Combinatorial Topology Research Articles

On the number of simplices required to triangulate a Lie group

We estimate the number of simplices required for triangulations of compact Lie groups. As in the previous work [11], our approach combines the estimation of the number of vertices by means of the covering type via a cohomological argument from [10], and application of the recent versions of the Lower Bound Theorem of combinatorial topology. For the exceptional Lie groups, we present a complete calculation using the description of their cohomology rings given by the first and third authors. For the infinite series of classical Lie groups of growing dimension d, we estimate the growth rate of number of simplices of the highest dimension, which extends to the case of simplices of (fixed) codimension d−i.

A topological perspective on distributed network algorithms

More than two decades ago, combinatorial topology was shown to be useful for analyzing distributed fault-tolerant algorithms in shared memory systems and in message passing systems. In this work, we show that combinatorial topology can also be useful for analyzing distributed algorithms in failure-free networks of arbitrary structure. To illustrate this, we analyze consensus, set-agreement, and approximate agreement in networks, and derive lower bounds for these problems under classical computational settings, such as the local model and dynamic networks.

Geometric, Algebraic, and Topological Combinatorics

The 2019 Oberwolfach meeting “Geometric, Algebraic and Topological Combinatorics” was organized by Gil Kalai (Jerusalem), Isabella Novik (Seattle), Francisco Santos (Santander), and Volkmar Welker (Marburg). It covered a wide variety of aspects of Discrete Geometry, Algebraic Combinatorics with geometric flavor, and Topological Combinatorics. Some of the highlights of the conference included (1) Karim Adiprasito presented his very recent proof of the g-conjecture for spheres (as a talk and as a “Q&A” evening session) (2) Federico Ardila gave an overview on “The geometry of matroids”, including his recent extension with Denham and Huh of previous work of Adiprasito, Huh and Katz.

Asynchronous Computability Theorem in Arbitrary Solo Models

In this paper, we establish the asynchronous computability theorem in d-solo system by borrowing concepts from combinatorial topology, in which we state a necessary and sufficient conditions for a task to be wait-free computable in that system. Intuitively, a d-solo system allows as many d processes to access it as if each were running solo, namely, without detecting communication from any peer. As an application, we completely characterize the solvability of the input-less tasks in such systems. This characterization also leads to a hardness classification of these tasks according to whether their output complexes hold a d-nest structure. As a byproduct, we find an alternative way to distinguish the computational power of d-solo objects for different d.

Chordality, d-collapsibility, and componentwise linear ideals

Using the concept of d-collapsibility from combinatorial topology, we define chordal simplicial complexes and show that their Stanley-Reisner ideals are componentwise linear. Our construction is inspired by and an extension of “chordal clutters” which was defined by Bigdeli, Yazdan Pour and Zaare-Nahandi in 2017, and characterizes Betti tables of all ideals with a linear resolution in a polynomial ring.We show d-collapsible and d-representable complexes produce componentwise linear ideals for appropriate d. Along the way, we prove that there are generators that when added to the ideal, do not change Betti numbers in certain degrees.We then show that large classes of componentwise linear ideals, such as Gotzmann ideals and square-free stable ideals have chordal Stanley-Reisner complexes, that Alexander duals of vertex decomposable complexes are chordal, and conclude that the Betti table of every componentwise linear ideal is identical to that of the Stanley-Reisner ideal of a chordal complex.

Jordan surface theorem for simple closed SST-surfaces

The present paper proposes a new type of Jordan surface theorem for simple closed SST-surfaces. To do this work, we use space set topological (referred to as SST-, for short) structures on compact or non-compact surfaces in Rn. Indeed, the SST-structure is a special kind of Alexandroff topological structure and further, it is indeed a locally finite (LF-, for brevity) topological one. Owing to the LF-topological structure, it can efficiently be used in studying objects from the viewpoints of pure and applied topology such as discrete geometry, combinatorial topology, computational topology, digital topology, digital geometry and so on. The present paper first establishes an SST-structure of a subdivided abstract cell (SAC-, for brevity) complex on X(⊂Rn) using a generalized face to face tessellation of X⊂Rn. Second, the paper introduces a strong SST-homeomorphism to classify SST-surfaces. Finally, unlike the typical Jordan surface theorem under Hausdorff topology or polyhedral geometry, the current theorem is formulated by using Alexandroff topological or SST-structure. Thus this theorem has strong advantages being compared with those associated with Euclidean topology, digital topology and polygonal geometry. Besides, based on the SST-structure on an SAC-complex, the paper suggests some remarks on digital manifolds, Khalimsky manifolds, (n−1)-dimensional polyhedral pseudomanifolds and so on.

The CAS‐extended model

SummaryWait‐freedom guarantees that all processes complete their operations in a finite number of steps regardless of the delay of any process. Combinatorial topology has been proposed in the literature as a formal verification technique to prove the wait‐free computability of decision tasks. Wait‐freedom is proved through the properties of a static topological structure that expresses all possible combinations of execution paths of the protocol solving the decision task. The practical application of combinatorial topology as a formal verification technique is limited because the existing theory only considers protocols in which the manner of communication between processes is through read‐write memory. This research proposes an extension to the existing theory, called the CAS‐extended model. The extended theory includes Compare‐And‐Swap (CAS) and Load‐Link/Store‐Conditional (LL/SC), which are atomic primitives used to achieve wait‐freedom in state‐of‐the‐art protocols. The CAS‐extended model theory can be used to formally verify wait‐free algorithms used in practice, such as concurrent data structures. We present new definitions detailing the construction of a protocol complex in the CAS‐extended model. As a proof‐of‐concept, we formally verify a wait‐free queue with three processes using the CAS‐extended combinatorial topology.

Development of ordinary harness for aircraft onboard cable networks

The article is devoted to one of the most important components of the onboard complex of aircraft equipment - the onboard cable network. The contents of the design documentation for the aircraft onboard cable network are disclosed. The statement of the problem of designing harnesses is defined in general terms. The main stages of designing aircraft onboard cable networks are described on the verbal level, as well as in the form of logical algorithms and graph-algorithms. Some theoretical aspects of designing aircraft onboard cable networks are presented. The concepts of topological space, topological structure, and continuous mapping of the harness structure into the aircraft structure are introduced. Geometric research of an ordinary cable harness of the onboard cable network led to the need to consider the harnesses as a geometric complex in the framework of combinatorial topology. An example of compiling a table of connections of ordinary harnesses for the aircraft onboard system of ultra-short wave communication is given. The rules and requirements for the information content of the table of connections of an ordinary harness to the aircraft on-board system are emphasized. Mention is made of the need to integrate ordinary harnesses into a complex one consisting of tens or even hundreds of ordinary harnesses to simplify the process of installation of the onboard cable network in the aircraft.

A combinatorial method to compute explicit homology cycles using Discrete Morse Theory

In this paper we shall describe a combinatorial method related to Discrete Morse Theory, which allows us to calculate explicit homology cycles in polyhedral complexes. These cycles form a basis, in the case when the critical cells are in an isolated dimension. We illustrate the use of this technique by several examples from combinatorial topology, including the complexes of multihomomorphisms between complete graphs. Our method is optimal from the computational complexity point of view, requiring execution time which is linear in the number of d-cells.

The Splendors and Miseries of Rounds

Take a stroll with me through the distributed computing literature: we could scour the proceedings of PODC, DISC, SIROCCO, OPODIS, and more; we could explore the issues of Distributed Com- puting; or we could skim through the multitude of preprints on arXiv. Yet, I can already predict that with good probability, the paper we choose to read will contain the word \round" or łayer".1 Is it a lower-bound paper? Then round complexity is probably one measure of performance considered. Is it a combinatorial topology paper? Almost all techniques developed from application of combinatorial topology assume layers of communication, such that an execution can be reduced as successive complexes; here again rounds are essential. Is it a paper on fault-tolerance? Then some broadcasting strategy to ensure decent replication probably uses rounds.

Algorithmic homeomorphism of 3-manifolds as a corollary of geometrization

In this paper we prove two results, one semi-historical and the other new. The semi-historical result, which goes back to Thurston and Riley, is that the geometrization theorem implies that there is an algorithm for the homeomorphism problem for closed, oriented, triangulated 3-manifolds. We give a self-contained proof, with several variations at each stage, that uses only the statement of the geometrization theorem, basic hyperbolic geometry, and old results from combinatorial topology and computer science. For this result, we do not rely on normal surface theory, methods from geometric group theory, nor methods used to prove geometrization. The new result is that the homeomorphism problem is elementary recursive, i.e., that the computational complexity is bounded by a bounded tower of exponentials. This result relies on normal surface theory, Mostow rigidity, and bounds on the computational complexity of solving algebraic equations.

Different versions of the nerve theorem and colourful simplices

Given a simplicial complex and a collection of subcomplexes covering it, the nerve theorem, a fundamental tool in topological combinatorics, guarantees a certain connectivity of the simplicial complex when connectivity conditions on the intersection of the subcomplexes are satisfied.We show that it is possible to extend this theorem by replacing some of these connectivity conditions on the intersection of the subcomplexes by connectivity conditions on their union. While this is interesting for its own sake, we use this extension to generalize in various ways the Meshulam lemma, a powerful homological version of the Sperner lemma. We also prove a generalization of the Meshulam lemma that is somehow reminiscent of the polytopal generalization of the Sperner lemma by De Loera, Peterson, and Su. For this latter result, we use a different approach and we do not know whether there is a way to get it via a nerve theorem of some kind.

On the Relationship Between Ehrhart Unimodality and Ehrhart Positivity

For a given lattice polytope, two fundamental problems within the field of Ehrhart theory are (1) to determine if its (Ehrhart) $$h^*$$ -polynomial is unimodal and (2) to determine if its Ehrhart polynomial has only positive coefficients. The former property of a lattice polytope is known as Ehrhart unimodality and the latter property is known as Ehrhart positivity. These two properties are often simultaneously conjectured to hold for interesting families of lattice polytopes, yet they are typically studied in parallel. As to answer a question posed at the 2017 Introductory Workshop to the MSRI Semester on Geometric and Topological Combinatorics, the purpose of this note is to show that there is no general implication between these two properties in any dimension greater than two. To do so, we investigate these two properties for families of well-studied lattice polytopes, assessing one property where previously only the other had been considered. Consequently, new examples of each phenomena are developed, some of which provide an answer to an open problem in the literature. The well-studied families of lattice polytopes considered include zonotopes, matroid polytopes, simplices of weighted projective spaces, empty lattice simplices, smooth polytopes, and $${\varvec{s}}$$ -lecture hall simplices.

The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg

We discuss five discrete results: the lemmas of Sperner and Tucker from combinatorial topology and the theorems of Carath\'eodory, Helly, and Tverberg from combinatorial geometry. We explore their connections and emphasize their broad impact in application areas such as game theory, graph theory, mathematical optimization, computational geometry, etc.

On $f$- and $h$-vectors of relative simplicial complexes

A relative simplicial complex is a collection of sets of the form $\Delta \setminus \Gamma$, where $\Gamma \subset \Delta$ are simplicial complexes. Relative complexes played key roles in recent advances in algebraic, geometric, and topological combinatorics but, in contrast to simplicial complexes, little is known about their general combinatorial structure. In this paper, we address a basic question in this direction and give a characterization of $f$-vectors of relative (multi)complexes on a ground set of fixed size. On the algebraic side, this yields a characterization of Hilbert functions of quotients of homogeneous ideals over polynomial rings with a fixed number of indeterminates. Moreover, we characterize $h$-vectors of fully Cohen--Macaulay relative complexes as well as $h$-vectors of Cohen--Macaulay relative complexes with minimal faces of given dimensions. The latter resolves a question of Bj\"orner.

The topology of look-compute-move robot wait-free algorithms with hard termination

Look-Compute-Move models for a set of autonomous robots have been thoroughly studied for over two decades. We consider the standard Asynchronous Luminous Robots (ALR) model, where robots are located in a graph G. Each robot, repeatedly Looks at its surroundings and obtains a snapshot containing the vertices of G, where all robots are located; based on this snapshot, each robot Computes a vertex (adjacent to its current position), and then Moves to it. Robots have visible lights, allowing them to communicate more information than only its actual position, and they move asynchronously, meaning that each one runs at its own arbitrary speed. We are also interested in a case which has been barely explored: the robots need not all be present initially, they might appear asynchronously. We call this the Extended Asynchronous Appearing Luminous Robots (EALR) model. A central problem in the mobile robots area is bringing the robots to the same vertex. We study several versions of this problem, where the robots move towards the same (or close to each other) vertices. And we concentrate on the requirement that each robot executes a finite number of Look-Compute-Move cycles, independently of the interleaving of other robot’s cycles, and then stops. Our main result is direct connections between the (ALR and) EALR model and the asynchronous wait-free multiprocess read/write shared memory (WFSM) model. General robot tasks in a graph are also provided, which include several version of gathering. Finally, using the connection between the EALR model and the WFSM model, a combinatorial topology characterization for the solvable robot tasks is presented.

Configuration Spaces of Graphs with Certain Permitted Collisions

If G is a graph with vertex set V, let $${{\mathrm{Conf}}}_n^{{{\mathrm{sink}}}}(G,V)$$ be the space of n-tuples of points on G, which are only allowed to overlap on elements of V. We think of $${{\mathrm{Conf}}}_n^{{{\mathrm{sink}}}}(G,V)$$ as a configuration space of points on G, where points are allowed to collide on vertices. In this paper, we attempt to understand these spaces from two separate, but closely related, perspectives. Using techniques of combinatorial topology we compute the fundamental groups and homology groups of $${{\mathrm{Conf}}}_n^{{{\mathrm{sink}}}}(G,V)$$ in the case where G is a tree. Next, we use techniques of asymptotic algebra to prove statements about $${{\mathrm{Conf}}}_n^{{{\mathrm{sink}}}}(G,V)$$ , for general graphs G, whenever n is sufficiently large. It is proven that, for general graphs, the homology groups exhibit generalized representation stability in the sense of Ramos ( arXiv:1606.02673 , 2016).

On tangent cones at infinity of algebraic varieties

In this paper, we define the geometric and algebraic tangent cones at infinity of algebraic varieties and establish the following version at infinity of Whitney’s theorem [Local properties of analytic varieties, in Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse) (Princeton University Press, Princeton, N. J., 1965), pp. 205–244; Tangents to an analytic variety, Ann. of Math. 81 (1965) 496–549]: The geometric and algebraic tangent cones at infinity of complex algebraic varieties coincide. The proof of this fact is based on a geometric characterization of the geometric tangent cone at infinity using the global Łojasiewicz inequality with explicit exponents for complex algebraic varieties. Moreover, we show that the tangent cone at infinity of a complex algebraic variety is actually the part at infinity of this variety [G.-M. Greuel and G. Pfister, A Singular Introduction to Commutative Algebra, 2nd extended edn. (Springer, Berlin, 2008)]. We also show that the tangent cone at infinity of a complex algebraic variety can be computed using Gröbner bases.

Combinatorial topology and constructive mathematics

Combinatorial topology and constructive mathematics

As-exact-as-possible repair of unprintable STL files

PurposeThe class of models that can be represented by STL files is larger than the class of models that can be printed using additive manufacturing technologies. Stated differently, there exist well-formed STL files that cannot be printed. This paper aims to formalize such a gap and describe a fully automatic procedure to turn any such file into a printable model.Design/methodology/approachBased on well-established concepts from combinatorial topology, this paper provide an unambiguous description of all the mathematical entities involved in the modeling-printing pipeline. Specifically, this paper formally defines the conditions that an STL file must satisfy to be printable, and, based on these, an as-exact-as-possible repairing algorithm is designed.FindingsIt has been found that, to cope with all the possible triangle configurations, the algorithm must distinguish between triangles that bind solid parts and triangles that constitute zero-thickness sheets. Only the former set can be fixed without distortion.Research limitations/implicationsOwing to the specific approach used that tracks the so-called “outer hull,” models with inner cavities cannot be treated.Practical implicationsThanks to this new method, the shift from a 3D model to a printed prototype is faster, easier and more reliable.Social implicationsThe availability of this easily accessible model preparation tool has the potential to foster a wider diffusion of home-made 3D printing in non-professional communities.Originality/valuePrevious methods that are guaranteed to fix all the possible configurations provide only approximate solutions with an unnecessary distortion. Conversely, this procedure is as exact as possible, meaning that no visible distortion is introduced unless it is strictly imposed by limitations of the printing device. Thanks to such unprecedented flexibility and accuracy, this algorithm is expected to significantly simplify the modeling-printing process, in particular within the continuously emerging non-professional “maker” communities.