In this paper we study parametrized topological complexity of bundles of real projective spaces which arise as projectivisations of vector bundles. We develop algebraic machinery for computing lower bounds for the parametrized topological complexity of projective bundles based on theory of Stiefel - Whitney characteristic classes. We establish sharp upper bounds for the parametrized topological complexity of projective bundles improving the general upper bounds. Combining the lower and the upper bounds we compute explicitly many specific examples.
We construct weak homotopy equivalences between the geometric realizations of directed Vietoris-Rips complexes and their underlying directed graphs, seen as pseudotopological spaces. Pseudotopological spaces are a generalization of (Čech) closure spaces which in turn generalize topological spaces, but they also include graphs and directed graphs as full subcategories, making them a natural bridge that connects classical algebraic topology with the more applied side of topology. This weak homotopy equivalence implies that singular homology groups of finite directed graphs can be efficiently calculated from finite combinatorial structures, despite their associated chain groups being infinite dimensional. Along the way, we establish analogues of classical results such as the existence of a long exact sequence for homotopy groups of pairs of pseudotopological spaces and that a weak homotopy equivalence induces isomorphisms for homology groups. This work is similar to the work of McCord for finite topological spaces but in the context of pseudotopological spaces. Our results also give a novel approach for studying (higher) homotopy groups of discrete mathematical structures such as (directed) graphs or digital images.
In this paper, we study a new construction which associates a combinatorial cubical complex (G) to an arbitrary undirected simple graph G. The vertices of (G) are indexed by all possible orientations of the edges of G. The cells of (G) are the sets of independent flexes, where a flex is a simultaneous change of orientations of the edges adjacent to a certain sink or a certain source in G. Accordingly, we call (G) the flex complex of the graph G. Our focus is on studying topology and combinatorics of the flex complexes. The main topological theorem says that for an arbitrary graph G, the flex complex (G) is homotopy equivalent to a disjoint union of tori. We also provide formulae for the number of these tori. Furthermore, we prove a much more precise combinatorial result saying that when G is connected, every connected component of (G) is either a collapsible cubical complex, or can be collapsed to a cycle whose length is equal to the number of vertices of G. We shall provide a combinatorial enumeration for the components of both types. Our study is motivated by the beauty and naturality of the graph construction, as well as by the mathematical modeling of the network evolution.
We study a simple type of modular robot, consisting of a collection of identical d-dimensional cubes in ℝ^d , which change position by sliding along each other (parallel to the axes). We describe the configuration space of such a mechanism, and show that it has a dense open smooth submanifold, and that book singularities are generic in its complement.
Previous works on lexicographic optimal chains have shown that they provide meaningful geometric homology representatives while being easier to compute than their l^1 -norm optimal counterparts. We present a novel algorithm to efficiently compute lexicographic optimal chains with a given boundary in a triangulation of 3-space, by leveraging a Lefschetz duality at the chain level and an augmented version of the classical disjoint-set data structure. We also show that the space of lexicographic optimal cycles forms a vector space isomorphic to the homology groups of the complex, a property suggesting a parallel with l^2 -norm optimal chains and Hodge theory. A canonical basis for this space of lexicographic optimal chains can be defined, called critical basis, and we show how to compute it using standard matrix reduction algorithms. In applications, we show how both computing optimal chains with a given boundary and critical bases offer new promising ways of efficiently reconstructing open surfaces in difficult acquisition scenarios.
A persistence diagram is a finite multiset of birth-death pairs representing the lifetimes of topological features across a filtration. Existing functional and kernel representations of persistence diagrams are typically constructed extrinsically through embeddings into auxiliary spaces. For filtrations with finite indexing sets, the associated virtual persistence diagram group obtained by Grothendieck completion of the persistence diagram monoid is a finitely generated lattice. We define a phase map sending each persistence interval to a circular coordinate and a character map aggregating the phases of intervals in a virtual persistence diagram. We introduce heat damping on characters of virtual persistence diagram groups to suppress the unstable frequencies. We derive Lipschitz bounds for the resulting kernels and apply them in a synthetic segmentation experiment.
Given a relation R ⊆ I × J between two sets, Dowker's Theorem (1952) states that the homology groups of two associated simplicial complexes, now known as Dowker complexes, are isomorphic. In its modern form, the full result asserts a functorial homotopy equivalence between the two Dowker complexes. What can be said about relations defined on three or more sets? We present a simple generalization to multiway relations of the form R ⊆ I_1 × I_2 ×⋯× I_m. The theorem asserts functorial homotopy equivalences between m multiway Dowker complexes and a variant of the rectangle complex of Brun and Salbu from their recent short proof of Dowker's Theorem. Our proof uses Smale's homotopy mapping theorem and factors through a cellular Dowker lemma that expresses the main idea in more general form. To make the geometry more transparent, we work with a class of spaces called prod-complexes then transfer the results to simplicial complexes through a simplexification process. We conclude with a detailed study of ternary relations, identifying seven functorially defined homotopy types and twelve natural transformations between them.
We study the higher (or sequential) topological complexity TC_s of manifolds with abelian fundamental group. We give sufficient conditions for TC_s to be non-maximal in both the orientable and non-orientable cases. In combination with cohomological lower bounds, we also obtain some exact values for certain families of manifolds.
Since its introduction as a computable approximation of the Reeb graph, the Mapper graph has become one of the most popular tools from topological data analysis for performing data visualization and inference. However, finding an appropriate metric (that is, a tractable metric with theoretical guarantees) for comparing Reeb and Mapper graphs, in order to, e.g., quantify the rate of convergence of the Mapper graph to the Reeb graph, is a difficult problem. While several metrics have been proposed in the literature, none is able to incorporate measure information, when data points are sampled according to an underlying probability measure. The resulting Reeb and Mapper graphs are therefore purely deterministic and combinatorial, and substantial effort is thus required to ensure their statistical validity. In this article, we handle this issue by treating Reeb and Mapper graphs as metric measure spaces. This allows us to use Gromov-Wasserstein metrics to compare these graphs directly in order to better incorporate the probability measures that data points are sampled from. Then, we describe the geometry that arises from this perspective, and we derive rates of convergence of the Mapper graph to the Reeb graph in this context. Finally, we showcase the usefulness of such metrics for Reeb and Mapper graphs in a few numerical experiments.
The shadow of an abstract simplicial complex 𝒦 with vertices in ℝ^N is a subset of ℝ^N defined as the union of the convex hulls of simplices of 𝒦 . The Vietoris–Rips complex of a metric space (𝒮,d) at scale β is an abstract simplicial complex whose each k-simplex corresponds to (k+1) points of 𝒮 within diameter β . In case 𝒮⊂ℝ^2 and d(a,b)=‖ a-b‖ the standard Euclidean metric, the natural shadow projection of the Vietoris–Rips complex is already proved by Chambers et al. to induce isomorphisms on π _0 and π _1 . We extend the result beyond the standard Euclidean distance on 𝒮⊂ℝ^N to a family of path-based metrics, d^ε _𝒮 . From the pairwise Euclidean distances of points in 𝒮 , we introduce a family (parametrized by ε ) of path-based Vietoris–Rips complexes ℛ^ε _β (𝒮) for a scale β >0 . If 𝒮⊂ℝ^2 is Hausdorff-close to a planar Euclidean graph 𝒢 , we provide quantitative bounds on scales β ,ε for the shadow projection map of the Vietoris–Rips complex of (𝒮,d^ε _𝒮) at scale β to induce π _1 -isomorphism. This paper first studies the homotopy-type recovery of 𝒢⊂ℝ^N using the abstract Vietoris–Rips complex of a Hausdorff-close sample 𝒮 under the d^ε _𝒮 metric. Then, our result on the π _1 -isomorphism induced by the shadow projection lends itself to providing also a geometrically close embedding for the reconstruction. Based on the length of the shortest loop and large-scale distortion of the embedding of 𝒢 , we quantify the choice of a suitable sample density ε and a scale β at which the shadow of ℛ^ε _β (𝒮) is homotopy-equivalent and Hausdorff-close to 𝒢 .
We introduce the monoidal Rips filtration, a filtered simplicial set for weighted directed graphs and other lattice-valued networks. Our construction generalizes the Vietoris-Rips filtration for metric spaces by replacing the maximum operator, determining the filtration values, with a more general monoidal product. We establish interleaving guarantees for the monoidal Rips persistent homology, capturing existing stability results for real-valued networks. When the lattice is a product of totally ordered sets, we are in the setting of multiparameter persistence. Here, the interleaving distance is bounded in terms of a generalized network distance. We use this to prove a novel stability result for the sublevel Rips bifiltration. Our experimental results show that our method performs better than Flagser in a graph regression task, and that combining different monoidal products in point cloud classification can improve performance.
Quantum invariants in low dimensional topology offer a wide variety of valuable invariants of knots and 3-manifolds, presented by explicit formulas that are readily computable. Their computational complexity has been actively studied and is tightly connected to topological quantum computing. In this article, we prove that for any 3-manifold quantum invariant in the Reshetikhin-Turaev model, there is a deterministic polynomial time algorithm that, given as input an arbitrary closed 3-manifold M, outputs a closed 3-manifold M' with same quantum invariant, such that M' is hyperbolic, contains no low genus embedded incompressible surface, and is presented by a strongly irreducible Heegaard diagram. Our construction relies on properties of Heegaard splittings and the Hempel distance. At the level of computational complexity, this proves that the hardness of computing a given quantum invariant of 3-manifolds is preserved even when severely restricting the topology and the combinatorics of the input. This positively answers a question raised by Samperton.
One common function class in machine learning is the class of ReLU neural networks. ReLU neural networks induce a piecewise linear decomposition of their input space called the canonical polyhedral complex. It has previously been established that it is decidable whether a ReLU neural network is piecewise linear Morse. In order to expand computational tools for analyzing the topological properties of ReLU neural networks, and to harness the strengths of discrete Morse theory, we introduce a schematic for translating between a given piecewise linear Morse function (e.g. parameters of a ReLU neural network) on a canonical polyhedral complex and a compatible (“relatively perfect") discrete Morse function on the same complex. Our approach is constructive, producing an algorithm that can be used to determine if a given vertex in a canonical polyhedral complex corresponds to a piecewise linear Morse critical point. Furthermore we provide an algorithm for constructing a consistent discrete Morse pairing on cells in the canonical polyhedral complex which contain this vertex. We additionally provide some new realizability results with respect to sublevel set topology in the case of shallow ReLU neural networks.
We study the concepts of the ℓ _p -Vietoris-Rips simplicial set and the ℓ _p -Vietoris-Rips complex of a metric space, where 1≤ p ≤∞ . This theory unifies two established theories: for p=∞ , this is the classical theory of Vietoris-Rips complexes, and for p=1, this corresponds to the blurred magnitude homology theory. We prove several results that are known for the Vietoris-Rips complex in the general case: (1) we prove a stability theorem for the corresponding version of the persistent homology; (2) we show that, for a compact Riemannian manifold and a sufficiently small scale parameter, all the “ ℓ _p -Vietoris-Rips spaces” are homotopy equivalent to the manifold; (3) we demonstrate that the ℓ _p -Vietoris-Rips spaces are invariant (up to homotopy) under taking the metric completion. Additionally, we show that the limit of the homology groups of the ℓ _p -Vietoris-Rips spaces, as the scale parameter tends to zero, does not depend on p; and that the homology groups of the ℓ _p -Vietoris-Rips spaces commute with filtered colimits of metric spaces.
Hypergraphs have seen widespread application in network and data science communities in recent years. We present a survey of recent work to construct auxiliary structures from hypergraphs—specifically simplicial, relative, and chain complexes—that can be used to build homology theories for hypergraphs. We define and describe nine different constructions and their associated homology theories. We discuss some interesting properties of each homology theory to show how various hypergraph properties imply properties of the homology groups. We also include discussion of functoriality for several of the homology theories. Finally, we provide a series of illustrative examples by computing many of these homology theories for small hypergraphs to show the variability of the methods and build intuition.
This note proves that only a linear number of holes in a Čech complex of n points in ℝ^d can persist over an interval of constant length. Specifically, for any fixed dimension p0 , the number of p-dimensional holes in the Čech complex at radius 1 that persist to radius 1+ε is bounded above by a constant times n, where n is the number of points. The proof uses a packing argument supported by relating the Čech complexes with corresponding snap complexes over the cells in a partition of space. The argument is self-contained and elementary, relying on geometric and combinatorial constructions rather than on the existing theory of sparse approximations or interleavings. The bound also applies to Alpha complexes and Vietoris–Rips complexes. While our result can be inferred from prior work on sparse filtrations, to our knowledge, no explicit statement or direct proof of this bound appears in the literature.
The matching complex (G) of a graph G is a simplicial complex whose simplices are matchings in G. These complexes appear in various places and found applications in many areas of mathematics including computational geometry, representation theory, combinatorics, etc. In this article, we consider the matching complexes of the categorical product P_n × P_m of path graphs P_n and P_m . For m = 1 , P_n × P_m is a discrete graph and therefore its matching complex is the void complex. For m = 2 , (P_n × P_m) has been proved to be homotopy equivalent to a wedge of spheres by Kozlov. We show that for n ≥ 2 and 3 ≤ m ≤ 5 , the matching complex of P_n × P_m is homotopy equivalent to a wedge of spheres. For m = 3 , we explicitly compute the number and dimension of spheres appearing in the wedge. Furthermore, for m ∈{4, 5} , we provide the minimum and maximum dimensions of spheres appearing in the wedge in the homotopy type of (P_n × P_m) .
We establish the first nontrivial lower bound on the (higher) topological complexity of the unordered configuration spaces of a general graph. As an application, we show that, for most graphs, the topological complexity eventually stabilizes at its maximal possible value, a direct analogue of a stability phenomenon in the ordered setting first conjectured by Farber. We estimate the stable range in terms of the number of trivalent vertices.
