Using spaces of homomorphisms and the descending central series of the free groups, simplicial spaces are constructed for each integer q>1 and every topological group G, with realizations B(q,G) that filter the classifying space BG. In particular for q=2 this yields a single space B(2,G) assembled from all the n-tuples of commuting elements in G. Homotopy properties of the B(q,G) are considered for finite groups. Cohomology calculations are provided for compact Lie groups. The spaces B(2,G) are described in detail for transitively commutative groups.
The so called Čech and Vietoris-Rips simplicial filtrations are designed to capture information about the topological structure of metric datasets. These filtrations are two of the workhorses in the field of topological data analysis. They enjoy stability with respect to the Gromov-Hausdorff (GH) distance, and this stability property allows us to estimate the GH distance between finite metric space representations of the underlying datasets. Via the concept of Gromov's curvature sets we construct a rich theoretical framework of valuation-induced stable filtration functors. This framework inclu…
We show that the Strong Novikov Conjecture for the maximal C*-algebra C*(G) of a discrete group G is equivalent to a statement in topological K-theory for which the corresponding statement in algebraic K-theory is always true. We also show that for any group G, rational injectivity of the full assembly map for the topological K-theory of C*(G) follows from rational injectivity of the restricted assembly map.
In topological data analysis, a point cloud data P extracted from a metric space is often analyzed by computing the persistence diagram or barcodes of a sequence of Rips complexes built on $P$ indexed by a scale parameter. Unfortunately, even for input of moderate size, the size of the Rips complex may become prohibitively large as the scale parameter increases. Starting with the Sparse Rips filtration introduced by Sheehy, some existing methods aim to reduce the size of the complex so as to improve the time efficiency as well. However, as we demonstrate, existing approaches still fall short o…
In this paper a parameterized generalization of a good cover filtration is introduced called an ε-good cover, defined as a cover filtration in which the reduced homology groups of the image of the inclusions between the intersections of the cover filtration at two scales ε apart are trivial. Assuming that one has an ε-good cover filtration of a finite simplicial filtration, we prove a tight bound on the bottleneck distance between the persistence diagrams of the nerve filtration and the simplicial filtration that is linear with respect to ε and the homology dimension. This bound is the result …
To any finite simplicial complex X, we associate a natural filtration starting from Chari and Joswig's discrete Morse complex and abutting to the matching complex of X. This construction leads to the definition of several homology theories, which we compute in a number of examples. We also completely determine the graded object associated to this filtration in terms of the homology of simpler complexes. This last result provides some connections to the number of vertex-disjoint cycles of a graph.
Shulman's spatial type theory internalizes the modalities of Lawvere's axiomatic cohesion in a homotopy type theory, enabling many of the constructions from Schreiber's modal approach to differential cohomology to be carried out synthetically. In spatial type theory, every type carries a spatial cohesion among its points and every function is continuous with respect to this. But in mathematical practice, objects may be spatial in more than one way at the same time; for example, a simplicial space has both topological and simplicial structures. In this paper, we put forward a type theory with "…
We use combinatorial group theory methods to extend the definition of a classical James-Hopf invariant to a simplicial group setting. This allow us to realize certain coalgebra idempotents at sSet -level and obtain a functorial decomposition of the spectral sequence, associated with the lower p-central series filtration on the free simplicial group.
Topological Data Analysis (TDA) studies the shape of data. A common topological descriptor is the persistence diagram, which encodes topological features in a topological space at different scales. Turner, Mukeherjee, and Boyer showed that one can reconstruct a simplicial complex embedded in R^3 using persistence diagrams generated from all possible height filtrations (an uncountably infinite number of directions). In this paper, we present an algorithm for reconstructing plane graphs K=(V,E) in R^2 , i.e., a planar graph with vertices in general position and a straight-line embedding, from a …
Rips complexes are important structures for analyzing topological features of metric spaces. Unfortunately, generating these complexes is expensive because of a combinatorial explosion in the complex size. For $n$ points in $\mathbb{R}^d$, we present a scheme to construct a $2$-approximation of the filtration of the Rips complex in the $L_\infty$-norm, which extends to a $2d^{0.25}$-approximation in the Euclidean case. The $k$-skeleton of the resulting approximation has a total size of $n2^{O(d\log k +d)}$. The scheme is based on the integer lattice and simplicial complexes based on the baryce…
Let $X_{\bullet}$ denote a simplicial space. The purpose of this note is to record a decomposition of the suspension of the individual spaces $X_n$ occurring in $X_{\bullet}$ in case the spaces $X_n$ satisfy certain mild topological hypotheses and where these decompositions are natural for morphisms of simplicial spaces. In addition, the summands of $X_n$ which occur after one suspension are stably equivalent to choices of filtration quotients of the geometric realization $|X_{\bullet}|$. The purpose of recording these decompositions is that they imply decompositions of the single suspension o…
The classical persistence algorithm computes the unique decomposition of a persistence module implicitly given by an input simplicial filtration. Based on matrix reduction, this algorithm is a cornerstone of the emergent area of topological data analysis. Its input is a simplicial filtration defined over the integers $\mathbb{Z}$ giving rise to a $1$-parameter persistence module. It has been recognized that multiparameter version of persistence modules given by simplicial filtrations over $d$-dimensional integer grids $\mathbb{Z}^d$ is equally or perhaps more important in data science applicat…
We study simplicial profinite groups with a view towards applications in profinite combinatorial group theory. This approach provides a natural framework to the concept of pro-$\mathfrak{C}$-presentation of a pro-$\mathfrak{C}$-group $G$ as a 1-truncation of its free simplicial pro-$\mathfrak{C}$-resolution. The category of simplicial pro-$\mathfrak{C}$-groups has a closed simplicial model category structure. This yields a possibility to define some old and new derived functors as left Quillen derived functors from this simplicial model category. When $\mathfrak{C}$ is L-groups, than may const…
A tower is a sequence of simplicial complexes connected by simplicial maps. We show how to compute a filtration, a sequence of nested simplicial complexes, with the same persistent barcode as the tower. Our approach is based on the coning strategy by Dey et al. (SoCG 2014). We show that a variant of this approach yields a filtration that is asymptotically only marginally larger than the tower and can be efficiently computed by a streaming algorithm, both in theory and in practice. Furthermore, we show that our approach can be combined with a streaming algorithm to compute the barcode of the to…
In this paper, we propose persistent spectral based machine learning (PerSpect ML) models for drug design. Persistent spectral models, including persistent spectral graph, persistent spectral simplicial complex and persistent spectral hypergraph, are proposed based on spectral graph theory, spectral simplicial complex theory and spectral hypergraph theory, respectively. Different from all previous spectral models, a filtration process, as proposed in persistent homology, is introduced to generate multiscale spectral models. More specifically, from the filtration process, a series of nested top…
We consider the orbit type filtration on a manifold $X$ with locally standard action of a compact torus and the corresponding homological spectral sequence $(E_X)^r_{*,*}$. If all proper faces of the orbit space $Q=X/T$ are acyclic, and the free part of the action is trivial, this spectral sequence can be described in full. The ranks of diagonal terms are equal to the $h'$-numbers of the Buchsbaum simplicial poset $S_Q$ dual to $Q$. Betti numbers of $X$ depend only on the orbit space $Q$ but not on the characteristic function. If $X$ is a slightly different object, namely the model space $X=(P…
This is the second in a trilogy of papers introducing and studying the notion of decomposition space as a general framework for incidence algebras and Möbius inversion, with coefficients in $\infty$-groupoids. A decomposition space is a simplicial $\infty$-groupoid satisfying an exactness condition weaker than the Segal condition. Just as the Segal condition expresses composition, the new condition expresses decomposition. In this paper, we introduce various technical conditions on decomposition spaces. The first is a completeness condition (weaker than Rezk completeness), needed to control si…
We study the inverse problem for persistent homology: For a fixed simplicial complex $K$, we analyse the fiber of the continuous map $\mathrm{PH}$ on the space of filters that assigns to a filter $f: K \to \mathbb R$ the total barcode of its associated sublevel set filtration of $K$. We find that $\mathrm{PH}$ is best understood as a map of stratified spaces. Over each stratum of the barcode space, the map $\mathrm{PH}$ restricts to a (trivial) fiber bundle with fiber a polyhedral complex. Amongst other we derive a bound for the dimension of the fiber depending on the number of distinct endpoi…
We define the Coxeter cochain complex of a Coxeter group (G,S) with coefficients in a Z[G]-module A. This is closely related to the complex of simplicial cochains on the abstract simplicial complex I(S) of the commuting subsets of S. We give some representative computations of Coxeter cohomology and explain the connection between the Coxeter cohomology for groups of type A, the (singular) homology of certain configuration spaces, and the (Tor) homology of certain local Artin rings.
Columbia University Irving Medical Center（哥伦比亚大学欧文医学中心）
The absence of a conventional association between the cell-cell cohabitation and its emergent dynamics into cliques during development has hindered our understanding of how cell populations proliferate, differentiate, and compete, i.e. the cell ecology. With the recent advancement of the single-cell RNA-sequencing (RNA-seq), we can potentially describe such a link by constructing network graphs that characterize the similarity of the gene expression profiles of the cell-specific transcriptional programs, and analyzing these graphs systematically using the summary statistics informed by the alg…
The Vietoris-Rips filtration for an $n$-point metric space is a sequence of large simplicial complexes adding a topological structure to the otherwise disconnected space. The persistent homology is a key tool in topological data analysis and studies topological features of data that persist over many scales. The fastest algorithm for computing persistent homology of a filtration has time $O(M(u)+u^2\log^2 u)$, where $u$ is the number of updates (additions or deletions of simplices), $M(u)=O(u^{2.376})$ is the time for multiplication of $u\times u$ matrices. For a space of $n$ points given by t…
In this master thesis, we extend results from classical simple homotopy theory to the world of stratified homotopy theory. To obtain a well-established framework to work in, we prove a series of results on two model categories of simplicial sets and topological spaces, both equipped with a notion of filtration, introduced by Sylvain Douteau in his PHD thesis. In particular, we show that there is a fully faithful embedding of homotopy categories from the (finite) filtered simplicial into the filtered topological setting. We also use these results to characterize the morphisms in the topological…
