoacán)Extremal stability for configuration spaceshttps://zbmath.org/1528.550122024-03-13T18:33:02.981707Z"Knudsen, Ben"https://zbmath.org/authors/?q=ai:knudsen.ben"Miller, Jeremy"https://zbmath.org/authors/?q=ai:miller.jeremy-a"Tosteson, Philip"https://zbmath.org/authors/?q=ai:tosteson.philipLet \(M\) be a manifold of dimension \(d \geq 2\). Classical work of \textit{D. McDuff} [Topology 14, 91--107 (1975; Zbl 0296.57001)], \textit{G. Segal} [Acta Math. 143, 39--72 (1979; Zbl 0427.55006)], and \textit{T. Church} [Invent. Math. 188, No. 2, 465--504 (2012; Zbl 1244.55012)] establishes rational homological stability for the unordered configuration spaces \(B_n(M)\). If \(d\) is even, then their results can be interpreted as saying that there exists a polynomial in \(n\) of degree at most \(\dim(H_0(M;\mathbb{Q}))-1\), which coincides with the \(i\)-th rational Betti number of \(B_n(M)\) for \(n\) large enough. In particular, this is a pattern in (constant) low homological dimension \(i\). If \(d\) is odd, then work of \textit{C. F. Bödigheimer} et al. [Topology 28, No. 1, 111--123 (1989; Zbl 0689.55012)] gives a closed form expression for the rational homology.
Focussing therefore on the case \(d\) even, the authors prove \textit{extremal stability}, a new phenomenon occurring in (constant) low homological \textit{codimension}. Precisely, letting \(v_n = n(d-1)+|\pi_0(M)|\), they show that for each \(i\) there are two polynomials in \(n\), each of degree at most \(H_{d-1}(M;\mathbb{Q}^{w})-1\), such that for \(n\) large, the dimension of \(H_{v_n-i}(B_n(M;\mathbb{Q}))\) coincides with the first polynomial if \(n\) is even and the second if \(n\) is odd. The example of the surface of genus 2 shows that their degree bound is sharp. The case \(M = \mathbb{C}P^3\) was already known, and is due to \textit{M. Maguire} [``Computing cohomology of configuration spaces'', Preprint, \url{arXiv:1612.06314}]. The only other known example of extremal stability, due to \textit{J. Miller} et al. [Compos. Math. 156, No. 4, 822--861 (2020; Zbl 1458.11085)], is for congruence subgroups of \(SL_n(\mathbb{Z})\).
Reviewer: Guy Boyde (Utrecht)Cellular cohomology in homotopy type theoryhttps://zbmath.org/1528.550132024-03-13T18:33:02.981707Z"Buchholtz, Ulrik"https://zbmath.org/authors/?q=ai:buchholtz.ulrik"Hou, Kuen-Bang"https://zbmath.org/authors/?q=ai:hou.kuen-bangSummary: We present a development of cellular cohomology in homotopy type theory. Cohomology associates to each space a sequence of abelian groups capturing part of its structure, and has the advantage over homotopy groups in that these abelian groups of many common spaces are easier to compute. Cellular cohomology is a special kind of cohomology designed for cell complexes: these are built in stages by attaching spheres of progressively higher dimension, and cellular cohomology defines the groups out of the combinatorial description of how spheres are attached. Our main result is that for finite cell complexes, a wide class of cohomology theories (including the ones defined through Eilenberg-MacLane spaces) can be calculated via cellular cohomology. This result was formalized in the Agda proof assistant.Differential and low-dimensional topologyhttps://zbmath.org/1528.570012024-03-13T18:33:02.981707Z"Juhász, András"https://zbmath.org/authors/?q=ai:juhasz.andras-p|juhasz.andrasThis book offers an excellent and concise introduction to fundamental concepts in differential and low-dimensional topology, catering to advanced undergraduate and early-stage graduate students. While other resources delve into individual topics with more depth, this textbook stands out by emphasizing connections between them. Each chapter comes with a selection of exercises.
The book is organized into six chapters, complemented by an appendix covering fiber bundles and characteristic classes. The initial chapter guides the reader from the definition of topological manifolds to smooth ones, also addressing classification results in low dimensions. Morse theory and cobordism, a concept weaker than homeomorphism, are explored, with numerous exercises for comprehension.
Chapter two delves into higher-dimensional manifolds, introducing Milnor's construction of an exotic 7-sphere. The Whitney trick and the \(h\)-cobordism theorem are presented, leading to a classification theorem for higher-dimensional manifolds using surgery theory. The chapter concludes with Cerf theory, contributing to Kirby's calculus of diagrams for 4-manifolds.
The third chapter focuses on 3-manifolds, while the fourth explores knots and links. It introduces significant knot classes, fibred knots, and various knot invariants, from classical ones like the knot group to modern ones like the Jones polynomial and the HOMFLY polynomial. This section covers concordance groups, Dehn surgery, and branched coverings.
Chapter five is dedicated to Heegaard Floer homology, providing invariants for oriented 3-manifolds, links, and oriented 4-manifolds. Knot Floer homology, an invariant for knots and links, detects Seifert genus and fibredness, offering lower bounds on the four-ball genus.
The concluding chapter addresses 4-manifolds, highlighting the differences from smaller dimensions in terms of smooth and topological classification. It demonstrates that every finitely presented group can be the fundamental group of a smooth manifold, utilizing Kirby's calculus for the construction and manipulation of 4-manifolds.
Reviewer: Valeriano Aiello (Genève)Extractive text summarization using topological featureshttps://zbmath.org/1528.683902024-03-13T18:33:02.981707Z"Kumar, Ankit"https://zbmath.org/authors/?q=ai:kumar.ankit"Sarkar, Apurba"https://zbmath.org/authors/?q=ai:sarkar.apurbaSummary: The amount of text data generated these days is increasing exponentially, and it is becoming a very tedious process to extract meaningful information from the huge amounts of text data. In this work, we propose two methods to summarize the texts using topological features that capture the information over the topological structure, such as connected components and holes in the text data. The first method uses the concept of minimum dominating set to group the sentences into multiple clusters and to find the similarities between clusters using topological features (connected components and tunnels). Sentence scoring and extraction of key sentences from each cluster are done by the existing method of TextRank. The second method uses the pretrained \textit{GloVe} (global vectors to represent the words) and to find the similarities between sentences using topological features. A classical set cover based algorithm has been used to extract the key sentences for the summary. Both methods are compared on the basis of rouge scores with the existing method, i.e., TextRank, and the results are satisfactory.
For the entire collection see [Zbl 1516.68006].Quantum sheaf cohomology on Grassmannianshttps://zbmath.org/1528.811582024-03-13T18:33:02.981707Z"Guo, Jirui"https://zbmath.org/authors/?q=ai:guo.jirui"Lu, Zhentao"https://zbmath.org/authors/?q=ai:lu.zhentao"Sharpe, Eric"https://zbmath.org/authors/?q=ai:sharpe.eric-rIn this article under reviewed, the authors describes and investigates quantum sheaf cohomolgy on Grassmanians with deformations of the tangent bundle. A ring structure is used to derive the main results. Quantum cohomology is an essential part in algebraic geometry and string theory. In particular, quantum sheaf cohomology is a generalization quantum cohomology. Many authors studied in details quantum sheaf cohomolgy on toric varieties. There are many stages to tackle and to deal with quantum sheaf cohomolgy. The first author of this article under reviewed \textit{J. Guo} has published an interesting and similar article in the title: [Commun. Math. Phys. 374, No. 2, 661--688 (2020; Zbl 1435.81130)]
A general description of the ring structure has been used and found from both physical prospective and mathematical prospectives. In fact, as was indicates in the article mentioned above, that one can study quantum sheaf cohomology by representing the theory with generators and relations.
The article is well written. It gives crucial and vital methods to focus on physics derivations with excellent examples. These examples describe the correlation functions and quantum cohomollogy in certain specific situation. For the purpose to explain such examples, the authors consider only some special cases.
The article contains interesting sections. Without going in the technical details, this article under reviewed introduces a novel study and a significant approach to quantum sheaf cohomology on Grassmannians. In this article, the introduction provides sufficient background. Such introduction includes the relevant references. There are very good references in the end of the article. The methods of the article are clearly and adequately described. The research design is excellent. In fact, theory of Grassmannians is presented clearly. Non-abelain case is recalled in a good way. Then an interesting section on ring structures of quantum sheaf cohomolgy is presented clearly. This section of ring structure approach contains three subsection. The first is about Gauge invariant operators. Second one is about quantum sheaf cohomology ring with specialization to ordinary classical cohomolgy as well as specialization to quantum cohomology. Then a very excellent section for examples. The section of conclusion is presented. It support the results of the paper. In fact, this theory can be investigated more in the future either by physical approach or mathematical approach.
Reviewer: Ahmad Alghamdi (Makkah)Pointillisme à la Signac and construction of a quantum fiber bundle over convex bodieshttps://zbmath.org/1528.811802024-03-13T18:33:02.981707Z"de Gosson, Maurice"https://zbmath.org/authors/?q=ai:de-gosson.maurice-a"de Gosson, Charlyne"https://zbmath.org/authors/?q=ai:de-gosson.charlyneSummary: We use the notion of polar duality from convex geometry and the theory of Lagrangian planes from symplectic geometry to construct a fiber bundle over ellipsoids that can be viewed as a quantum-mechanical substitute for the classical symplectic phase space. The total space of this fiber bundle consists of geometric quantum states, products of convex bodies carried by Lagrangian planes by their polar duals with respect to a second transversal Lagrangian plane. Using the theory of the John ellipsoid we relate these geometric quantum states to the notion of ``quantum blobs'' introduced in previous work; quantum blobs are the smallest symplectic invariant regions of the phase space compatible with the uncertainty principle. We show that the set of equivalence classes of unitarily related geometric quantum states is in a one-to-one correspondence with the set of all Gaussian wavepackets. We emphasize that the uncertainty principle appears in this paper as geometric property of the states we define, and is not expressed in terms of variances and covariances, the use of which was criticized by \textit{J. B. M. Uffink} and \textit{J. Hilgevoord} [Found. Phys. 15, No. 9, 925--944 (1985; \url{doi:10.1007/BF00739034})].Localization of U(1) gauge field by non-minimal coupling with gravity in braneworldshttps://zbmath.org/1528.831742024-03-13T18:33:02.981707Z"Zhao, Zhen-Hua"https://zbmath.org/authors/?q=ai:zhao.zhenhua"Xie, Qun-Ying"https://zbmath.org/authors/?q=ai:xie.qun-ying"Fu, Chun-E."https://zbmath.org/authors/?q=ai:fu.chun-e"Zhou, Xiang-Nan"https://zbmath.org/authors/?q=ai:zhou.xiangnan(no abstract)