Combinatorial Manifolds Research Articles

Geometrical Theory on Combinatorial Manifolds

For an integer m ≥ 1, a combinatorial manifold fM is defined to be a geometrical object fM such that for (...), there is a local chart (see paper) where Bnij is an nij -ball for integers 1 ≤ j ≤ s(p) ≤ m. Topological and differential structures such as those of d-pathwise connected, homotopy classes, fundamental d-groups in topology and tangent vector fields, tensor fields, connections, Minkowski norms in differential geometry on these finitely combinatorial manifolds are introduced. Some classical results are generalized to finitely combinatorial manifolds. Euler-Poincare characteristic is discussed and geometrical inclusions in Smarandache geometries for various geometries are also presented by the geometrical theory on finitely combinatorial manifolds in this paper.

Positively Curved Combinatorial 3-Manifolds

We present two theorems in the "discrete differential geometry" of positively curved spaces. The first is a combinatorial analog of the Bonnet-Myers theorem: $\bullet$ A combinatorial 3-manifold whose edges have degree at most five has edge-diameter at most five. When all edges have unit length, this degree bound is equivalent to an angle-deficit along each edge. It is for this reason we call such spaces positively curved. Our second main result is analogous to the sphere theorems of Toponogov and Cheng: $\bullet$ A positively curved 3-manifold, as above, in which vertices $v$ and $w$ have edge-distance five is a sphere whose triangulation is completely determined by the structure of $Lk(v)$ or $Lk(w)$. In fact, we provide a procedure for constructing a maximum diameter sphere from a suitable $Lk(v)$ or $Lk(w)$. The compactness of these spaces (without an explicit diameter bound) was first proved via analytic arguments in a 1973 paper by David Stone. Our proof is completely combinatorial, provides sharp bounds, and follows closely the proof strategy for the classical results.

A property that characterizes Euler characteristic among invariants of combinatorial manifolds

If a real valued invariant of compact combinatorial manifolds (with or without boundary) depends only on the number of simplices in each dimension in the manifold, then the invariant is completely determined by the Euler characteristic of the manifold and its boundary. So essentially, the Euler characteristic is the unique invariant of this type.

Deforming Stanley–Reisner schemes

We study the deformation theory of projective Stanley–Reisner schemes associated to combinatorial manifolds. We achieve detailed descriptions of first order deformations and obstruction spaces. Versal base spaces are given for certain Stanley–Reisner surfaces.

On converting sets of tetrahedra to combinatorial and PL manifolds

We investigate the problem of removing singularities from a non-manifold tetrahedral mesh so as to convert it to a more exploitable manifold representation. Given the twofold combinatorial and geometrical nature of a 3D simplicial complex, we propose two conversion algorithms that, depending on the targeted application, modify either its connectivity only or both its connectivity and its geometry. In the first case, the tetrahedral mesh is converted to a combinatorial 3-manifold, whereas in the second case it becomes a piecewise linear (PL) 3-manifold. For both the approaches, the conversion takes place while using only local modifications around the singularities. We outline sufficient conditions on the mesh to guarantee the feasibility of the approaches and we show how singularities can be both identified and removed according to the configuration of their neighborhoods. Furthermore, besides adapting and extending surface-based approaches to a specific class of full-dimensional simplicial complexes in 3D, we show that our algorithms can be implemented using a flexible data structure for manifold tetrahedral meshes which is suitable for general applications. In order to exclude pathological configurations while providing sound guarantees, the input mesh is required to be a sub-complex of a combinatorial ball; this makes it possible to assume that all the singularities are part of the mesh boundary.

The construction of combinatorial manifolds with prescribed sets of links of vertices

To every oriented closed combinatorial manifold we assign the set (with repetitions) of isomorphism classes of links of its vertices. The resulting transformation is the main object of study in this paper. We pose an inversion problem for and show that this problem is closely related to Steenrod's problem on the realization of cycles and to the Rokhlin-Schwartz-Thom construction of combinatorial Pontryagin classes. We obtain a necessary condition for a set of isomorphism classes of combinatorial spheres to belong to the image of . (Sets satisfying this condition are said to be balanced.) We give an explicit construction showing that every balanced set of isomorphism classes of combinatorial spheres falls into the image of after passing to a multiple set and adding several pairs of the form , where is the sphere with the orientation reversed. Given any singular simplicial cycle of a space , this construction enables us to find explicitly a combinatorial manifold and a map such that for some positive integer . The construction is based on resolving singularities of . We give applications of the main construction to cobordisms of manifolds with singularities and cobordisms of simple cells. In particular, we prove that every rational additive invariant of cobordisms of manifolds with singularities admits a local formula. Another application is the construction of explicit (though inefficient) local combinatorial formulae for polynomials in the rational Pontryagin classes of combinatorial manifolds.

Topological finiteness for edge-vertex enumeration

The number of PL-homeomorphism types of combinatorial manifolds in a fixed dimension with an upper bound on g 2 is finite.

Three-Dimensional Pseudomanifolds on Eight Vertices

A normal pseudomanifold is a pseudomanifold in which the links of simplices are also pseudomanifolds. So, a normal 2-pseudomanifold triangulates a connected closed 2-manifold. But, normald-pseudomanifolds form a broader class than triangulations of connected closedd-manifolds ford≥3. Here, we classify all the 8-vertex neighbourly normal 3-pseudomanifolds. This gives a classification of all the 8-vertex normal 3-pseudomanifolds. There are 74 such 3-pseudomanifolds, 39 of which triangulate the 3-sphere and other 35 are not combinatorial 3-manifolds. These 35 triangulate six distinct topological spaces. As a preliminary result, we show that any 8-vertex 3-pseudomanifold is equivalent by proper bistellar moves to an 8-vertex neighbourly 3-pseudomanifold. This result is the best possible since there exists a 9-vertex nonneighbourly 3-pseudomanifold which does not allow any proper bistellar moves.

Combinatorial fiber bundles and fragmentation of a fiberwise PL homeomorphism

With a compact PL manifold X we associate a category $$\mathfrak{T}(X)$$ . The objects of $$\mathfrak{T}(X)$$ are all combinatorial manifolds of type X, and morphisms are combinatorial assemblies. We prove that the homotopy equivalence $$B\mathfrak{T}(X) \approx BPL(X)$$ holds, where PL(X) is the simplicial group of PL homeomorphisms. Thus the space $$B\mathfrak{T}(X)$$ is a canonical countable (as a CW-complex) model of BPL (X). As a result, we obtain functorial pure combinatorial models for PL fiber bundles with fiber X and a PL polyhedron B as the base. Such a model looks like a $$\mathfrak{T}(X)$$ -coloring of some triangulation K of B. The vertices of K are colored by objects of $$\mathfrak{T}(X)$$ , and the arcs are colored by morphisms in such a way that the diagram arising from the 2-skeleton of K is commutative. Comparing with the classical results of geometric topology, we obtain combinatorial models of the real Grassmannian in small dimensions: $$B\mathfrak{T}(S^{n - 1} ) \approx BO(n)$$ for n = 1, 2, 3, 4. The result is proved in a sequence of results on similar models of BPL (X). Special attention is paid to the main noncompact case X = ℝn and to the tangent bundle and Gauss functor of a combinatorial manifold. The trick that makes the proof possible is a collection of lemmas on “fragmentation of a fiberwise homeomorphism,” a generalization of the folklore lemma on fragmentation of an isotopy. Bibliography: 34 titles.

Uniqueness of Walkup's 9-vertex 3-dimensional Klein bottle

Via a computer search, Altshuler and Steinberg found that there are 1296 + 1 combinatorial 3-manifolds on nine vertices, of which only one is non-sphere. This exceptional 3-manifold K 9 3 triangulates the twisted S 2 -bundle over S 1 . It was first constructed by Walkup. In this paper, we present a computer-free proof of the uniqueness of this non-sphere combinatorial 3-manifold. As opposed to the computer-generated proof, ours does not require wading through all the 9-vertex 3-spheres. As a preliminary result, we also show that any 9-vertex combinatorial 3-manifold is equivalent by proper bistellar moves to a 9-vertex neighbourly 3-manifold.

Higher-dimensional Bruckner–Garg type theorem

In this paper, we investigate several properties of maps from a compactum X to an n-dimensional (combinatorial) manifold M n . We introduce the notions of stable point and locally extreme point of map, and we prove a higher-dimensional Bruckner–Garg type theorem for the fiber structure of a generic map in the space C ( X , M n ) of maps from a compactum X with dim X ⩾ n to an n-dimensional manifold M n ( n ⩾ 1 ). As applications, we also study the spaces of Bing maps, Lelek maps, k-dimensional maps and Krasinkiewicz maps in C ( X , M n ) .

Curve Matching, Time Warping, and Light Fields: New Algorithms for Computing Similarity between Curves

The problem of curve matching appears in many application domains, like time series analysis, shape matching, speech recognition, and signature verification, among others. Curve matching has been studied extensively by computational geometers, and many measures of similarity have been examined, among them being the Frechet distance (sometimes referred in folklore as the "dog-man" distance). A measure that is very closely related to the Frechet distance but has never been studied in a geometric context is the Dynamic Time Warping measure (DTW), first used in the context of speech recognition. This measure is ubiquitous across different domains, a surprising fact because notions of similarity usually vary significantly depending on the application. However, this measure suffers from some drawbacks, most importantly the fact that it is defined between sequences of points rather than curves. Thus, the way in which a curve is sampled to yield such a sequence can dramatically affect the quality of the result. Some attempts have been made to generalize the DTW to continuous domains, but the resulting algorithms have exponential complexity. In this paper we propose similarity measures that attempt to capture the "spirit" of dynamic time warping while being defined over continuous domains, and present efficient algorithms for computing them. Our formulation leads to a very interesting connection with finding short paths in a combinatorial manifold defined on the input chains, and in a deeper sense relates to the way light travels in a medium of variable refractivity.

Discrete Surfaces and Frontier Orders

Many applications require the extraction of an object boundary from a discrete image. In most cases, the result of such a process is expected to be, topologically, a surface, and this property might be required in subsequent operations. However, only through careful design can such a guarantee be provided. In the present article we will focus on partially ordered sets and the notion of n-surfaces introduced by Evako et al. to deal with this issue. Partially ordered sets are topological spaces that can represent the topology of a wide range of discrete spaces, including abstract simplicial complexes and regular grids. It will be proved in this article that (in the framework of simplicial complexes) any n-surface is an n-pseudomanifold, and that any n-dimensional combinatorial manifold is an n-surface. Moreover, given a subset of an n-surface (an object), we show how to build a partially ordered set called frontier order, which represents the boundary of this object. Similarly to the continuous case, where the boundary of an n-manifold, if not empty, is an (n?1)-manifold, we prove that the frontier order associated to an object is a union of disjoint (n?1)-surfaces. Thanks to this property, we show how topologically consistent Marching Cubes-like algorithms can be designed using the framework of partially ordered sets.

On Social Statement Spaces

In this fundamental draft we shall introduce the concept of social propositional spaces as a general notion of social science and in particular language, logic and methodology. This should help us bridge an old gap between extension and cognition, or said in more specific words, between space as conceived in mathematics or physics and space as is brought in by social location, by the meaning of predicates, their explanation and logic. 1 We shall not rely on the idea of factor analysis where some “main components” are fit into the data. Because the notion of euclidean normalization and orthogonality, in this context, is a bit artificial. We first show how combinatorial manifolds of statements can be represented in finite vector spaces over the Galois field F2. This space which we denote as logic statement space can be spanned by generators of alternating codes. In this way the old concepts of truth tables and interaction attributes can be linked to informatics and in particular code theory and geometry. Extending the logic statement space onto a real vector space by mapping units of the finite rings onto units of the real rings we obtain logistic statement spaces. It turns out that vectors in logistic statement spaces represent cross-tables of logits connected with logistic regression analysis. As a completion of the theory we show that the logit is the first derivative of entropy with respect to probability.

Local formulae for combinatorial Pontryagin classes

Let be the characteristic class of a combinatorial manifold given by a polynomial in the rational Pontryagin classes of . We prove that for any polynomial there is a function taking each combinatorial manifold to a cycle in its rational simplicial chains such that: 1) the Poincaré dual of represents the cohomology class ; 2) the coefficient of each simplex in the cycle is determined solely by the combinatorial type of . We explicitly describe all such functions for the first Pontryagin class. We obtain estimates for the denominators of the coefficients of the simplices in the cycles .

Markov's theorem and algorithmically non-recognizable combinatorial manifolds

We prove the theorem of Markov on the existence of an algorithmically non-recognizable combinatorial n-dimensional manifold for every n≥4. We construct for the first time a concrete manifold which is algorithmically non-recognizable. A strengthened form of Markov's theorem is proved using the combinatorial methods of regular neighbourhoods and handle theory. The proofs coincide for all n≥4. We use Borisov's group [8] with insoluble word problem. It has two generators and twelve relations. The use of this group forms the base for proving the strengthened form of Markov's theorem. (The author is indebted to S.I. Adian for this idea.)

Oriented matroids, complex manifolds, and a combinatorial model for BU

We introduce a new notion of complex oriented matroid and develop some basic properties of this object. Our definition of complex oriented matroids bears the same relationship to classical oriented matroids that the stratification of the complex plane into nine components corresponding to the signs of the complex and real parts has with the three-component sign stratification of the real line. We then use these complex oriented matroids to set up the foundations of a combinatorial version of complex geometry analogous to MacPherson's combinatorial differential manifolds; in this world, the representing object for the functor of (combinatorial) complex vector bundles is the nerve of a poset of complex oriented matroids. We conclude by showing that this space is homotopy equivalent to the complex Grassmannian, thus deducing that our combinatorial world is able to completely capture the notion of complex vector bundles.

UNUSUAL FORMULAE FOR THE EULER CHARACTERISTIC

Everyone knows that the Euler characteristic of a combinatorial manifold is given by the alternating sum of its numbers of simplices. It is shown that there are other linear combinations of the numbers of simplices which are combinatorial invariants, but that all such invariants are multiples of the Euler characteristic.

Projectivities in simplicial complexes and colorings of simple polytopes

For each strongly connected finite-dimensional (pure) simplicial complex $\Delta$ we construct a finite group $\Pi(\Delta)$ , the group of projectivities of $\Delta$ , which is a combinatorial but not a topological invariant of $\Delta$ . This group is studied for combinatorial manifolds and, in particular, for polytopal simplicial spheres. The results are applied to a coloring problem for simplicial (or, dually, simple) polytopes which arises in the area of toric manifolds.

Combinatorial manifold mesh reconstruction and optimization from unorganized points with arbitrary topology

In this paper, a novel approach is proposed to reliably reconstruct the geometric shape of a physically existing object based on unorganized point cloud sampled from its boundary surface. The proposed approach is composed of two steps. In the first step, triangle mesh structure is reconstructed as a continuous manifold surface by imposing explicit relationship among the discrete data points. For efficient reconstruction, a growing procedure is employed to build the 2-manifold directly without intermediate 3D representation. Local and global topological operations with ensured completeness and soundness are defined to incrementally construct the 2-manifold with arbitrary topology. In addition, a novel criterion is proposed to control the growing process for ensured geometric integrity and automatic boundary detection with a non-metric threshold. The reconstructed manifold surface captures the object topology with the built-in combinatorial structure and approximates the object geometry to the first order. In the second step, new methods are proposed to efficiently obtain reliable curvature estimation for both the object surface and the reconstructed mesh surface. The combinatorial structure of the triangle mesh is then optimized by changing its local topology to minimize the curvature difference between the two surfaces. The optimized triangle mesh achieves second order approximation to the object geometry and can serve as a basis for many applications including virtual reality, computer vision, and reverse engineering.