arXiv subjects

Dejan Govc

Publications and source records attributed to Dejan Govc.

14 recordsLinked to original sources

arXiv2025

Fundamental groups of small simplicial complexes

The number of nonisomorphic simplicial complexes with up to $n$ vertices increases super-exponentially with $n$, which makes exhaustive computation of invariants associated with such complexes a daunting task. In this paper we provide a complete list of groups that arise as fundamental groups of simplicial complexes with at most $8$ vertices. In addition we give many examples of fundamental groups of complexes with $9$ vertices although the complete classification seems to be beyond reach at the moment. Our results lead to many applications, including progress on the Bj\"orner-Lutz conjecture regarding vertex-minimal triangulations of the Poincar\'e homology sphere, improved recognition criteria for PL triangulations of manifolds and computation of the Karoubi-Weibel invariant for many groups.

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arXiv2023

Equivariant covering type and the number of vertices in equivariant triangulations

We introduce the notion of the \emph{equivariant covering type} of a space $X$ on which a finite group $G$ acts, and study its properties. The equivariant covering type measures the size of $G$-equivariant good covers of $X$ and is thus an extension of the \emph{covering type} of a space, introduced by Karoubi and Weibel. We show that the equivariant covering type is a $G$-homotopy invariant and describe its relation with other $G$-invariants, like the equivariant LS-category, $G$-genus and the multiplicative structures of equivariant cohomology theories. We also compute the $G$-covering type of regular $G$-graphs, give estimates for orientation-preserving actions on surfaces and for the projectivizations of complex representations of $G$ and cohomology spheres. As an application, we derive estimates of sizes of minimal $G$-triangulations for various $G$-spaces.

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arXiv2021

An application of neighbourhoods in digraphs to the classification of binary dynamics

A binary state on a graph means an assignment of binary values to its vertices. For example, if one encodes a network of spiking neurons as a directed graph, then the spikes produced by the neurons at an instant of time is a binary state on the encoding graph. Allowing time to vary and recording the spiking patterns of the neurons in the network produces an example of binary dynamics on the encoding graph, namely a one-parameter family of binary states on it. The central object of study in this article is the closed neighbourhood of a vertex $v$ in a graph $\mathcal{G}$, namely the subgraph of $\mathcal{G}$ that is induced by $v$ and all its neighbours in $\mathcal{G}$. We present a topological/graph theoretic method for extracting information out of binary dynamics on a graph, based on a selection of a relatively small number of vertices and their neighbourhoods. As a test case we demonstrate an application of the method to binary dynamics that arises from sample activity on the Blue Brain Project reconstruction of cortical tissue of a rat.

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arXiv2021

Estimates of covering type and minimal triangulations based on category weight

In a recent publication (D. Govc, W. Marzantowicz, P. Pavesic, Estimates of covering type and the number of vertices of minimal triangulations, Discr. Comp. Geom. 63 (2019), 31-48) we have introduced a new method, based on the Lusternik-Schnirelmann category and the cohomology ring of a space X, that yields lower bounds for the size of a triangulation of X. In this paper we present an important extension that takes into account the fundamental group of X. In fact, if it contains elements of finite order, then one can often find cohomology classes of high 'category weight', which in turn allow for much stronger estimates of the size of triangulations of X. We develop several weighted estimates and then apply our method to compute explicit lower bounds for the size of triangulations of orbit spaces of cyclic group actions on a variety of spaces including products of spheres, Stiefel manifolds, Lie groups and highly-connected manifolds.

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arXiv2021

Complexes of Tournaments, Directionality Filtrations and Persistent Homology

Complete digraphs are referred to in the combinatorics literature as tournaments. We consider a family of semi-simplicial complexes, that we refer to as "tournaplexes", whose simplices are tournaments. In particular, given a digraph $\mathcal{G}$, we associate with it a "flag tournaplex" which is a tournaplex containing the directed flag complex of $\mathcal{G}$, but also the geometric realisation of cliques that are not directed. We define several types of filtrations on tournaplexes, and exploiting persistent homology, we observe that flag tournaplexes provide finer means of distinguishing graph dynamics than the directed flag complex. We then demonstrate the power of these ideas by applying them to graph data arising from the Blue Brain Project's digital reconstruction of a rat's neocortex.

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arXiv2020

Asymptotic Behaviour of the Containment of Certain Mesh Patterns

We present some results on the proportion of permutations of length $n$ containing certain mesh patterns as $n$ grows large, and give exact enumeration results in some cases. In particular, we focus on mesh patterns where entire rows and columns are shaded. We prove some general results which apply to mesh patterns of any length, and then consider mesh patterns of length four. An important consequence of these results is to show that the proportion of permutations containing a mesh pattern can take a wide range of values between $0$ and $1$.

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arXiv2020

Computing Homotopy Types of Directed Flag Complexes

Combinatorially and stochastically defined simplicial complexes often have the homotopy type of a wedge of spheres. A prominent conjecture of Kahle quantifies this precisely for the case of random flag complexes. We explore whether such properties might extend to graphs arising from nature. We consider the brain network (as reconstructed by Varshney & al.) of the Caenorhabditis elegans nematode, an important model organism in biology. Using an iterative computational procedure based on elementary methods of algebraic topology, namely homology, simplicial collapses and coning operations, we show that its directed flag complex is homotopy equivalent to a wedge of spheres, completely determining, for the first time, the homotopy type of a flag complex corresponding to a brain network. We also consider the corresponding flag tournaplex and show that torsion can be found in the homology of its local directionality filtration. As a toy example, directed flag complexes of tournaments from McKay's collection are classified up to homotopy. Moore spaces other than spheres occur in this classification. As a tool, we prove that the fundamental group of the directed flag complex of any tournament is free by considering its cell structure.

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arXiv2020

How many simplices are needed to triangulate a Grassmannian?

We compute a lower bound for the number of simplices that are needed to triangulate the Grassmann manifold $G_k(\mathbb{R}^n)$. In particular, we show that the number of top-dimensional simplices grows exponentially with $n$. More precise estimates are given for $k=2,3,4$. Our method can be used to estimate the minimal size of triangulations for other spaces, like Lie groups, flag manifolds, Stiefel manifolds etc.

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arXiv2019

Persistent Magnitude

In this paper we introduce the persistent magnitude, a new numerical invariant of (sufficiently nice) graded persistence modules. It is a weighted and signed count of the bars of the persistence module, in which a bar of the form $[a,b)$ in degree $d$ is counted with weight $(e^{-a}-e^{-b})$ and sign $(-1)^d$. Persistent magnitude has good formal properties, such as additivity with respect to exact sequences and compatibility with tensor products, and has interpretations in terms of both the associated graded functor, and the Laplace transform. Our definition is inspired by Otter's notion of blurred magnitude homology: we show that the magnitude of a finite metric space is precisely the persistent magnitude of its blurred magnitude homology. Turning this result on its head, we obtain a strategy for turning existing persistent homology theories into new numerical invariants by applying the persistent magnitude. We explore this strategy in detail in the case of persistent homology of Morse functions, and in the case of Rips homology.

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arXiv2019

Estimates of covering type and the number of vertices of minimal triangulations

The covering type of a space $X$ is defined as the minimal cardinality of a good cover of a space that is homotopy equivalent to $X$. We derive estimates for the covering type of $X$ in terms of other invariants of $X$, namely the ranks of the homology groups, the multiplicative structure of the cohomology ring and the Lusternik-Schnirelmann category of $X$. By relating the covering type to the number of vertices of minimal triangulations of complexes and combinatorial manifolds, we obtain, within a unified framework, several estimates which are either new or extensions of results that have been previously obtained by ad hoc combinatorial arguments. Moreover, our methods give results that are valid for entire homotopy classes of spaces.

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arXiv2019

Computing persistent homology of directed flag complexes

We present a new computing package Flagser, designed to construct the directed flag complex of a finite directed graph, and compute persistent homology for flexibly defined filtrations on the graph and the resulting complex. The persistent homology computation part of Flagser is based on the program Ripser [Bau18a], but is optimized specifically for large computations. The construction of the directed flag complex is done in a way that allows easy parallelization by arbitrarily many cores. Flagser also has the option of working with undirected graphs. For homology computations Flagser has an Approximate option, which shortens compute time with remarkable accuracy. We demonstrate the power of Flagser by applying it to the construction of the directed flag complex of digital reconstructions of brain microcircuitry by the Blue Brain Project and several other examples. In some instances we perform computation of homology. For a more complete performance analysis, we also apply Flagser to some other data collections. In all cases the hardware used in the computation, the use of memory and the compute time are recorded.

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arXiv2017

Unimodal Category and the Monotonicity Conjecture

We completely characterize the unimodal category for functions $f:\mathbb R\to[0,\infty)$ using a decomposition theorem obtained by generalizing the sweeping algorithm of Baryshnikov and Ghrist. We also give a characterization of the unimodal category for functions $f:S^1\to[0,\infty)$ and provide an algorithm to compute the unimodal category of such a function in the case of finitely many critical points. We then turn to the monotonicity conjecture of Baryshnikov and Ghrist. We show that this conjecture is true for functions on $\mathbb R$ and $S^1$ using the above characterizations and that it is false on certain graphs and on the Euclidean plane by providing explicit counterexamples. We also show that it holds for functions on the Euclidean plane whose Morse-Smale graph is a tree using a result of Hickok, Villatoro and Wang.

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arXiv2017

An Approximate Nerve Theorem

The Nerve Theorem relates the topological type of a suitably nice space with the nerve of a good cover of that space. It has many variants, such as to consider acyclic covers and numerous applications in topology including applied and computational topology. The goal of this paper is to relax the notion of a good cover to an approximately good cover, or more precisely, we introduce the notion of an $\varepsilon$-acyclic cover. We use persistent homology to make this rigorous and prove tight bounds between the persistent homology of a space endowed with a function and the persistent homology of the nerve of an $\varepsilon$-acyclic cover of the space. Using the Mayer-Vietoris spectral sequence, we upper bound how local non-acyclicity can affect the global homology. To prove the best possible bound we must introduce special cases of interleavings between persistence modules called left and right interleavings. Finally, we provide examples which achieve the bound proving the lower bound and tightness of the result.

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arXiv2013

On the definition of homological critical value

We point out that there is a problem with the definition of homological critical value (as defined in the widely cited paper \cite{stability} by Cohen-Steiner, Edelsbrunner and Harer). Under that definition, the critical value lemma of \cite{stability} in fact fails. We provide several counterexamples and a definition (due to Bubenik and Scott \cite{categorification}) we feel should be preferred and under which the critical value lemma does indeed hold. One of the counterexamples we have found is a height function on a compact smooth manifold. In the end we prove that, despite all this, a modified version of the critical value lemma remains valid under the original definition.

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