Recent zbMATH articles in MSC 55https://www.zbmath.org/atom/cc/552021-04-16T16:22:00+00:00WerkzeugTopological data analysis of zebrafish patterns.https://www.zbmath.org/1456.920042021-04-16T16:22:00+00:00"Mcguirl, Melissa R."https://www.zbmath.org/authors/?q=ai:mcguirl.melissa-r"Volkening, Alexandria"https://www.zbmath.org/authors/?q=ai:volkening.alexandria"Sandstede, Björn"https://www.zbmath.org/authors/?q=ai:sandstede.bjornSummary: Self-organized pattern behavior is ubiquitous throughout nature, from fish schooling to collective cell dynamics during organism development. Qualitatively these patterns display impressive consistency, yet variability inevitably exists within pattern-forming systems on both microscopic and macroscopic scales. Quantifying variability and measuring pattern features can inform the underlying agent interactions and allow for predictive analyses. Nevertheless, current methods for analyzing patterns that arise from collective behavior capture only macroscopic features or rely on either manual inspection or smoothing algorithms that lose the underlying agent-based nature of the data. Here we introduce methods based on topological data analysis and interpretable machine learning for quantifying both agent-level features and global pattern attributes on a large scale. Because the zebrafish is a model organism for skin pattern formation, we focus specifically on analyzing its skin patterns as a means of illustrating our approach. Using a recent agent-based model, we simulate thousands of wild-type and mutant zebrafish patterns and apply our methodology to better understand pattern variability in zebrafish. Our methodology is able to quantify the differential impact of stochasticity in cell interactions on wild-type and mutant patterns, and we use our methods to predict stripe and spot statistics as a function of varying cellular communication. Our work provides an approach to automatically quantifying biological patterns and analyzing agent-based dynamics so that we can now answer critical questions in pattern formation at a much larger scale.Higher dimensional simplicial complexity.https://www.zbmath.org/1456.550012021-04-16T16:22:00+00:00"Borat, Ayse"https://www.zbmath.org/authors/?q=ai:borat.ayseTopological complexity was introduced by \textit{M. Farber} [Discrete Comput. Geom. 29, No. 2, 211--221 (2003; Zbl 1038.68130)] to measure how far is a topological space from admitting a motion planner. Let \(X\) be a topological space, its topological complexity, denoted \(TC(X)\), can be defined as the sectional category (Schwarz genus) of the path space fibration \(\pi\colon X^{[0,1]}\to X\times X\), \(\pi(\gamma)=(\gamma(0),\gamma(1))\).
For a topological space \(X\), even a smooth manifold or a complex, its topological complexity \(TC(X)\) is very difficult to compute. That is the reason why several discretizations of the concept were introduced. \textit{K. Tanaka} [Algebr. Geom. Topol. 18, No. 2, 779--796 (2018; Zbl 1394.55005)] developed a combinatorial approach from the point of view of finite \(T_0\)-spaces. Both \textit{D. Fernández-Ternero} et al. [Proc. Am. Math. Soc. 146, No. 10, 4535--4548 (2018; Zbl 1402.55007)] and \textit{J. González} [New York J. Math. 24, 279--292 (2018; Zbl 1394.55004)] adressed the problem in the simplicial setting. However, note that the discrete topological complexity defined by Fernández-Ternero et al. is different from the simplicial complexity by González.
The notion of topological complexity was generalised by \textit{Y. B. Rudyak} [Topology Appl. 157, No. 5, 916--920 (2010; Zbl 1187.55001)] to define the higher dimensional topological complexity of a space \(X\), denoted \(TC_{n}(X)\), as the Schwarz genus of the map \(\pi\colon X^{[0,1]}\to X^n\), where \(\pi\) assigns \(n\) intermediate points to a path \(\gamma\). As a consequence, \(TC_2(X)=TC(X)\).
In the paper under review, the author introduces the notion of higher dimensional simplicial complexity, as an analogue of Rudyak's generalisation of topological complexity in the simplicial setting. This analogy is made concrete by Theorem 2.9, which guarantees that for a finite simplicial complex \(K\), its higher dimensional simplicial complexity is equal to the higher dimensional topological complexity of the geometric realization of \(K\). Then, some properties of the higher dimensional simplicial complexity are studied. Finally, a fully worked-out example is provided.
It may be also of interest to read, due to its relation with the paper under review, the work on higher analogs of simplicial and combinatorial complexity by \textit{A. K. Paul} [Topology Appl. 267, Article ID 106859, 12 p. (2019; Zbl 1425.57015)].
Reviewer: David Mosquera-Lois (Santiago de Compostela)Results on the homotopy type of the spaces of locally convex curves on $\mathbb{S}^3$.https://www.zbmath.org/1456.570242021-04-16T16:22:00+00:00"Alves, Emília"https://www.zbmath.org/authors/?q=ai:alves.emilia"Saldanha, Nicolau C."https://www.zbmath.org/authors/?q=ai:saldanha.nicolau-corcaoA smooth curve \(\gamma:[0,1]\to \mathbb{S}^3\) in \(4\)-dimensional Euclidean space \(\mathbb{R}^4\)
with image on the sphere \(\mathbb{S}^3\), is said to be locally convex, if the set of vectors
\({\gamma}(t),{\gamma}'(t), {\gamma}''(t),{\gamma}'''(t)\) is a positive basis in \(\mathbb{R}^4\)
for all \(t\in[0,1]\). By the Gram-Schmidt procedure, we can turn this basis into an orthonormal basis and thereby associate a Frenet frame curve \(\mathcal{F}_{\gamma}: [0,1]\to SO_4\) in the special orthogonal
group \(SO_4\) to a locally convex curve.
For any matrix \(Q\in SO_4\), let \(\mathcal{L}\mathbb{S}^3(Q)\) denote the space of all locally
convex curves \(\gamma:[0,1]\to \mathbb{S}^3\) where \(\mathcal{F}_{\gamma}(0)=I\) (the identity matrix)
and \(\mathcal{F}_{\gamma}(1)=Q\). It was proved by [\textit{N. C. Saldanha} and \textit{B. Shapiro}, J. Singul. 4, 1--22 (2012; Zbl 1292.58002)] that there are at most 3 different homeomorphism types among the path components of \(\mathcal{L}\mathbb{S}^3(Q)\). But otherwise very little seems to be known about these components and their generalizations to higher dimensional spheres \(\mathbb{S}^n\).
In the present paper the authors prove several interesting theorems on the homotopy and homology of these path components, in particular for the case \(Q=-I\). The results are technical and cannot be given in detail.
Reviewer: Vagn Lundsgaard Hansen (Lyngby)Classifying spaces for families of subgroups for systolic groups.https://www.zbmath.org/1456.200472021-04-16T16:22:00+00:00"Osajda, Damian"https://www.zbmath.org/authors/?q=ai:osajda.damian"Prytuła, Tomasz"https://www.zbmath.org/authors/?q=ai:prytula.tomaszSummary: We determine the large scale geometry of the minimal displacement set of a hyperbolic isometry of a systolic complex. As a consequence, we describe the centraliser of such an isometry in a systolic group. Using these results, we construct a low-dimensional classifying space for the family of virtually cyclic subgroups of a group acting properly on a systolic complex. Its dimension coincides with the topological dimension of the complex if the latter is at least four. We show that graphical small cancellation complexes are classifying spaces for proper actions and that the groups acting on them properly admitthree-dimensional classifying spaces with virtually cyclic stabilisers. This is achieved by constructing a systolic complex equivariantly homotopy equivalent to a graphical small cancellation complex. On the way we develop a systematic approach to graphical small cancellation complexes. Finally, we construct low-dimensional models for the family of virtually abelian subgroups for systolic, graphical small cancellation, and some CAT(0) groups.A characterization of relatively hyperbolic groups via bounded cohomology.https://www.zbmath.org/1456.200522021-04-16T16:22:00+00:00"Franceschini, Federico"https://www.zbmath.org/authors/?q=ai:franceschini.federicoSummary: It was proved by \textit{L. Mineyev} and \textit{A. Yaman} [``Relative hyperbolicity and bounded cohomology'', Preprint] that, if \((\Gamma,\Gamma')\) is a relatively hyperbolic pair, the comparison map
\[
H_b^k(\Gamma,\Gamma';V)\to H^k(\Gamma,\Gamma';V)
\]
is surjective for every \(k\geq2\), and any bounded \(\Gamma\)-module \(V\). By exploiting results of \textit{D. Groves} and \textit{J. F. Manning} [Isr. J. Math. 168, 317--429 (2008; Zbl 1211.20038)], we give another proof of this result. Moreover, we prove the opposite implication under weaker hypotheses than the ones required by Mineyev and Yaman [loc. cit.] .Equivariant dimensions of graph \(C^\ast\)-algebras.https://www.zbmath.org/1456.190032021-04-16T16:22:00+00:00"Chirvasitu, Alexandru"https://www.zbmath.org/authors/?q=ai:chirvasitu.alexandru"Passer, Benjamin"https://www.zbmath.org/authors/?q=ai:passer.benjamin-w"Tobolski, Mariusz"https://www.zbmath.org/authors/?q=ai:tobolski.mariuszLet \(E = (E^0, E^1, r, s)\) be a graph with countable vertex set \(E^0\), countable edge set \(E^1\), the rival and source maps \(r,s : E^1 \to E^0\) and the adjacency matrix \(A_E = (A_{vw}), A_{vw} = \#\{ \mbox{edges with source \textit{v} and rival \textit{w}}\}\). One associates to \(E\) a g\textit{raph \(C^\ast\)-algebra} \(C^*(E)\) generated by the mutually orthogonal projections \(P_v, v\in E^0\) corresponding to vertices and the mutually orthogonal partial isometries \(S_e, e\in E^1\) corresponding to edges satisfying the conditions: for each \(e\in E^1\),
\(S_e^*S_e= P_{r(e)}\), \(S_eS_e^* = P_{s(e)}\), and for each \(v\in E^0\), \(P_v = \sum_{e\in s^{-1}(v)} S_eS^*_e\). The gauge action \(\mathbb S^1 \curvearrowright C^*(E)\) is defined by \(S_e \mapsto \lambda E_e\) and \(P_v \mapsto P_v, \forall \lambda\in \mathbb S^1\). The restriction of the \textit{gauge action} to subgroup \(\mathbb Z/k \hookrightarrow \mathbb S^1\). For a subgroup \(G\) acting on a unital \(C^\ast\)-algebra \(A\), the \textit{local-triviality dimension} \(\dim_{LT}^G(A)\) is the smallest \(n\) for which there exist \(G\)-equivalent *-homomorphism \(\rho_0, \dots, \rho_n: C_0((0,1]) \otimes C(G) \to A\) such that \(\sum_{i=0}^n \rho_i(t\otimes 1) = 1\). The weak (resp., strong) local-triviality dimension \(\dim_{WLT}^G(A)\)(resp., \(\dim_{SLT}^G(A)\)) is the smallest \(n\) for which there exist \(G\)-equivalent *-homomorphism \(\rho_0, \dots, \rho_n: C_0((0,1]) \otimes C(G) \to A\) such that \(\sum_{i=0}^n \rho_i(t\otimes 1)\) is invertible (resp. there is a unital *-homomorphism \(C(E_nG) \to A\), \(E_nG := E_{n-a}G*G, E_0G:= G)\). It is clear that \(\dim_{WLT}^G(A) \leq \dim_{LT}^G(A) \leq \dim_{SLT}^G(A) \).
For \(C^\ast\)-algebras of finite acyclic graphs and finite cycles, as the main result, the authors
\textit{characterize the finiteness of these dimensions} (Theorems 3.4, 4.1, 4.4), and then
study the gauge actions on various examples of graph \(C^\ast\)-algebras, including Cuntz algebras (\S5.1), the Toeplitz algebra (\S5.2), and the antipodal actions on quantum spheres (\S5.4).
Reviewer: Do Ngoc Diep (Hanoi)Discrete noncommutative Gel'fand Naĭmark duality.https://www.zbmath.org/1456.460612021-04-16T16:22:00+00:00"Bertozzini, Paolo"https://www.zbmath.org/authors/?q=ai:bertozzini.paolo"Conti, Roberto"https://www.zbmath.org/authors/?q=ai:conti.roberto.1"Pitiwan, Natee"https://www.zbmath.org/authors/?q=ai:pitiwan.nateeSummary: We present, in a simplified setting, a non-commutative version of the well-known Gel'fand-Naĭmark duality (between the categories of compact Hausdorff topological spaces and commutative unital \(C^*\)-algebras), where ``geometric spectra'' consist of suitable finite bundles of one-dimensional \(C^*\)-categories equipped with a transition amplitude structure satisfying saturation conditions. Although this discrete duality actually describes the trivial case of finite-dimensional \(C^*\)-algebras, the structures are here developed at a level of generality adequate for the formulation of a general topological/uniform Gel'fand-Naĭmark duality, fully addressed in a companion work.Cup-product for equivariant Leibniz cohomology and Zinbiel algebras.https://www.zbmath.org/1456.170032021-04-16T16:22:00+00:00"Mukherjee, Goutam"https://www.zbmath.org/authors/?q=ai:mukherjee.goutam"Saha, Ripan"https://www.zbmath.org/authors/?q=ai:saha.ripanThe goal of the the article under review is the introduction of equivariant Leibniz cohomology and a Zinbiel product on it.
Recall that a (right) \textit{Leibniz algebra} is is a \(k\)-vector space \({\mathfrak g}\) with a \(k\)-bilinear bracket \([,]\) such that for all \(x,y,z\in{\mathfrak g}\)
\[[x,[y,z]]=[[x,y],z]-[x,z],y].\]
Jean-Louis Loday introduced Leibniz algebras to study the failure of periodicity in algebraic K-theory. He noticed that the Chevalley-Eilenberg coboundary operator lifts to tensor powers (when putting the bracket in the \(i\)th place) to give the coboundary operator \(d:{\mathfrak g}^{\otimes n}\to {\mathfrak g}^{\otimes(n-1)}\) for the homology of Leibniz algebras
The authors consider linear actions of a finite group \(G\) by automorphisms on a Leibniz algebra \({\mathfrak g}\). They develop in detail a differential geometric example (where they naturally switch to \textit{left} Leibniz algebras), namely the Leibniz algebroid structure on \(\Lambda^{n-1}T^*M\) for a Nambu-Poisson manifold \(M\) of order \(n\) with a smooth action of a finite group \(G\).
Then follows the definition of the equivariant cohomology. For this, the authors use Bredon's equivariant cohomology set-up. Namely, to \(G\), one associates a category \(O_G\) whose objects are the left cosets \(G/H\) for \(H\) running through all subgroups \(H\) of \(G\), and morphisms \(G/H\to G/K\) being \(G\)-maps (for the \(G\)-action on \(G/H\) given by left-translation). An \(O_G\)-module is then a contravariant functor \(O_G\to k\)-mod. In the symmetric monoidal category of \(O_G\)-modules, one can consider associative commutative algebras \(A\), but also (right) Leibniz algebras. In fact, the data of a Leibniz algebra \({\mathfrak g}\) in \(k\)-modules equipped with the action of a finite group \(G\) gives rise to a Leibniz algebra in \(O_G\)-modules.
The equivariant cohomology is then defined with the help of standard complexes. On the one hand, one can transpose the setting of the Loday standard complex for Leibniz cohomology into the category of \(O_G\)-modules. On the other hand, for an associative commutative \(O_G\)-algebra \(A\), one can consider in the collection of
\[S^n({\mathfrak g},A):=\bigoplus_{H < G}CL^n({\mathfrak g},A(G/H))\]
the subcomplex of invariant cochains. The authors show in their Theorem 4.5 that both complexes compute the same cohomology, the \textit{\(G\)-equivariant Leibniz cohomology} of \({\mathfrak g}\).
The last section is then devoted to the construction to the graded Zinbiel cup product on equivariant Leibniz cohomology.
Reviewer: Friedrich Wagemann (Nantes)Quantitative Tamarkin theory.https://www.zbmath.org/1456.530082021-04-16T16:22:00+00:00"Zhang, Jun"https://www.zbmath.org/authors/?q=ai:zhang.jun.8In 1980's, Kashiwara and Schapira developed a powerful theory, called the microlocal sheaf theory,
connecting analysis, symplectic geometry, and partial differential equations.
In symplectic geometry, a central topic is the non-displaceability problems.
In his pioneering work [Invent. Math. 82, 307--347 (1985; Zbl 0592.53025)], \textit{M. Gromov} proved the non-squeezing theorem,
which can be thought of as a classical result concerning non-displaceability.
It was \textit{D. Tamarkin} who first illustrated how to use the microlocal sheaf theory to solve non-displaceability problems [Springer Proc. Math. Stat. 269, 99--223 (2018; Zbl 1416.35019)].
Since then, aiming at translating more objects in symplectic geometry into the language of sheaves, extensive works have been done.
The purpose of the book under review is to provide an exposition of
the fast development of this topic, which focuses on the relations
between symplectic geometry and Tamarkin category theory, especially the Guillermou-Kashiwara-Schapira sheaf quantization
based on microlocal sheaf theory.
The book is divided into four parts.
The first part introduces the basic objects in symplectic geometry and
the key concept of singular support in microlocal sheaf theory.
The second part centers on the concepts of derived
category, persistence \textbf{k}-module, and singular support which serve as preparations
for the topics in later chapters.
The third part deals with the Tamarkin category theory.
The fourth part discusses various applications of Tamarkin categories in
symplectic geometry.
A more detailed review of the contents is given below.
The book starts with an introductory Chapter 1, that provides a quite readable overview of the whole book.
It contains a brief review of symplectic geometry and a sheaf-theoretic topics related to symplectic geometry, such as the singular support of a sheaf, the Tamarkin category, and the Hofer norm.
Chapter 2 is about the derived categories and the derived functors.
In particular, it includes an important result called the microlocal Morse lemma, a generalization of the classical Morse lemma to a microlocal formulation.
Based on the microlocal Morse lemma, the Tamarkin category is constructed at the beginning of Chapter 3.
This chapter devotes to a detailed study of the Tamarkin category theory.
Many symplectic-related topics are presented, for instance, the sheaf convolution and composition, Lagrangian Tamarkin categories and so on.
The last chapter is about the applications of Tamarkin categories in
symplectic geometry.
Starting with a presentation of the Guillermou-Kashiwara-Schapira sheaf quantization,
the author introduces many sheaf theoretic objects related to the symplectic geometry, especially, the symplectic geometry of the cotangent bundle.
At last, using sheaf invariants developed in the book, the author presents a new proof of Gromov's non-squeezing theorem.
The book contains an appendix, which presents some details on the relation between persistence modules and constructible sheaves, the computation of the sheaf hom, and the dynamical interpretation of the Guillermou-Kashiwara-Schapira sheaf quantization from the perspective of semi-classical analysis.
Reviewer: Xiaojun Chen (Chengdu)Limit theorems for process-level Betti numbers for sparse and critical regimes.https://www.zbmath.org/1456.600442021-04-16T16:22:00+00:00"Owada, Takashi"https://www.zbmath.org/authors/?q=ai:owada.takashi"Thomas, Andrew M."https://www.zbmath.org/authors/?q=ai:thomas.andrew-mA (random) simplicial complex \(\check{C} (\mathcal{X}, t)\), \( t \geq 0\), on a (random) point set \(\mathcal{X} \subset \mathbb{R}^d\) is constructed such that (i) the \(0\)-simplices are the points of \(\mathcal{X}\); (ii) the \(k\)-simplices \([x_0, \ldots, x_k]\) are included if every pair of points in \(\{ x_0, \ldots, x_k \} \subset \mathcal{X}\) lie within distance \(t\).
The probabilistic model is to take \(\mathcal{X} = \mathcal{P}_n\), a Poisson point process of intensity \(n \mu\), where \(\mu\) is a measure on \(\mathbb{R}^d\) with a bounded and continuous density, and to consider topological statistics
of the process \( ( \check{C} ( \mathcal{P}_n , s_n t ) )_{t \geq 0}\) as \(n \to \infty\) for appropriate scaling sequences \(s_n\). In particular, the subject of the present paper is the \(k\)th Betti number process given by \(\beta_{k,n} (t) = \beta_k ( \check{C} ( \mathcal{P}_n , s_n t ) )\). Motivation arises in part from the recent interest in \emph{persistent homology} and topological data analysis.
As in the classical case of the random geometric graph [\textit{M. Penrose}, Random geometric graphs. Oxford: Oxford University Press (2003; Zbl 1029.60007)], the asymptotic behaviour is different in the three regimes \(ns_n^d \to 0\) (sparse), \(n s_n^d \to \infty\) (dense), and \(n s_n^d \to \lambda \in (0,\infty)\) (critical).
The authors first consider the sparse regime where \(n s_n^d \to 0\) but \(n^{k+2} s_n^{d(k+1)} \to \infty\), and show that
\(\beta_{k,n}\) converges in finite-dimensional distributions to a Gaussian process associated with connected components on \(k+2\) points. In an ultra-sparse regime where \(n s_n^d \to 0\) very fast, there is instead a Poisson limit theorem.
In the critical regime with \(s_n = n^{-1/d}\), there is also convergence to a Gaussian limit, but now with a much more complicated structure: it is an infinite sum of Gaussian processes that are associated with components on \(k+2, k+3, \ldots\) points.
Proofs make use of local dependence, Stein's method, and Poisson process technology. The process-versions of these results are new, although marginal central limit theorems were known in some cases: e.g., [\textit{R. Iwasa}, Homology Homotopy Appl. 22, No. 1, 343--374 (2020; Zbl 1435.13013)].
Reviewer: Andrew Wade (Durham)A counterexample to Las Vergnas' strong map conjecture on realizable oriented matroids.https://www.zbmath.org/1456.520292021-04-16T16:22:00+00:00"Wu, Pei"https://www.zbmath.org/authors/?q=ai:wu.peiSummary: The Las Vergnas strong map conjecture asserts that any strong map of oriented matroids \(f : \mathcal{M}_1 \rightarrow \mathcal{M}_2\) can be factored into extensions and contractions. This conjecture is known to be false due to a construction by Richter-Gebert, who finds a strong map that is not factorizable; however, in his example, \(\mathcal{M}_1\) is not realizable. The question of whether there exists a non-factorizable strong map between realizable oriented matroids remains open. In this paper, we provide a counterexample to the strong map conjecture on realizable oriented matroids, which is a strong map \(f : \mathcal{M}_1 \rightarrow \mathcal{M}_2\), where \(\mathcal{M}_1\) is an alternating oriented matroid of rank 4 and \(f\) has corank 2. We prove that the map is not factorizable by showing that there is no uniform oriented matroid \(\mathcal{M}^{\prime}\) of rank 3 such that \(\mathcal{M}_1 \rightarrow \mathcal{M}^{\prime} \rightarrow \mathcal{M}_2\).On LS-category and topological complexity of some fiber bundles and Dold manifolds.https://www.zbmath.org/1456.550022021-04-16T16:22:00+00:00"Naskar, Bikramaditya"https://www.zbmath.org/authors/?q=ai:naskar.bikramaditya"Sarkar, Soumen"https://www.zbmath.org/authors/?q=ai:sarkar.soumenThe Lusternik-Schnirelmann category (or just category)
\(\mbox{cat}(X)\) of a space \(X\) is a classical important numerical homotopy invariant not only in algebraic topology but also in differential topology and dynamical systems. Another important, but more recent, numerical homotopy invariant is the topological complexity \(\mbox{TC}(X)\) of a space \(X\), which was introduced by \textit{M. Farber} [Discrete Comput. Geom. 29 No. 2, 211--221 (2003; Zbl 1038.68130)] in order to study the motion planning problem in robotics from a topological perspective. Under mild conditions, both invariants are known to satisfy the product inequality:
\(\mbox{cat}(X\times Y)\leq \mbox{cat}(X)+\mbox{cat}(Y)-1\)
and \(\mbox{TC}(X\times Y)\leq \mbox{TC}(X)+\mbox{TC}(Y)-1\), respectively.
Starting from the fact that the product space is a trivial bundle, the authors of the paper under review face the problem of extending the above inequalities to some
classes of more general fiber bundles. In this sense they prove that under certain conditions for a fiber bundle \(F\rightarrow E\stackrel{p}{\rightarrow }B\) the inequality
\(\mbox{cat}(E)\leq \mbox{cat}(B)+\mbox{cat}(F)-1\) holds. A similar formula is also given for topological complexity. Then, taking into account these results and considering usual cohomological lower bounds they are able to give, on the one hand, the computation of the category of Dold manifolds; on the other, they establish tight bounds for the topological complexity of a wide class of Dold manifolds.
Finally, some discussion about the equivariant topological complexity of some spaces related to Dold manifolds is also given.
Reviewer: José Calcines (La Laguna)Differential geometry and Lie groups. A second course.https://www.zbmath.org/1456.530012021-04-16T16:22:00+00:00"Gallier, Jean"https://www.zbmath.org/authors/?q=ai:gallier.jean-h"Quaintance, Jocelyn"https://www.zbmath.org/authors/?q=ai:quaintance.jocelynThis book is written as a second course on differential geometry. So the reader is supposed to be familiar with some themes from the first course on differential geometry -- the theory of manifolds and some elements of Riemannian geometry.
In the first two chapters here some topics from linear algebra are provided -- a detailed exposition of tensor algebra and symmetric algebra, exterior tensor products and exterior algebra. These chapters may be useful when studying the material of this book for those students, who did not study these topics in their algebraic course.
Some themes, which are covered in this book, are rather standard for books on differential geometry - they are differential forms, de Rham cohomology, integration on manifolds, connections and curvature in vector bundles, fibre bundles, principal bundles and metrics on bundles. But a number of topics discussed in this book are not always included in courses on differential geometry and are rarely contained in textbooks on differential geometry. The presence of these topics makes this book especially interesting for modern students. Here is a list of some such topics: an introduction to Pontrjagin
classes, Chern classes, and the Euler class, distributions and the Frobenius theorem. Three chapters need to be highlighted separately. Chapter 7 -- spherical harmonics and an introduction to the representations of compact Lie groups. Chapter 8 -- operators on Riemannian manifolds: Hodge Laplacian, Laplace-Beltrami Laplacian, Bochner
Laplacian. Chapter 11 -- Clifford algebras and groups, groups Pin\((n)\), Spin\((n)\).
Not all statements in this book are given with proofs, for some only links to other textbooks are given. But the most important results are given here with complete proofs and accompanied by examples. Each chapter of this book ends with a list of interesting and sometimes very important problems. At the end of the book there is a very detailed list of the notation used (symbol index) and a detailed list (index) of the terms used.
Reviewer: V. V. Gorbatsevich (Moskva)Matrix method for persistence modules on commutative ladders of finite type.https://www.zbmath.org/1456.550042021-04-16T16:22:00+00:00"Asashiba, Hideto"https://www.zbmath.org/authors/?q=ai:asashiba.hideto"Escolar, Emerson G."https://www.zbmath.org/authors/?q=ai:escolar.emerson-g"Hiraoka, Yasuaki"https://www.zbmath.org/authors/?q=ai:hiraoka.yasuaki"Takeuchi, Hiroshi"https://www.zbmath.org/authors/?q=ai:takeuchi.hiroshiA persistence module \(M\) on a commutative ladder over the field \(K\) can be identified with a commutative diagram of \(K\)-vector spaces and \(K\)-linear maps of the form
\[\begin{array}{cccccccc}
W_1 & \xleftrightarrow{\phi_1} & W_2 & \xleftrightarrow{\phi_2} & \cdots
& W_{n-1} & \xleftrightarrow{\phi_{n-1}} & W_n\\
\hspace{6pt}\uparrow{F_1} & & \hspace{6pt}\uparrow{F_2} &
& & \hspace{6pt}\uparrow{F_{n-1}} & & \hspace{6pt}\uparrow{F_n}\\
V_1 & \xleftrightarrow{\psi_1} & V_2 & \xleftrightarrow{\psi_2} & \cdots
& V_{n-1} & \xleftrightarrow{\psi_{n-1}} & V_n
\end{array}\]
where each horizontal arrow has a specified direction, namely either \(\leftarrow\) or \(\rightarrow\). The direction \(\tau_i\) of the arrow \(\phi_i\) is required to be the same as that of \(\psi_i\), and the sequence \(\tau=(\tau_1,\ldots,\tau_{n-1})\) is called the orientation of the ladder. In the language of zigzag persistent homology, each horizontal row in the diagram is a \(\tau\)-module [\textit{G. Carlsson} and \textit{V. de Silva}, Found. Comput. Math. 10, No. 4, 367--405 (2010; Zbl 1204.68242)]. Any such module can be written as the direct sum of interval \(\tau\)-modules of the form \({\mathbb I}[a,b]\), where \({\mathbb I}[a,b]_i=K\) if \(a\leq i\leq b\) and trivial otherwise, and all non-trivial arrows are the identity. Using this basis, the map from the bottom row of the diagram to the top row determines a matrix \(\Phi(M)\). Under the assumption that \(n\leq 4\), the authors provide a Smith normal form style algorithm for reducing \(\Phi(M)\). The persistence diagram of \(M\) can then be extracted from the resulting matrix.
Reviewer: Jason Hanson (Redmond)Genera of the torsion-free polyhedra.https://www.zbmath.org/1456.550052021-04-16T16:22:00+00:00"Kolesnyk, P. O."https://www.zbmath.org/authors/?q=ai:kolesnyk.p-oSummary: We study the genera of polyhedra (finite cell complexes), i.e., the classes of polyhedra such that all their localizations are stably homotopically equivalent. More precisely, we describe the genera of the torsion-free polyhedra of dimensions not greater than 11. In particular, we find the number of stable homotopy classes in these genera.The trace of the local \(\mathbb{A}^1\)-degree.https://www.zbmath.org/1456.140272021-04-16T16:22:00+00:00"Brazelton, Thomas"https://www.zbmath.org/authors/?q=ai:brazelton.thomas"Burklund, Robert"https://www.zbmath.org/authors/?q=ai:burklund.robert"McKean, Stephen"https://www.zbmath.org/authors/?q=ai:mckean.stephen"Montoro, Michael"https://www.zbmath.org/authors/?q=ai:montoro.michael"Opie, Morgan"https://www.zbmath.org/authors/?q=ai:opie.morganThe theory of \(\mathbb{A}^1\)-enumerative geometry is a relatively young topic, defined as an application of the \(\mathbb{A}^1\)-homotopy theory of \textit{F. Morel} and \textit{V. Voevodsky} [Publ. Math., Inst. Hautes Étud. Sci. 90, 45--143 (1999; Zbl 0983.14007)] to enumerative geometry over arbitrary base fields \(\kappa\); see [\textit{B. Williams} and \textit{K. Wickelgren}, in: Handbook of homotopy theory. Boca Raton, FL: CRC Press. 42 p. (2020; Zbl 07303335)].
In this short article, the authors study the local \(\mathbb{A}^1\)-degree, a foundational tool in \(\mathbb{A}^1\)-enumerative geometry. Analogous to the classical Brouwer degree of a continuous map \(\mathbb{R}^n\to \mathbb{R}^n\) with an isolated zero at the origin taking values in the integers, the local \(\mathbb{A}^1\)-degree is an invariant of a map \(f\colon\mathbb{A}^n_\kappa\to \mathbb{A}^n_\kappa\) at an isolated zero \(p\) taking values in the Grothendieck--Witt group \(\mathrm{GW}(\kappa)\) of the field \(\kappa\). The main theorem [Theorem 1.3] states that if the map \(\kappa\to\kappa(p)\) from the residue field of \(p\) is separable and of finite degree, then the local \(\mathbb{A}^1\)-degree of \(f\) can be computed as the image of the local \(\mathbb{A}^1\)-degree of the base change of \(f\) over \(\kappa(p)\) under the natural transfer map of Grothendieck-Witt groups. This result is somewhat inspired by Morel's original work in defining the local \(\mathbb{A}^1\)-degree [\textit{F. Morel}, in: Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22--30, 2006. Volume II: Invited lectures. Zürich: European Mathematical Society (EMS). 1035--1059 (2006; Zbl 1097.14014)] and more recent work by \textit{J. L. Kass} and \textit{K. Wickelgren} [Duke Math. J. 168, No. 3, 429--469 (2019; Zbl 1412.14014)]. The authors also obtain a corollary of Theorem 1.3, generalising a statement of Kass and Wickelgren [loc. cit.] relating the local \(\mathbb{A}^1\)-degree with the Scheja-Storch bilinear form.
The authors begin in Section 2 by reviewing the necessary background in \(\mathbb{A}^1\)-homotopy theory, including the construction of the local \(\mathbb{A}^1\)-degree, and some of the six functor formalism of \(\mathbb{A}^1\)-stable homotopy theory, as detailed by \textit{M. Hoyois} [Algebr. Geom. Topol. 14, No. 6, 3603--3658 (2014; Zbl 1351.14013)], for example. The six functor formalism allows for interpretations of the various maps defining the local \(\mathbb{A}^1\)-degree and the transfer of Grothendieck-Witt groups internally in \(\mathcal{SH}(\kappa)\), the stable motivic homotopy category of \(\kappa\).
In Section 3, the authors prove Theorem 1.3. To do this, a commutative diagram [Diagram (3)] is constructed in the unstable motivic homotopy category of \(\kappa\). Upon passage to the stable category \(\mathcal{SH}(\kappa)\), Diagram (3) yields Diagram (4) containing the local \(\mathbb{A}^1\)-degree as an endomorphism of the motivic sphere \(\mathbb{P}^n_\kappa/\mathbb{P}^{n-1}_\kappa\). Using the purity theorem of Morel-Voevodsky and an analysis of certain motivic Thom spaces, various arrows in Diagram (4) are recast and shown to be invertible; see Diagram (7). Theorem 1.3 now follows from Diagram (7) and the fact that \(\kappa\to \kappa(p)\) is finite and separable. As a final paragraph, the authors show the Scheja-Storch form agrees with the local \(\mathbb{A}^1\)-degree when \(\kappa\to\kappa(p)\) is separable and of finite degree, extending a previous result of Wickelgren and Kass [loc. cit.], where one assumes that \(p\) is \(\kappa\)-rational.
Reviewer: Jack Davies (Utrecht)Groups acting on trees and the Eilenberg-Ganea problem for families.https://www.zbmath.org/1456.570212021-04-16T16:22:00+00:00"Sánchez Saldaña, Luis Jorge"https://www.zbmath.org/authors/?q=ai:sanchez-saldana.luis-jorgeLet \(G\) be a discrete group and \(\mathcal{F}\) be a family of subgroups of \(G\), by this we mean, a collection of subgroups of \(G\) closed under taking subgroups and conjugation by elements of \(G\). Now let \(E_{\mathcal{F}}G\) be a \textit{universal space} for actions with isotropy in \(\mathcal{F}\). The \(\mathcal{F}\)-\textit{geometric dimension} of \(G\), \(gd_{\mathcal{F}}(G)\), is the minimum nonnegative integer, \(n\), such that there exists an \(n\)-dimensional model for \(E_{\mathcal{F}}G\). On the other hand, let \(H^{*}_{\mathcal{F}}(G;M)\) denote the Bredon cohomology of \(G\) with coefficients in an \(\mathcal{O}_{\mathcal{F}}\)-module \(M\). The \(\mathcal{F}\)-cohomological dimension of \(G\), denoted by \(cd_{\mathcal{F}}(G),\) is the largest nonnegative integer, \(n\), such that \(H^{n}_{\mathcal{F}}(G;M)\) is nontrivial for some \(\mathcal{O}_{\mathcal{F}}\)-module \(M\). It is easy to verify the inequality \(cd_{\mathcal{F}}(G)\leq gd_{\mathcal{F}}(G)\) and it is known
that this is an equality if \(cd_{\mathcal{F}}(G)\geq 3\) and for any family \(\mathcal{F}\). There are some examples, constructed by \textit{N. Brady} et al. [J. Lond. Math. Soc., II. Ser. 64, No. 2, 489--500 (2001; Zbl 1016.20035)] where \(cd_{\mathcal{F}}(G)=2\) and \( gd_{\mathcal{F}}(G)=3\) for the families of finite and virtually cyclic subgroups. Whether this is the case for all families and groups is still an open problem. The purpose of the paper under review is the construction of more examples of groups such that \(cd_{\mathcal{F}}(G)=2\) and \( gd_{\mathcal{F}}(G)=3\) for the families of finite,
virtually abelian of bounded rank and virtually poly-cyclic subgroups of bounded rank. The main technique is the Bass-Serre theory of groups acting on trees.
In order to state the main theorem, we need some definitions. A group \(G\) is called an \(\mathcal{F}\)-\textit{Eilenberg-Ganea} group if \(cd_{\mathcal{F}}(G)=2\) and \( gd_{\mathcal{F}}(G)=3\). Let \(\mathbf{Y}\) be a graph of groups with fundamental group \(G\) (in the Bass-Serre terminology) with associated tree \(T\) and
let \(\mathcal{P}\subset G\) be a collection of subgroups of \(G\). These data are referred to as a splitting of \(G\). The splitting is called
\(\mathcal{P}\)-\textit{acylindrical} if there exists an integer \(k\) such that the stabilizer \(G_{c}\) of a path \(c\subset T\) belongs to \(\mathcal{P}\) for every \(c\) of length \(k\). Furthermore, \(\mathbf{Y}\) is called \(\mathcal{F}\) admissible if the following conditions hold: (1) There exists
a vertex \(P\) such that its vertex group \(Y_{P}\) is a \((Y_{P}\cap\mathcal{F})\)-\textit{Eilenberg-Ganea} group, (2) for all vertices \(P\in \mathbf{Y}\), it follows that \(gd_{\mathcal{F}\cap Y_{P}}(Y_{P})\leq 3\) and \( cd_{\mathcal{F}\cap Y_{P}}(Y_{P})\leq 2\), and (3) for all edges \(y\in \mathbf{Y}\),
we have \( gd_{\mathcal{F}\cap Y_{y}}(Y_{y})\leq 2\) and \( cd_{\mathcal{F}\cap Y_{y}}(Y_{y})\leq 1\). The Main Theorem is as follows:
Theorem. Let \(\mathbf{Y}\) be a graph of groups with fundamental group \(G\) and Bass-Serre tree \(T\). Let \(\mathcal{F}\) be a family of subgroups of \(G\). Let \(\mathcal{F}_{0}\) be the collection of subgroups of \(\mathcal{F}\) that fix a vertex and \(\mathcal{F}_{1}\) those subgroups of \(\mathcal{F}\) that act cocompactly on a geodesic line of \(T\). Assume:
\begin{enumerate}
\item \(\mathbf{Y}\) is \(\mathcal{F}\)-admissible,
\item\(\mathcal{F}=\mathcal{F}_{0}\sqcup \mathcal{F}_{1}\),
\item the splitting of \(G\) is \(\mathcal{P}\)-acylindrical for some collection \(\mathcal{P}\subseteq \mathcal{F}_{0}\) closed under taking subgroups,
\item every \(\mathcal{P}\)-by \(V\) subgroup of \(G\) belongs to \(\mathcal{F}\), where \(V=\mathbb{Z}\) or \(D_{\infty}\),
\item every \(\mathcal{F}_{0}\)-by-\(\mathbb{Z}/2\) subgroup of \(G\) belongs to \(\mathcal{F}_{0}\).
\end{enumerate}
Then \(G\) is an \(\mathcal{F}\)-\textit{Eilenberg-Ganea} group.
In the above, a \(\mathcal{P}\)-by \(Q\) group is a group \(L\) that fits in a short exact sequence \(1\to N\to L\to Q\to 1\) with \(N\in \mathcal{P}\).
Reviewer: Daniel Juan Pineda (Michoacán)Elliptic classes of Schubert varieties.https://www.zbmath.org/1456.140602021-04-16T16:22:00+00:00"Kumar, Shrawan"https://www.zbmath.org/authors/?q=ai:kumar.shrawan"Rimányi, Richárd"https://www.zbmath.org/authors/?q=ai:rimanyi.richard"Weber, Andrzej"https://www.zbmath.org/authors/?q=ai:weber.andrzejSummary: We introduce new notions in elliptic Schubert calculus: the (twisted) Borisov-Libgober classes of Schubert varieties in general homogeneous spaces \(G/P\). While these classes do not depend on any choice, they depend on a set of new variables. For the definition of our classes we calculate multiplicities of some divisors in Schubert varieties, which were only known for full flag varieties before. Our approach leads to a simple recursions for the elliptic classes. Comparing this recursion with R-matrix recursions of the so-called elliptic weight functions of Rimanyi-Tarasov-Varchenko we prove that weight functions represent elliptic classes of Schubert varieties.Dynamics near an idempotent.https://www.zbmath.org/1456.370232021-04-16T16:22:00+00:00"Shaikh, Md. Moid"https://www.zbmath.org/authors/?q=ai:shaikh.md-moid"Patra, Sourav Kanti"https://www.zbmath.org/authors/?q=ai:patra.sourav-kanti"Ram, Mahesh Kumar"https://www.zbmath.org/authors/?q=ai:ram.mahesh-kumarSummary: \textit{N. Hindman} and \textit{I. Leader} [Semigroup Forum 59, No. 1, 33--55 (1999; Zbl 0942.22003)]
first introduced the notion of the semigroup of ultrafilters converging to zero for a dense subsemigroup of \(((0, \infty), +)\). Using the algebraic structure of the Stone-Čech compactification, \textit{M. A. Tootkaboni} and \textit{T. Vahed} [Topology Appl. 159, No. 16, 3494--3503 (2012; Zbl 1285.54017)]
generalized and extended this notion to an idempotent instead of zero, that is a semigroup of ultrafilters converging to an idempotent \(e\) for a dense subsemigroup of a semitopological semigroup \((R, +)\) and they gave the combinatorial proof of the Central Sets Theorem near \(e\). Algebraically one can define quasi-central sets near \(e\) for dense subsemigroups of \((R, +)\). In a dense subsemigroup of \((R, +)\), C-sets near \(e\) are the sets, which satisfy the conclusions of the Central Sets Theorem near \(e\). \textit{S. K. Patra} [Topology Appl. 240, 173--182 (2018; Zbl 1392.37008)]
gave dynamical characterizations of these combinatorially rich sets near zero. In this paper, we shall establish these dynamical characterizations for these combinatorially rich sets near \(e\). We also study minimal systems near \(e\) in the last section of this paper.Topology change of level sets in Morse theory.https://www.zbmath.org/1456.370612021-04-16T16:22:00+00:00"Knauf, Andreas"https://www.zbmath.org/authors/?q=ai:knauf.andreas"Martynchuk, Nikolay"https://www.zbmath.org/authors/?q=ai:martynchuk.nikolayThis paper considers Morse functions \(f \in C^2(M, \mathbb{R})\) on a smooth \(m\)-dimensional manifold without boundary. For such functions and for every critical point \(x \in M\) of \(f\), the Hessian of \(f\) at \(x\) is nondegenerate.
The authors are interested in the topology of the level sets \(f^{-1}(a) = \partial{M^a}\) where \(M^a = \{ x \in M : f(x) \le a\}\) is a so-called sublevel set. The authors address the question of under what conditions the topology of \(\partial{M^a}\) can change when the function \(f\) passes a critical level with one or more critical points. ``Change in topology'' here means roughly that if \(a\) and \(b\) are regular values with \(a < b\), then \(H_k(\partial{M^a};G) \ne H_k(\partial{M^b};G)\) where \(k\) is an integer and the \(H_k\) are homology groups over some abelian group \(G\) when the function \(f\) passes through a level with one or more critical points. A motivation for this question derives from the \(n\)-body problem: does the topology of the integral manifolds always change when passing through a bifurcation level?
The first part of the paper considers level sets of abstract Morse functions that satisfy the Palais-Smale condition and whose level sets have finitely generated homology groups. They show that for such functions the topology of \(f^{-1}(a)\) changes when passing a single critical point if the index of the critical point is different from \(m/2\) where \(m\) is the dimension of the manifold \(M\). Then they move on to consider the case where \(M\) is a vector bundle of rank \(n\) over a manifold \(N\) of dimension \(n\), and where (up to a translation) \(f\) is a Morse function that is the sum of a positive definite quadratic form on the fibers and a potential function that is constant on the fibers. Here the authors have in mind Hamiltonian functions \(H\) on the cotangent bundle of the base manifold \(N\). In this case they show that the topology of \(H^{-1}(h)\) always changes when passing a single critical point if the Euler characteristic of \(N\) is not \(\pm 1\).
Finally the authors apply their results to examples from Hamiltonian and celestial mechanics, with emphasis on the planar three-body problem. There they show that the topology always changes for the planar three-body problem provided that the reduced Hamiltonian is a Morse function with at most two critical points on each level set.
Reviewer: William J. Satzer Jr. (St. Paul)Torsion in homology of dihedral quandles of even order.https://www.zbmath.org/1456.570102021-04-16T16:22:00+00:00"Yang, Seung Yeop"https://www.zbmath.org/authors/?q=ai:yang.seung-yeopQuandles are algebraic structures with axioms motivated by the Reidemeister moves from knot theory. Rack and quandle homology theories have been studied in the past and present [\textit{J. S. Carter} et al., Trans. Am. Math. Soc. 355, No. 10, 3947--3989 (2003; Zbl 1028.57003); \textit{T. Nosaka}, Topology Appl. 158, No. 8, 996--1011 (2011; Zbl 1227.57020); \textit{S. Y. Yang}, J. Knot Theory Ramifications 26, No. 3, Article ID 1741010, 13 p. (2017; Zbl 1373.57035)]. In this paper the author has studied the torsion subgroups of rack and quandle homology of non-connected quandles in order to partially prove the conjecture given below.
\textbf{Conjecture}: [\textit{M. Niebrzydowski} and \textit{J. H. Przytycki}, J. Pure Appl. Algebra 213, No. 5, 742--755 (2009; Zbl 1178.57015); J. Algebra 324, No. 7, 1529--1548 (2010; Zbl 1219.55003)] The number \(k\) annihilates Tor\(H_n^W(R_{2k}),\) unless \(k=2^k, t>1\) and the number \(2k\) is the smallest number annihilating Tor\(H_n^W(R_{2k})\) for \(k = 2^t, t>1,\) where \(W= R, Q.\)
The main Theorem 2.2 partially proves the above conjecture and Table 2 provides some computational results on the homology of dihedral quandles of order \(2k\) when \(k\) is ``odd''.
The author leaves the readers two open questions at the end for future research.
Reviewer: Indu R. U. Churchill (Oswego)The Galois action and cohomology of a relative homology group of Fermat curves.https://www.zbmath.org/1456.112172021-04-16T16:22:00+00:00"Davis, Rachel"https://www.zbmath.org/authors/?q=ai:davis.rachel"Pries, Rachel"https://www.zbmath.org/authors/?q=ai:pries.rachel-j"Stojanoska, Vesna"https://www.zbmath.org/authors/?q=ai:stojanoska.vesna"Wickelgren, Kirsten"https://www.zbmath.org/authors/?q=ai:wickelgren.kirsten-gLet \(p\) be a prime satisfying Vandiver's conjecture, i.e., such that \(p\) does not divide the order of \(h^+\) of the class group of \(\mathbb{Q}(\zeta+\zeta^{-1})\), where \(\zeta\) is a \(p\)-th root of unity. Let \(X\) be the degree \(p\) Fermat curve \(x^p+y^p=z^p\). Let \(U\subset X\) be the affine open given by \(z\neq 0\). Consider the closed subscheme \(Y\subset U\) defined by \(xy=0\). Let \(H_1(U,Y;\mathbb{Z}/p)\) denote the étale homology group with \(\mathbb{Z}/p \) coefficients, of the pair \((U\otimes \bar{K},Y\otimes\bar{K})\). By [\textit{G. W. Anderson}, Duke Math. J. 54, 501--561 (1987; Zbl 1370.11069)], the group \(H_1(U,Y;\mathbb{Z}/p)\) is a free rank-one \(\mathbb{Z}/p[\mu_p\times\mu_p]\)-module with generator \(\beta\). The Galois action of \(\sigma\in G_{\mathbb{Q}(\zeta)}\) is then determined by \(\sigma\beta=B_\sigma\beta\), for some \(B_\sigma\in \mathbb{Z}/p[\mu_p\times\mu_p]\). Anderson theoretically described \(B_\sigma\). In this paper, a closed form formula for \(B_\sigma\) is given. Intermediate results by the same authors [\textit{R. Davis} et al., Assoc. Women Math. Ser. 3, 57--86 (2016; Zbl 1416.11045)] about the isomorphism class of the Galois group of the field extension through the action of \(G_{\mathbb{Q}(\zeta)}\) factors, are strongly used.
The first application of this formula is that the norm of \(B_\sigma\) is \(0\) for almost all \(\sigma\). This is important in computing Galois cohomology as in Section 6 where a method for the efficient computation of the first cohomology group \(H^1(G_{\mathbb{Q}(\eta)}, H_1(U,Y;\mathbb{Z}/p))\) is given. This will eventually play a key role in understanding obstructions for rational points on Fermat curves as Ellenberg's obstruction related to the non-abelian Chabauty method.
A second application of the main formula is a proof of the fact that \(H_1(U;\mathbb{Z}/p)\) is trivialized by the product of \(\lfloor 2p/3\rfloor\) terms of the form \((B_\sigma-1)\).
Reviewer: Elisa Lorenzo García (Rennes)An application of cubical cohomology to Adinkras and supersymmetry representations.https://www.zbmath.org/1456.814292021-04-16T16:22:00+00:00"Doran, Charles F."https://www.zbmath.org/authors/?q=ai:doran.charles-f"Iga, Kevin M."https://www.zbmath.org/authors/?q=ai:iga.kevin-m"Landweber, Gregory D."https://www.zbmath.org/authors/?q=ai:landweber.gregory-dSummary: An Adinkra is a class of graphs with certain signs marking its vertices and edges, which encodes off-shell representations of the super Poincaré algebra. The markings on the vertices and edges of an Adinkra are cochains for cubical cohomology. This article explores the cubical cohomology of Adinkras, treating these markings analogously to characteristic classes on smooth manifolds.Realization of graph symmetries through spatial embeddings into the 3-sphere.https://www.zbmath.org/1456.570232021-04-16T16:22:00+00:00"Ikeda, Toru"https://www.zbmath.org/authors/?q=ai:ikeda.toru.1|ikeda.toru.2Approaches to the realization problem of abstract symmetries by symmetries of the \(3\)-sphere through spatial embeddings are known. For example, topologists have worked with certain families of graphs such as complete graphs, complete bipartite graphs, or Möbius ladders, as can be seen in [\textit{E. Flapan} and \textit{E. D. Lawrence}, J. Knot Theory Ramifications 23, No. 14, Article ID 1450077, 13 p. (2014; Zbl 1320.57008); \textit{E. Flapan} et al., J. Lond. Math. Soc., II. Ser. 73, No. 1, 237--251 (2006; Zbl 1091.57001)] and in [\textit{K. Hake} et al., Tokyo J. Math. 39, No. 1, 133--156 (2016; Zbl 1357.57012)].
Proving the existence of a spatial embedding requires creating an embedding of the vertex set of a given graph into \(S^3\) based upon singularities, then extending it by placing edges while they are setwise invariant under a finite subgroup action of self-diffeomorphisms of \(S^3\). However, these results are applicable to a certain family of graphs, and it would be difficult to deal with all possible graphs in this way due to their dependence on graph structures. The aim of this paper is to provide a technique which is applicable to any graph, and below is the main result of this article.
Theorem. Let \(\Gamma\) be a finite simple graph and \(G\) a subgroup of \(Aut(\Gamma)\). Suppose there is an isomorphism \(\varphi\) from \(G\) to a subgroup of \(O(4)\), and that \(G\) has an \(S^3\)-type singularity with respect to \(\varphi\). Then, \(G\) is realizable by \(\varphi(G)\) through a spatial embedding of \(\Gamma\) into \(S^3\).
Reviewer: Ryo Ohashi (Wilkes-Barre)A generalized Blakers-Massey theorem.https://www.zbmath.org/1456.180172021-04-16T16:22:00+00:00"Anel, Mathieu"https://www.zbmath.org/authors/?q=ai:anel.mathieu"Biedermann, Georg"https://www.zbmath.org/authors/?q=ai:biedermann.georg"Finster, Eric"https://www.zbmath.org/authors/?q=ai:finster.eric"Joyal, André"https://www.zbmath.org/authors/?q=ai:joyal.andreThe classical \textit{Blakers-Massey theorem} [\textit{A. L. Blakers} and \textit{W. S. Massey}, Proc. Natl. Acad. Sci. USA 35, 322--328 (1949; Zbl 0040.25801); Ann. Math. (2) 53, 161--205 (1951; Zbl 0042.17301); Ann. Math. (2) 55, 192--201 (1952; Zbl 0046.40604); Ann. Math. (2) 58, 409--417 (1953; Zbl 0053.12901)], aka the \textit{homotopy excision theorem}, is one of the most fundamental facts in homotopy theory, claiming that, given a homotopy pushout diagram of spaces
\[
\begin{array} [c]{ccc} A & \begin{array} [c]{c} g\\
\rightarrow\\
\end{array} & C\\
\begin{array} [c]{cc} f & \downarrow \end{array} & _{\lrcorner} & \downarrow\\
B & \rightarrow & D \end{array}
\]
such that the map \(f\)\ is \(m\)-connected and the map \(g\)\ is \(n\)-connected, the canonical map \(A\rightarrow B\times_{D}C\)\ to the homotopy pullback is in fact \(\left( m+n\right) \)-connected, giving rise to the \textit{Freudenthal suspension theorem} [\textit{H. Freudenthal}, Compos. Math. 5, 299--314 (1937; Zbl 0018.17705)] and therefore paving the way to \textit{stable homotopy theory}.
A new proof of this theorem [\textit{K.-B. Hou (Favonia)} et al., in: Proceedings of the 2016 31st annual ACM/IEEE symposium on logic in computer science, LICS 2016, New York City, NY, USA, July 5--8, 2016. New York, NY: Association for Computing Machinery (ACM). 565--574 (2016; Zbl 1395.55011)] was found in the context of \textit{homotopy type theory} providing an elementary axiomatization of homotopy-theoretic reasoning [\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)], which is generally thought to serve as an internal language for the \(\infty\)-topoi as developed by Rezk [\url{https://faculty.math.illinois.edu/\symbol{126}rezk/homotopy-topos-sketch.pdf}] and [\textit{J. Lurie}, Higher topos theory. Princeton, NJ: Princeton University Press (2009; Zbl 1175.18001)]. The original proof in [\textit{K.-B. Hou (Favonia)} et al., in: Proceedings of the 2016 31st annual ACM/IEEE symposium on logic in computer science, LICS 2016, New York City, NY, USA, July 5--8, 2016. New York, NY: Association for Computing Machinery (ACM). 565--574 (2016; Zbl 1395.55011)] was translated into the language of higher category theory by Rezk [\url{https://faculty.math.illinois.edu/\symbol{126} rezk/freudenthal-and-blakers-massey.pdf}]. The main result of this paper is a much generalized theorem, applying not only to spaces, but to an arbitrary \(\infty\)-topos. Indeed, as was shown in [\textit{M. Anel} et al., J. Topol. 11, No. 4, 1100--1132 (2018; Zbl 1423.18009)], the generalized theorem is to be applied to an appropriate presheaf topos, yielding an analogue of the Blakers-Massey theorem in the context of Goodwille's calculus of functors.
The authors observe that the \(n\)-connected maps form the left class of a \textit{factorization system} \(\left( \mathfrak{L},\mathfrak{R}\right) \)\ on the category of spaces with the additional property that the left class \(\mathfrak{L}\) is stable under base change, referring to a factorization system obedient to this condition as a \textit{modality}. The main result of this paper goes as follows.
Theorem. Let \(\mathcal{E}\)\ be an \(\infty\)-topos and \(\left( \mathfrak{L} ,\mathfrak{R}\right) \) a modality on \(\mathcal{E}\). Write \(\Delta h:A\rightarrow A\times_{B}A\) for the diagonal of a map \(h:A\rightarrow B\in\mathcal{E}\) and \(-\square_{Z}-\)\ for the pushout product in the slice category \(\mathcal{E}_{/Z}\). Given a pushout square
\[
\begin{array} [c]{ccc} Z & \begin{array} [c]{c} g\\
\rightarrow\\
\end{array} & Y\\
\begin{array} [c]{cc} f & \downarrow \end{array} & _{\lrcorner} & \downarrow\\
X & \rightarrow & W \end{array}
\]
in \(\mathcal{E}\)\ with \(\Delta f\square_{Z}\Delta g\in\mathfrak{L}\), the canonical map \(\left( f,g\right) :Z\rightarrow X\times_{W}Y\) is also in \(\mathfrak{L}\).
A similar generalization of the Blakers-Massey theorem was obtained by \textit{W. Chachólski} et al. [Ann. Inst. Fourier 66, No. 6, 2641--2665 (2016; Zbl 1368.55006)], though their method involves the manipulation of \textit{weak cellular inequalities} of spaces, as introduced in [\textit{E. Dror Farjoun}, Cellular spaces, null spaces and homotopy localization. Berlin: Springer-Verlag (1995; Zbl 0842.55001)], whereas the authors focus on \(\infty \)-topos-theoretic tools such as descent.
A synopsis of the paper consisting of four sections goes as follows. \S 2 fixes higher-categorical conventions and recalls some elementary facts. \S 3 proceeds as follows.
\begin{itemize}
\item It begins by introducing the notion of a factorization system as well as the pushout product and pullback hom.
\item It then gives a short treatment of the \(n\)-connected/\(n\)-truncated factorization system in an \(\infty\)-topos.
\item It introduces the notion of a modality itself, providing a number of examples and deriving some elementary properties, including the dual Blakers-Massey theorem.
\item It concludes with the descent theorem for \(\mathfrak{L}\)-cartesian squares.
\end{itemize}
\S 4 turns to the proof of the generalized Blakers-Massey theorem itself, finishing with the derivation of the classical theorem as well as that of Chacholski-Scherner-Werndli.
Reviewer: Hirokazu Nishimura (Tsukuba)Selected papers. Edited by César Camacho.https://www.zbmath.org/1456.570012021-04-16T16:22:00+00:00"Lima, Elon"https://www.zbmath.org/authors/?q=ai:lages-lima.elonPublisher's description: This book contains all research papers published by the distinguished Brazilian mathematician Elon Lima. It includes the papers from his PhD thesis on homotopy theory, which are hard to find elsewhere. Elon Lima wrote more than 40 books in the field of topology and dynamical systems. He was a profound mathematician with a genuine vocation to teach and write mathematics.Marao, about Hopf fibration.https://www.zbmath.org/1456.550082021-04-16T16:22:00+00:00"Berishvili, Guram"https://www.zbmath.org/authors/?q=ai:berishvili.guramSummary: A marao is a cover of a vector space by a set of equidimensional subspaces with pairwise trivial intersections. Such structures give rise to fibrations of particular kind. Naturally occurring examples are described. In particular, it is explained how the classical Hopf fibrations can be uniformly obtained from maraos.Spectral algebra models of unstable \(v_n\)-periodic homotopy theory.https://www.zbmath.org/1456.550072021-04-16T16:22:00+00:00"Behrens, Mark"https://www.zbmath.org/authors/?q=ai:behrens.mark-joseph"Rezk, Charles"https://www.zbmath.org/authors/?q=ai:rezk.charlesLet \(X\) be a topological space and let \({\mathbb{S}}^X\) denote the Spanier-Whitehead
dual of \(X\). This is a commutative ring spectrum. A long standing question in topology is
to what extent one can recover \(X\) from \({\mathbb{S}}^X\). Classical rational homotopy
theory implies that one can essentially recover the rational homotopy type of \(X\) from
the rational homotopy type of \({\mathbb{S}}^X\). Moreover, there exists an algebra over the
Lie operad which is a kind of a Koszul dual to \({\mathbb{S}}^X\) whose rational homotopy groups
are isomorphic to the rational homotopy groups of \(X\).
A theorem of \textit{M. A. Mandell} [Topology 40, No. 1, 43--94 (2001; Zbl 0974.55004)] gives a partial \(p\)-adic analogue of these results. Mandell showed that the
\(p\)-adic homotopy type of \(X\) is essentially determined by the \(p\)-adic homotopy type of
\({\mathbb{S}}^X\). However, there is no known \(p\)-adic analogue of Lie algebra models
for homotopy types.
In this paper the authors develop a chromatic analogue of these theories. Namely, they show
that the topological André-Quillen cohomology spectrum of \({\mathbb{S}}^X\) serves
as a kind of Lie algebra model for the \(K(n)\)-localization of \(X\). More precisely, they show that
there is a natural map
\[
c^{K(n)}_X\colon \Phi_{K(n)}(X)\to\mathrm{TAQ}_{{\mathbb{S}}_{K(n)}}({\mathbb{S}}_{K(n)}^X).
\]
Here \(K(n)\) is Morava \(K\)-theory, \({\mathbb{S}}_{K(n)}\) denotes the \(K(n)\)-localization
of the sphere spectrum and \(\Phi\) is the Bousfield-Kuhn functor. The authors prove that
\(c^{K(n)}_X\) induces an isomorphism on homotopy groups when \(X\) is a sphere or a
product of spheres or most generally, when the Goodwillie tower of the identity converges in
\(v_n\)-periodic homotopy for \(X\).
For the entire collection see [Zbl 07174496].
Reviewer: Marja Kankaanrinta (Helsinki)Natural models of homotopy type theory.https://www.zbmath.org/1456.030232021-04-16T16:22:00+00:00"Awodey, Steve"https://www.zbmath.org/authors/?q=ai:awodey.steveSummary: The notion of a \textit{natural model} of type theory is defined in terms of that of a \textit{representable natural transfomation} of presheaves. It is shown that such models agree exactly with the concept of a \textit{category with families} in the sense of Dybjer, which can be regarded as an algebraic formulation of type theory. We determine conditions for such models to satisfy the inference rules for dependent sums \(\Sigma\), dependent products \(\Pi\) and intensional identity types , as used in homotopy type theory. It is then shown that a category admits such a model if it has a class of maps that behave like the abstract fibrations in axiomatic homotopy theory: They should be stable under pullback, closed under composition and relative products, and there should be weakly orthogonal factorizations into the class. It follows that many familiar settings for homotopy theory also admit natural models of the basic system of homotopy type theory.Truncation of unitary operads.https://www.zbmath.org/1456.180152021-04-16T16:22:00+00:00"Bao, Yan-Hong"https://www.zbmath.org/authors/?q=ai:bao.yanhong"Ye, Yu"https://www.zbmath.org/authors/?q=ai:ye.yu"Zhang, James J."https://www.zbmath.org/authors/?q=ai:zhang.james-yiming|zhang.james-jUnitary operads over a fixed field \(\Bbbk\) are those one-dimensional in arities zero and one; let \(Op_+\) denote the category of such.
A unitary operad \(P\) has a natural family of restriction operators \(P(n) \to P(s)\) for \(s \leq n\), obtained by composing with the \(0\)-ary operation in \(n-s\) places.
These restrictions are used to define a sequence of ideals \(\vphantom{\Upsilon}^k\Upsilon\) of \(P\) by intersecting some of their kernels.
An analogue of the Gelfand-Kirillov dimension of \(\Bbbk\)-algebras is given.
Namely, the GK dimension of a (locally finite-dimensional) operad \(P\) is defined by the formula
\[
\mathrm{GKdim}(P) = \limsup_{n\to \infty} \left( \log_n \left( \sum_{i=0}^n \dim_{\Bbbk} P(i)\right) \right).
\]
The authors also introduce and study other invariants, including the exponent, the signature, and the Hilbert series.
Many of the results are concerned with `2-unitary operads', which are related to `operads with multiplication' in the sense of Gerstenhaber-Voronov
[\textit{M. Gerstenhaber} and \textit{A. Voronov}, Int. Math. Res. Not. 1995, No. 3, 141--153 (1995; Zbl 0827.18004)].
More precisely, a \textbf{2-unitary} operad is an object of the comma category \(Mag\downarrow Op_+\) of operads under the magma operad, that is a unitary operad with a specified non-associative multiplication \(\mu\) (there are also associative and commutative variants of this notion).
If \(P\) is 2-unitary and has finite GK dimension, then the GK dimension must be an integer.
Further, this GK dimension can be characterized by the vanishing of the ideals \(\vphantom{\Upsilon}^k\Upsilon\).
The authors classify the 2-unitary operads of small GK dimension.
There is only a single 2-unitary operad of GK dimension 1, namely the commutative operad \(Com\).
Every augmented \(\Bbbk\)-algebra \(\Lambda\) generates a 2-unitary operad which is \(\Lambda\) in arity one and which has dimension greater than one in all higher arities so long as \(\Lambda\) is different from \(\Bbbk\).
This construction is part of an equivalence of categories between 2-unitary operads with GK dimension at most 2 and finite-dimensional, augmented \(\Bbbk\)-algebras.
The quotient operads of the associative operad \(Ass\) with fixed GK dimension at most 4 are classified when \(\Bbbk\) has characteristic zero.
There are only three such quotient operads, namely \(Ass / \vphantom{\Upsilon}^k\Upsilon\) for \(k=1,3,4\) of GK dimension \(k\).
In particular, there is no quotient operad of \(Ass\) with GK dimension 2.
It is shown that the situation is more complicated for \(k=5\).
There are several other interesting directions in the paper as well, including characterizations of artinian, semiprime operads (in the reduced, unitary, or 2-unitary cases), the fact that every signature can be realized by an object of \(Com\downarrow Op_+\), and the relationship between the GK dimension of operads and of their algebras.
Reviewer: Philip Hackney (Lafayette)Lusternik-Schnirelmann category of relation matrices on finite spaces and simplicial complexes.https://www.zbmath.org/1456.550032021-04-16T16:22:00+00:00"Tanaka, Kohei"https://www.zbmath.org/authors/?q=ai:tanaka.koheiLet \(\phi\colon K\to L\) be a simplicial map between finite simplicial complexes. The simplicial LS category of \(f\), denoted \(\mathrm{scat}(f)\) is the smallest non-negative integer \(n\) such that there exists a cover \(\{U_i\}_{i=0}^n\) of \(X\) where \(f|U_i\) is contiguous to the constant map for every \(0\leq i\leq n\). This is a simplicial analogue to the classic Lusternik-Schnirelmann category, defined in a similar way, of a continuous map between topological spaces. A recent result of \textit{J. González} [New York J. Math. 24, 279--292 (2018; Zbl 1394.55004)] shows that if \(\lambda_k\colon \mathrm{sd}^k(K)\to K\) is the simplicial approximation from the \(k^{th}\) barycentric subdivision of \(K\), then \(\mathrm{scat}(f\circ \lambda_K)=\mathrm{cat}(|f|)\) for sufficiently large \(k\). In other words, \(\mathrm{scat}(f)\) can be thought of as approximating \(\mathrm{cat}(|f|)\).
Using the result of González as well as some of the work of \textit{G. Raptis} [Homology Homotopy Appl. 12, No. 2, 211--230 (2010; Zbl 1215.18017)] on the homotopy theory of posets, the author of the paper under review develops a combinatorial method to approximate the LS category of a continuous map \(\phi\colon |K|\to |L|\) in practice. This is accomplished by studying the principal relational matrix of a map between finite spaces associated to the complexes. Once this matrix is obtained, one may perform ``operations'' on it, akin to elementary row operations, in order to reduce the matrix and compute the simplicial category. Some computations are given and it is pointed out that although in theory this method is promising since the number of open sets is theoretically finite, this may be difficult in practice since the barycentric subdivision grows factorially in the number of simplices.
Reviewer: Nicholas A. Scoville (Collegeville)Area-dependent quantum field theory.https://www.zbmath.org/1456.814112021-04-16T16:22:00+00:00"Runkel, Ingo"https://www.zbmath.org/authors/?q=ai:runkel.ingo"Szegedy, Lóránt"https://www.zbmath.org/authors/?q=ai:szegedy.lorantSummary: Area-dependent quantum field theory is a modification of two-dimensional topological quantum field theory, where one equips each connected component of a bordism with a positive real number -- interpreted as area -- which behaves additively under glueing. As opposed to topological theories, in area-dependent theories the state spaces can be infinite-dimensional. We introduce the notion of regularised Frobenius algebras in Hilbert spaces and show that area-dependent theories are in one-to-one correspondence to commutative regularised Frobenius algebras. We also provide a state sum construction for area-dependent theories. Our main example is two-dimensional Yang-Mills theory with compact gauge group, which we treat in detail.Brasselet number and Newton polygons.https://www.zbmath.org/1456.140622021-04-16T16:22:00+00:00"Dalbelo, Thaís M."https://www.zbmath.org/authors/?q=ai:dalbelo.thais-maria"Hartmann, Luiz"https://www.zbmath.org/authors/?q=ai:hartmann.luizAuthors' abstract: We present a formula to compute the Brasselet number of \( f:(Y,0)\rightarrow (\mathbb{C},0)\) where \(Y\subset X\) is a non-degenerate complete intersection in a toric variety \(X\). As applications we establish several results concerning invariance of the Brasselet number for families of non-degenerate complete intersections. Moreover, when \((X,0)=(\mathbb{C} ^{n},0)\) we derive sufficient conditions to obtain the invariance of the Euler obstruction for families of complete intersections with an isolated singularity which are contained in \(X\).
Reviewer: Tadeusz Krasiński (Łódź)Rational homotopy of mapping spaces between complex Grassmannians.https://www.zbmath.org/1456.550062021-04-16T16:22:00+00:00"Gatsinzi, Jean Baptiste"https://www.zbmath.org/authors/?q=ai:gatsinzi.jean-baptiste"Otieno, Paul Antony"https://www.zbmath.org/authors/?q=ai:otieno.paul-antony"Onyango-Otieno, Vitalis"https://www.zbmath.org/authors/?q=ai:onyango-otieno.vitalisThe authors use Sullivan models to describe the rational homotopy type of the component of the inclusion in the mapping space \(\mathrm{map}(Gr(2,n),Gr(2,n+r);i_n)\). They show that the homotopy type of \(\mathrm{map}(Gr(2,n),Gr(2,n+r);i_n)\) is that of a product of odd spheres. They go on to show that the cohomology algebra of \(\mathrm{map}(Gr(2,n),Gr(2,n+r);i_{n},r)\) for \(r>3n-6\) contains a polynomial algebra over a generator of degree \(2\).
Reviewer: Rugare Kwashira (Johannesburg)Implicit differential inclusions with acyclic right-hand sides: an essential fixed points approach.https://www.zbmath.org/1456.340152021-04-16T16:22:00+00:00"Andres, Jan"https://www.zbmath.org/authors/?q=ai:andres.jan"Górniewicz, Lech"https://www.zbmath.org/authors/?q=ai:gorniewicz.lechSummary: Effective criteria are given for the solvability of initial as well as boundary valueproblems to implicit ordinary differential inclusions whose right-hand sides are governed by compactacyclic maps. Cauchy and periodic implicit problems are also considered on proximate retracts. Ournew approach is based on the application of the topological essential fixed point theory. Implicitproblems for partial differential inclusions are only indicated.Higher Auslander algebras of type \(\mathbb{A}\) and the higher Waldhausen \(\mathsf{S}\)-constructions.https://www.zbmath.org/1456.180112021-04-16T16:22:00+00:00"Jasso, Gustavo"https://www.zbmath.org/authors/?q=ai:jasso.gustavoSummary: These notes are an expanded version of my talk at the ICRA 2018 in Prague, Czech Republic; they are based on joint work with \textit{T. Dyckerhoff} et al. [Adv. Math. 355, Article ID 106762, 73 p. (2019; Zbl 07107210)]. In them we relate Iyama's higher Auslander algebras of type \(\mathbb{A}\) to Eilenberg-Mac Lane spaces in algebraic topology and to higher-dimensional versions of the Waldhausen \(\mathsf{S}\)-construction from algebraic \(K\)-theory.
For the entire collection see [Zbl 07314259].Recognizing quasi-categorical limits and colimits in homotopy coherent nerves.https://www.zbmath.org/1456.180032021-04-16T16:22:00+00:00"Riehl, Emily"https://www.zbmath.org/authors/?q=ai:riehl.emily"Verity, Dominic"https://www.zbmath.org/authors/?q=ai:verity.dominicThe authors prove that various quasi-categories whose objects are \(\infty\)-categories in a very general sense are complete: admitting limits indexed by all simplicial sets. This result and others of a similar flavor follow from a general theorem in which they characterize the data that is required to define a limit cone in a quasi-category constructed as a homotopy coherent nerve. Since all quasi-categories arise this way up to equivalence, this analysis covers the general case. Namely, they show that quasi-categorical limit cones may be modeled at the point-set level by pseudo homotopy limit cones, whose shape is governed by the weight for pseudo limits over a homotopy coherent diagram but with the defining universal property up to equivalence, rather than isomorphism, of mapping spaces. The applications follow from the fact that the \((\infty,1)\)-categorical core of an \(\infty\)-cosmos admits weighted homotopy limits for all flexible weights, which includes in particular the weight for pseudo cones.
Reviewer: Philippe Gaucher (Paris)The fullness axiom and exact completion of homotopy categories.https://www.zbmath.org/1456.180072021-04-16T16:22:00+00:00"Emmenegger, Jacopo"https://www.zbmath.org/authors/?q=ai:emmenegger.jacopoThrough the use of a category-theoretic formulation of Aczel's Fullness Axiom from Constructive Set Theory, in this paper the author derives local cartesian closure for an exact completion of a category with weak finite limits. A characterisation of locally cartesian closed ex/wlex completions is given by the author in a previous paper, however it is not much suited for the study of ex/wlex completions of homotopy categories. ``The present paper provides a condition ensuring the local cartesian closure \((\mathrm{Ho}\mathbb{M})_{\mathrm{ex}}\) for a large class of model categories; this condition turns out to be what \textit{A. Carboni} and \textit{G. Rosolini} [J. Pure Appl. Algebra 154, No. 1--3, 103--116 (2000; Zbl 0962.18001)] named weak local cartesian closure, that is, simply existence of weak dependent products''.
In terms of \textit{full (dependent) diagrams} (Definitions 2.2 and 3.1), which arise as homotopy quotients of weak (dependent) products in \(\mathrm{Ho}\mathbb{M}\), characterisations of (local) cartesian closure of the exact completion \(\mathbb{C}_{\mathrm{ex}}\) are given in Theorems 2.7 and 3.7. Then, in order to transfer such results to the homotopy category \(\mathrm{Ho}\mathbb{M}\) of a Quillen model category \(\mathbb{M}\), the notion of cofibrant objects is used, culminating in the following:
Theorem 4.10. Let \(\mathbb{M}\) be a right proper model category where every object is cofibrant. If \(\mathbb{M}\) has weak dependent products, then \(\mathrm{Ho}\mathbb{M}\) has dependent full diagrams and, in turn, \((\mathrm{Ho}\mathbb{M})_{\mathrm{ex}}\) is locally cartesian closed.
As an application of the latter, local cartesian closure of the exact completion of the homotopy category of two standard model structures on the category of topological spaces (by Quillen and by Strøm) is proved, also answering a question left open in a paper by \textit{M. Gran} and \textit{E. M. Vitale} [Cah. Topologie Géom. Différ. Catégoriques 39, No. 4, 287--297 (1998; Zbl 0918.18003)].
.
Reviewer: Willian Ribeiro (Coimbra)