Recent zbMATH articles in MSC 55Nhttps://zbmath.org/atom/cc/55N2023-01-20T17:58:23.823708ZWerkzeugShellability and homology of \(q\)-complexes and \(q\)-matroidshttps://zbmath.org/1500.050622023-01-20T17:58:23.823708Z"Ghorpade, Sudhir R."https://zbmath.org/authors/?q=ai:ghorpade.sudhir-r"Pratihar, Rakhi"https://zbmath.org/authors/?q=ai:pratihar.rakhi"Randrianarisoa, Tovohery H."https://zbmath.org/authors/?q=ai:randrianarisoa.tovohery-hajatianaSummary: We consider a \(q\)-analogue of abstract simplicial complexes, called \(q\)-complexes, and discuss the notion of shellability for such complexes. It is shown that \(q\)-complexes formed by independent subspaces of a \(q\)-matroid are shellable. Further, we explicitly determine the homology of \(q\)-complexes corresponding to uniform \(q\)-matroids. We also outline some partial results concerning the determination of homology of arbitrary shellable \(q\)-complexes.\(L^2\)-Betti numbers of \(C^*\)-tensor categories associated with totally disconnected groupshttps://zbmath.org/1500.460492023-01-20T17:58:23.823708Z"Valvekens, Matthias"https://zbmath.org/authors/?q=ai:valvekens.matthiasThe present article studies homology and \(L^2\)-Betti numbers of two classes of rigid C*-tensor categories related to Hecke pairs.
First, the author studies normal and almost-normal inclusions of rigid C*-tensor categories \(\mathcal{C} \subseteq \mathcal{D}\) and shows in Theorem 3.8 that vanishing of \(L^2\)-Betti numbers of \(\mathcal{C}\) up to degree \(N\) implies the vanishing of \(L^2\)-Betti numbers of \(\mathcal{D}\) up to degree \(N\). This result applies to show that a rigid C*-tensor category with an amenable almost-normal subcategory must have vanishing \(L^2\)-Betti numbers. Key examples of such categories come from the inclusion of a compact normal subgroup into a totally disconnected locally compact group and were first considered in work of \textit{Y. Arano} and \textit{S. Vaes} [Abel Symp. 12, 1--43 (2016; Zbl 1375.46055)].
The second kind of results obtained in the article concern the homology of quasi-regular inclusions of \(\mathrm{II}_1\) factors arising from actions of discrete Hecke pairs. If \(\Lambda \leq \Gamma\) is a unimodular discrete Hecke pair, which is a commensurable inclusion of discrete groups such that every \(\Lambda\) double coset in \(\Gamma\) contains the same number of left and right \(\Lambda\) cosets, and \(\Gamma\) acts on a \(\mathrm{II}_1\) factor \(P\), then Proposition 4.8 and its Corollary 4.10 describe the homology of the inclusion \(P \rtimes \Lambda \subseteq P \rtimes \Gamma\) with certain coefficients in terms of the group cohomology of the associated Schlichting completion \(G = \Gamma {/\!\!/} \Lambda\). This result is used in the proof of Theorem~4.1, which calculates the \(L^2\)-Betti numbers of the inclusion \(P \rtimes \Lambda \subseteq P \rtimes \Gamma\) identifying them with the \(L^2\)-Betti numbers of \(G\) with respect to the Haar measure normalised on the (compact) closure of \(\Lambda\) in \(G\). More applications of the identification of homology can be obtained in the first degree, leading in Proposition 4.14 to the result that property~(T), the Haagerup property and amenability of the inclusion \(P \rtimes \Lambda \subseteq P \rtimes \Gamma\) are characterised by the respective properties for \(G\).
The article ends with a discussion of open problems, adding to a question already raised in Remark~3.7.
Reviewer: Sven Raum (Stockholm)Amenable covers and integral foliated simplicial volumehttps://zbmath.org/1500.550022023-01-20T17:58:23.823708Z"Löh, Clara"https://zbmath.org/authors/?q=ai:loh.clara"Moraschini, Marco"https://zbmath.org/authors/?q=ai:moraschini.marco"Sauer, Roman"https://zbmath.org/authors/?q=ai:sauer.romanA well-known theorem of Gromov, the proof of which has been worked out by \textit{N. V. Ivanov} [J. Sov. Math. 37, 1090--1115 (1987; Zbl 0612.55006)], says: If a space \(X\) admits an amenable covering, such that each point of \(X\) is contained in no more than \(n\) sets of the covering, then the canonical morphism from bounded to ordinary cohomology \(H_b^i(X)\to H^i(X)\) vanishes in degrees \(i\ge n\). (A covering is called amenable if for each set of the covering the image of its fundamental group in \(\pi_1X\) is amenable, e.g., if each set has amenable fundamental group.) As a consequence, a closed \(n\)-manifold \(M\), that allows such a covering, must have vanishing simplicial volume \(\Vert M\Vert=0\).
The paper under review shows that for a closed, aspherical \(n\)-manifold \(M\), that allows such a covering, also the parametrized simplicial volume \(\vert M\vert^\alpha\) vanishes for every essentially free standard \(\pi_1M\)-space \(\alpha=(X,\mu)\). As a consequence, if \(M\) is a closed, aspherical \(n\)-manifold with residually finite fundamental group, that allows a covering as above, then \(\Vert M\Vert_{\mathbf Z}^\infty=\Vert M\Vert=0\).
Reviewer: Thilo Kuessner (Augsburg)Persistent homology in \(\ell_\infty\) metrichttps://zbmath.org/1500.550032023-01-20T17:58:23.823708Z"Beltramo, Gabriele"https://zbmath.org/authors/?q=ai:beltramo.gabriele"Skraba, Primoz"https://zbmath.org/authors/?q=ai:skraba.primozBased on the \(l_{\infty}\)-metric, the paper examines a number of classical complexes, including the Čech, Vietoris-Rips, and Alpha complexes. ``We define two new families of flag complexes, the Alpha flag and Minibox complexes, and prove their equivalence to Čech complexes in homological degrees zero and one. Moreover, we provide algorithms for finding Minibox edges of two, three, and higher-dimensional points.'' Besides, using this algorithm, the paper presents computational experiments on random points. This approach can be useful to compute persistent homology of a complex.
Reviewer: Sang-Eon Han (Jeonju)Generalized persistence diagrams for persistence modules over posetshttps://zbmath.org/1500.550042023-01-20T17:58:23.823708Z"Kim, Woojin"https://zbmath.org/authors/?q=ai:kim.woojin"Mémoli, Facundo"https://zbmath.org/authors/?q=ai:memoli.facundoThe field of Topological Data Analysis (TDA) can be argued to have started with the introduction of persistence diagrams. The input data is that of a persistence module \(\mathcal{V} = (\{V_a\},\{\phi_{a,b}\})\) consisting of vector spaces \(\{V_a \mid a \in \mathbb{R}\}\) with morphisms \(\varphi_{a,b}:V_a \to V_b\) satisfying composition laws \(\varphi_{a,c} = \varphi_{b, c} \circ \varphi_{a,b}\). Under certain finiteness assumptions (see e.g.~[\textit{S. Y. Oudot}, Persistence theory. From quiver representations to data analysis. Providence, RI: American Mathematical Society (AMS) (2015; Zbl 1335.55001)]), these modules always decompose into so-called interval modules, which are defined as follows. Fixing a connected interval \(I \subseteq \mathbb{R}\), the interval module \(\mathcal{I}\) is one for which every vector space is either the one-dimensional vector space \(V_a = k\) if \(a \in I\); or is 0 otherwise. The maps are defined to be isomorphisms if possible, and the 0-map otherwise. The decomposition \(\mathcal{V} = \bigoplus_{I \in B} \mathcal{I}_I \) is unique up to isomorphism; so we call the collection \(B\) the barcode of \(\mathcal{V}\). These are often visualized as a collection of points \((a,b)\) for each interval \(I=(a,b)\) in \(B\), called a persistence diagram.
Following [\textit{P. Bubenik} and \textit{J. A. Scott}, Discrete Comput. Geom. 51, No. 3, 600--627 (2014; Zbl 1295.55005)], we can view the persistence module as a functor, \(\mathcal{V}: (\mathbb{R}, \leq) \to \mathbf{vec}\). With this vantage point, it is immediate to generalize the study of persistence modules to functors \(F: \mathbf{P} \to \mathcal{C}\) from a poset to an arbitrary category. Pairs of choices for \(\mathbf{P}\) and \(\mathcal{C}\) result in many objects of study in the TDA literature: persistence modules of course, but also multiparameter persistence, Reeb graphs, mapper graphs, zigzag persistence, and merge trees to name a few.
This paper provides a category theoretic version of the rank invariant (Definition 3.5), which generalizes the standard rank invariant \(rk(\phi_{a,b})\). This paper also extends the work of \textit{A. Patel} [J. Appl. Comput. Topol. 1, No. 3--4, 397--419 (2018; Zbl 1398.18015)] in by providing a combinatorial computation for Patel's generalized persistence diagram (Definition 3.13). By way of application, the authors further show that this can be used to compute 0-dimensional levelset persistence in the special case of Reeb graphs (realized when \(\mathcal{C} = \mathbf{Set}\)) without needing to pass to vector space representations.
Reviewer: Elizabeth Munch (East Lansing)Bayesian topological signal processinghttps://zbmath.org/1500.550052023-01-20T17:58:23.823708Z"Oballe, Christopher"https://zbmath.org/authors/?q=ai:oballe.christopher"Cherne, Alan"https://zbmath.org/authors/?q=ai:cherne.alan"Boothe, Dave"https://zbmath.org/authors/?q=ai:boothe.dave"Kerick, Scott"https://zbmath.org/authors/?q=ai:kerick.scott"Franaszczuk, Piotr J."https://zbmath.org/authors/?q=ai:franaszczuk.piotr-j"Maroulas, Vasileios"https://zbmath.org/authors/?q=ai:maroulas.vasileiosThis paper works on giving an interpretable framework for signal processing via sublevel homology. The paper is trying to establish interpretable links between sublevel set persistence diagrams of signals and their frequency domain which are used to study time series analysis. The authors explore Bayesian framework for the time series classification, from which they find that the Bayesian topological features are pretty useful just as other well-known features, such as those from power spectral densities and continuous wavelets. Finally, they apply their results to electroencephalography which is a neuroimaging technique wherein electrodes are placed on a subject's head to measure local changes in voltage over time, which are reported as a collection of time series.
Reviewer: Qingyun Zeng (Philadelphia)Stratified surgery and K-theory invariants of the signature operatorhttps://zbmath.org/1500.580042023-01-20T17:58:23.823708Z"Albin, Pierre"https://zbmath.org/authors/?q=ai:albin.pierre"Piazza, Paolo"https://zbmath.org/authors/?q=ai:piazza.paoloIn the paper under review, the authors generalize Higson-Roe's analytic interpretation of surgery exact sequence to the setting of stratified spaces.
More precisely, for every \(m\)-dimensional, oriented, smoothly stratified Cheeger space \(\hat {X}\) with fundamental group \(\Gamma\), they establish the following commutative diagram
\[
\begin{tikzcd}
L_{\mathrm{BQ}, d \hat{X} \times I}(\hat{X} \times I) \ar[r]\ar[d]& S_{\mathrm{BQ}}(\hat{X}) \ar[r]\ar[d]& N_{\mathrm{BQ}}(\hat{X}) \ar[r]\ar[d]&L_{\mathrm{BQ}, d \hat{X}}(\hat{X})\ar[d]\\
K_{m+1}\left(C_r^* \Gamma\right)\left[\frac{1}{2}\right] \ar[r] &K_{m+1}\left(D^*(\hat{X}_r^*)^n\right)\left[\frac{1}{2}\right] \ar[r] &K_m(\hat{X})\left[\frac{1}{2}\right] \ar[r]& K_m\left(C_r^* \Gamma\right)\left[\frac{1}{2}\right]
\end{tikzcd}
\]
Surgery theory is a subject of studying manifolds via cutting and pasting. As a sophisticated abstraction of surgery theory, surgery exact sequence [\textit{C. T. C. Wall}, Surgery on compact manifolds. 2nd ed. Providence, RI: American Mathematical Society (1999; Zbl 0935.57003)] allows people to convert concrete geometric operations into formal diagram chasing.
In a series of papers, \textit{N. Higson} and \textit{J. Roe} [\(K\)-Theory 33, No. 4, 277--299 (2004; Zbl 1083.19002); \(K\)-Theory 33, No. 4, 301--324 (2004; Zbl 1083.19003); \(K\)-Theory 33, No. 4, 325--346 (2004; Zbl 1085.19002)] established the remarkable result that there is a natural commutative diagram sending the surgery exact sequence of Wall to an ``analytic surgery sequence'' involving \(K\)-theory groups of certain \(C^*\)-algebras.
It is a meaningful task to extend the above results to singular spaces.
In the topological aspect, \textit{W. Browder} and \textit{F. Quinn} [Proc. int. Conf. Manifolds relat. Top. Topol., Tokyo 1973, 27--36 (1975; Zbl 0343.57017)] introduced a surgery exact sequence for stratified spaces (these are manifold-like singular spaces with each pure stratum being a manifold and endowed with tubular control data). In the analytic aspect, restricted to Witt or Cheeger spaces , the authors of the current paper developed the theory of signature operators [\textit{P. Albin} et al., Ann. Sci. Éc. Norm. Supér. (4) 45, No. 2, 241--310 (2012; Zbl 1260.58012); J. Noncommut. Geom. 11, No. 2, 451--506 (2017; Zbl 1375.57034); J. Reine Angew. Math. 744, 29--102 (2018; Zbl 1434.58001)]. This allows them to map the Browder-Quinn stratified surgery exact sequence to the analytic one and verify the commutativity of the diagram. A substantial technical point is the definition of ``ideal boundary condition''.
Besides the main theorem, the authors contribute a detailed account of Browder-Quinn's surgery exact sequence (which was not seen before in the literature). A geometric example on the cardinality of the stratified structure set is also included (in the spirit of [\textit{S. Chang} and \textit{S. Weinberger}, Geom. Topol. 7, 311--319 (2003; Zbl 1037.57028)]).
Reviewer: Hailiang Hu (Dalian)Explicit construction of \(N = 2\) SCFT orbifold models. Spectral flow and mutual localityhttps://zbmath.org/1500.810632023-01-20T17:58:23.823708Z"Belavin, Alexander"https://zbmath.org/authors/?q=ai:belavin.aleksandr-abramovich"Belavin, Vladimir"https://zbmath.org/authors/?q=ai:belavin.vladimir-a"Parkhomenko, Sergey"https://zbmath.org/authors/?q=ai:parkhomenko.sergei-evgenevichSummary: In this work we present a new approach to constructing Calabi-Yau orbifold models required for compactification in superstring theory. We use the connection of CY orbifolds with the class of exactly solvable \(N = 2\) SCFT models to explicitly construct a complete set of fields in these models using the twisting of the spectral flow and the requirement of mutual locality of the fields.Persistent topology of protein spacehttps://zbmath.org/1500.920752023-01-20T17:58:23.823708Z"Hamilton, W."https://zbmath.org/authors/?q=ai:hamilton.william-o|hamilton.wesley|hamilton.william-l|hamilton.william-edwin.1|hamilton.william-t|hamilton.wallace-l"Borgert, J. E."https://zbmath.org/authors/?q=ai:borgert.j-e"Hamelryck, T."https://zbmath.org/authors/?q=ai:hamelryck.thomas"Marron, J. S."https://zbmath.org/authors/?q=ai:marron.james-stephenSummary: Protein fold classification is a classic problem in structural biology and bioinformatics. We approach this problem using persistent homology. In particular, we use alpha shape filtrations to compare a topological representation of the data with a different representation that makes use of knot-theoretic ideas. We use the statistical method of angle-based joint and individual variation explained (AJIVE) to understand similarities and differences between these representations.
For the entire collection see [Zbl 1487.55002].