Combinatorial Manifolds Research Articles

Balanced shellings and moves on balanced manifolds

A classical result by Pachner states that two d-dimensional combinatorial manifolds with boundary are PL homeomorphic if and only if they can be connected by a sequence of shellings and inverse shellings. We prove that for balanced, i.e., properly (d+1)-colored, manifolds such a sequence can be chosen such that balancedness is preserved in each step. As a key ingredient we establish that any two balanced PL homeomorphic combinatorial manifolds with the same boundary are connected by a sequence of basic cross-flips, as was shown recently by Izmestiev, Klee and Novik for balanced manifolds without boundary. Moreover, we enumerate combinatorially different basic cross-flips and show that roughly half of these suffice to relate any two PL homeomorphic manifolds.

Complete and computable orbit invariants in the geometry of the affine group over the integers

The subject matter of this paper is the geometry of the affine group over the integers, $${\mathsf {GL}}(n,{\mathbb {Z}})\ltimes {\mathbb {Z}}^n$$ . Turing-computable complete $${\mathsf {GL}}(n,{\mathbb {Z}})\ltimes {\mathbb {Z}}^n$$ -orbit invariants are constructed for rational affine spaces, angles, segments, triangles and ellipses. In rational affine $${\mathsf {GL}}(n,{\mathbb {Q}})\ltimes {\mathbb {Q}}^n$$ -geometry, ellipses are classified by the Clifford–Hasse–Witt invariant, via the Hasse–Minkowski theorem. We classify ellipses in $${\mathsf {GL}}(n,{\mathbb {Z}})\ltimes {\mathbb {Z}}^n$$ -geometry combining results by Apollonius of Perga and Pappus of Alexandria with the Hirzebruch–Jung continued fraction algorithm. We then consider rational polyhedra, i.e., finite unions of simplexes in $${\mathbb {R}}^n$$ with rational vertices. Markov’s unrecognizability theorem for combinatorial manifolds states the undecidability of the problem whether two rational polyhedra P and $$P'$$ are continuously $${\mathsf {GL}}(n,{\mathbb {Q}})\ltimes {\mathbb {Q}}^n$$ -equidissectable. The same problem for the continuous $${\mathsf {GL}}(n,{\mathbb {Z}})\ltimes {\mathbb {Z}}^n$$ -equidissectability of P and $$P'$$ is open. We prove the decidability of the problem whether two rational polyhedra P and $$P'$$ in $${\mathbb {R}}^n$$ have the same $${\mathsf {GL}}(n,{\mathbb {Z}})\ltimes {\mathbb {Z}}^n$$ -orbit.

Balanced Triangulations on few Vertices and an Implementation of Cross-Flips

A $d$-dimensional simplicial complex is balanced if the underlying graph is $(d+1)$-colorable. We present an implementation of cross-flips, a set of local moves introduced by Izmestiev, Klee and Novik which connect any two PL-homeomorphic balanced combinatorial manifolds. As a result we exhibit a vertex minimal balanced triangulation of the real projective plane, of the dunce hat and of the real projective space, as well as several balanced triangulations of surfaces and 3-manifolds on few vertices. In particular we construct small balanced triangulations of the 3-sphere that are non-shellable and shellable but not vertex decomposable.

A 15-Vertex Triangulation of the Quaternionic Projective Plane

In 1992, Brehm and Kühnel constructed an 8-dimensional simplicial complex \(M^8_{15}\) with 15 vertices as a candidate to be a minimal triangulation of the quaternionic projective plane. They managed to prove that it is a manifold “like a projective plane” in the sense of Eells and Kuiper. However, it was not known until now if this complex is PL homeomorphic (or at least homeomorphic) to \({\mathbb {H}}P^2\). This problem was reduced to the computation of the first rational Pontryagin class of this combinatorial manifold. Realizing an algorithm due to Gaifullin, we compute the first Pontryagin class of \(M^8_{15}\). As a result, we obtain that it is indeed a minimal triangulation of \({\mathbb {H}}P^2\).

On a Vertex-Minimal Triangulation of $\mathbb R \mathrm P ^4$

We give three constructions of a vertex-minimal triangulation of $4$-dimensional real projective space $\mathbb{R}\mathrm{P}^4$. The first construction describes a $4$-dimensional sphere on $32$ vertices, which is a double cover of a triangulated $\mathbb{R}\mathrm{P}^4$ and has a large amount of symmetry. The second and third constructions illustrate approaches to improving the known number of vertices needed to triangulate $n$-dimensional real projective space. All three constructions deliver the same combinatorial manifold, which is also the same as the only known $16$-vertex triangulation of $\mathbb{R}\mathrm{P}^4$. We also give a short, simple construction of the $22$-point Witt design, which is closely related to the complex we construct.

On combinatorial Gauss-Bonnet Theorem for general Euclidean simplicial complexes

For a finitely triangulated closed surface M2, let αx be the sum of angles at a vertex x. By the well-known combinatorial version of the 2- dimensional Gauss-Bonnet Theorem, it holds Σx(2π - αx) = 2πχ(M2), where χ denotes the Euler characteristic of M2, αx denotes the sum of angles at the vertex x, and the sum is over all vertices of the triangulation. We give here an elementary proof of a straightforward higher-dimensional generalization to Euclidean simplicial complexes K without assuming any combinatorial manifold condition. First, we recall some facts on simplicial complexes, the Euler characteristics and its local version at a vertex. Then we define δ(τ) as the normed dihedral angle defect around a simplex τ. Our main result is Στ (-1)dim(τ)δ(τ) = χ(K), where the sum is over all simplices τ of the triangulation. Then we give a definition of curvature κ(x) at a vertex and we prove the vertex-version Σx∈K0 κ(x) = χ(K) of this result. It also possible to prove Morse-type inequalities. Moreover, we can apply this result to combinatorial (n + 1)-manifolds W with boundary B, where we prove that the difference of Euler characteristics is given by the sum of curvatures over the interior of W plus a contribution from the normal curvature along the boundary B: $$\chi \left( W \right) - \frac{1}{2}\chi \left( B \right) = \sum {_{\tau \in W - B}} {\left( { - 1} \right)^{\dim \left( \tau \right)}} + \sum {_{\tau \in B}} {\left( { - 1} \right)^{\dim \left( \tau \right)}}\rho \left( \tau \right)$$ .

THE CONCEPT OF MULTI-ALTERNATIVE IN THE ANIMATE AND INANIMATE STRUCTURES

The paper concentrates on some mechanisms of evolution that forms the constructive-oriented approach to the existence and development process of both wildlife and inanimate nature. This approach, also known as multi-alternative concept, is taken as a basis for the discussion of some interrelated and complementary principles of the diversity, discreteness and hierarchy in evolutionary processes and structures. First of all, it is demonstrated, that the possibility of evolution depends mainly on the heterogeneity of matter forms that implicates the diversity of its acquired characteristics. It can also be shown that the specified diversity is essentially enabled with discreteness of elements, which constitutes the combinatorial manifold of patterns and modes in physical and biological systems. Special attention is given to the issue of correlation between process hierarchy and structures, on one side, and its stability and homeostasis on the other. Finally, there are some examples provided to demonstrate the application of the considered principles in solving different technological problems, such as development of high-reliable power systems for space stations or designing evolutionary algorithms for decision-making systems. It is shown that the evolutionary concept of community multi-alternative opens prospects for its practical appliance to application analysis and control of complex systems, both of natural and anthropogenic origin.

A necessary condition for the tightness of odd-dimensional combinatorial manifolds

We present a necessary condition for (ℓ−1)-connected combinatorial (2ℓ+1)-manifolds to be tight. As a corollary, we show that there is no tight combinatorial three-manifold with first Betti number at most two other than the boundary of the four-simplex and the nine-vertex triangulation of the three-dimensional Klein bottle.

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A remarkable achievement of algorithmic topology is A.A.Markov’s theorem on the unsolvability of the homeomorphism problem for manifolds. Boone, Haken and Poenaru extended Markov’s original proof to the case of closed smooth manifolds. One of their initial difficulties was the introduction of a natural finite representation of a differentiable and/or combinatorial manifold. In this paper we extend this representation to compact smooth manifolds and propose an extension to smooth manifolds. Keywords : Computability and recursion theory, algorithmic topology, smooth manifolds. To cite this article: C.M Parra, J. Suarez Ramirez, Representacion finita de variedades compactas, Rev. Integr. Temas Mat. 33 (2015), No. 2, 97–105.

The mu vector, Morse inequalities and a generalized lower bound theorem for locally tame combinatorial manifolds

In a recent work (Bagchi and Datta, 2014) with Datta, we introduced the mu-vector (with respect to a given field) of simplicial complexes and used it to study tightness and lower bounds. In this paper, we modify the definition of mu-vectors. With the new definition, most results of Bagchi and Datta (2014) become correct without the hypothesis of 2-neighbourliness. In particular, the combinatorial Morse inequalities of Bagchi and Datta (2014) are now true of all simplicial complexes.As an application, we prove the following generalized lower bound theorem (GLBT) for connected locally tame combinatorial manifolds. If M is such a manifold of dimension d, then for 1≤ℓ≤d−12 and any field F,gℓ+1(M)≥(d+2ℓ+1)∑i=1ℓ(−1)ℓ−iβi(M;F). Equality holds here if and only if M is ℓ-stacked.We conjecture that, more generally, this theorem is true of all triangulated connected and closed homology manifolds. A conjecture on the sigma-vectors of triangulated homology spheres is proposed, whose validity will imply this GLB Conjecture for homology manifolds. We also prove the GLB Conjecture for all connected and closed combinatorial 3-manifolds. Thus, any connected closed combinatorial manifold M of dimension three satisfies g2(M)≥10β1(M;F), with equality iff M is 1-stacked. This result settles a question of Novik and Swartz (2009) in the affirmative.

Tight combinatorial manifolds and graded Betti numbers

In this paper, we study the conjecture of Kuhnel and Lutz, who state that a combinatorial triangulation of the product of two spheres $$\mathbb S^i \times \mathbb S^j$$ with $$j \ge i$$ is tight if and only if it has exactly $$i+2j+4$$ vertices. To approach this conjecture, we use graded Betti numbers of Stanley–Reisner rings. By using recent results on graded Betti numbers, we prove that the only if part of the conjecture holds when $$j>2i$$ and that the if part of the conjecture holds for triangulations all whose vertex links are simplicial polytopes. We also apply this algebraic approach to obtain lower bounds on the numbers of vertices and edges of triangulations of manifolds and pseudomanifolds.

A Note on the Tangent Bundle and Gauss Functor of Posets and Manifolds

We introduce a notion of the tangent bundle of a poset. In the case where the poset is the poset of simplices of a combinatorial manifold, the construction produces the best possible combinatorial model for the geometric compactified tangent bundle.

Exceptional surgeries on knots with exceptional classes

We survey the aspects of classical combinatorial sutured manifold theory and show how they can be adapted to study exceptional Dehn fillings and 2-handle additions. As a consequence, we show that if a hyperbolic knot \(\beta \) in a compact, orientable, hyperbolic 3-manifold, \(M\) has the property that winding number and wrapping number are not equal with respect to a non-trivial class in \(H_2(M,\partial M)\), and then, with only a few possible exceptions, every 3-manifold \(M'\) obtained by Dehn surgery on \(\beta \) with surgery distance \(\Delta \ge 2\) will be hyperbolic.

Localizable invariants of combinatorial manifolds and Euler characteristic

It is shown that if a real-valued PL-invariant of closed combinatorial manifolds admits a local formula that depends only on the f-vector of the link of each vertex, then the invariant must be a constant times the Euler characteristic.

Combinatorial 3-Manifolds with Transitive Cyclic Symmetry

In this article we give combinatorial criteria to decide whether a transitive cyclic combinatorial d-manifold can be generalized to an infinite family of such complexes, together with an explicit construction in the case that such a family exists. In addition, we substantially extend the classification of combinatorial 3-manifolds with transitive cyclic symmetry up to 22 vertices. Finally, a combination of these results is used to describe new infinite families of transitive cyclic combinatorial manifolds and in particular a family of neighborly combinatorial lens spaces of infinitely many distinct topological types.

On stellated spheres and a tightness criterion for combinatorial manifolds

We introduce k-stellated spheres and consider the class Wk(d) of triangulated d-manifolds, all of whose vertex links are k-stellated, and its subclass Wk∗(d), consisting of the (k+1)-neighbourly members of Wk(d). We introduce the mu-vector of any simplicial complex and show that, in the case of 2-neighbourly simplicial complexes, the mu-vector dominates the vector of Betti numbers componentwise; the two vectors are equal precisely for tight simplicial complexes. We are able to estimate/compute certain alternating sums of the components of the mu-vector of any 2-neighbourly member of Wk(d) for d≥2k. As a consequence of this theory, we prove a lower bound theorem for such triangulated manifolds, and we determine the integral homology type of members of Wk∗(d) for d≥2k+2. As another application, we prove that, when d≠2k+1, all members of Wk∗(d) are tight. We also characterize the tight members of Wk∗(2k+1) in terms of their kth Betti numbers. These results more or less answer a recent question of Effenberger, and also provide a uniform and conceptual tightness proof for all except two of the known tight triangulated manifolds.We also prove a lower bound theorem for homology manifolds in which the members of W1(d) provide the equality case. This generalizes a result (the d=4 case) due to Walkup and Kühnel. As a consequence, it is shown that every tight member of W1(d) is strongly minimal, thus providing substantial evidence in favour of a conjecture of Kühnel and Lutz asserting that tight homology manifolds should be strongly minimal.

Physics inspired algorithms for (co)homology computations of three-dimensional combinatorial manifolds with boundary

The issue of computing (co)homology generators of a cell complex is gaining a pivotal role in various branches of science. While this issue may be rigorously solved in polynomial time, it is still overly demanding for large scale problems. Drawing inspiration from low-frequency electrodynamics, this paper presents a physics inspired algorithm for first cohomology group computations on three-dimensional complexes. The algorithm is general and exhibits orders of magnitude speed up with respect to competing ones, allowing to handle problems not addressable before. In particular, when generators are employed in the physical modeling of magneto-quasistatic problems, this algorithm solves one of the most long-lasting problems in low-frequency computational electromagnetics. In this case, the effectiveness of the algorithm and its ease of implementation may be even improved by introducing the novel concept of lazy cohomology generators.

Non-homogeneous combinatorial manifolds

In this paper we extend the classical theory of combinatorial manifolds to the non-homogeneous setting. NH-manifolds are polyhedra which are locally like Euclidean spaces of varying dimensions. We show that many of the properties of classical manifolds remain valid in this wider context. NH-manifolds appear naturally when studying Pachner moves on (classical) manifolds. We introduce the notion of NH-factorization and prove that PL-homeomorphic manifolds are related by a finite sequence of NH-factorizations involving NH-manifolds.

Centrally symmetric manifolds with few vertices

A centrally symmetric 2 d-vertex combinatorial triangulation of the product of spheres S i × S d − 2 − i is constructed for all pairs of nonnegative integers i and d with 0 ⩽ i ⩽ d − 2 . For the case of i = d − 2 − i , the existence of such a triangulation was conjectured by Sparla. The constructed complex admits a vertex-transitive action by a group of order 4 d. The crux of this construction is a definition of a certain full-dimensional subcomplex, B ( i , d ) , of the boundary complex of the d-dimensional cross-polytope. This complex B ( i , d ) is a combinatorial manifold with boundary and its boundary provides a required triangulation of S i × S d − i − 2 . Enumerative characteristics of B ( i , d ) and its boundary, and connections to another conjecture of Sparla are also discussed.

Normal surfaces as combinatorial slicings

We investigate slicings of combinatorial manifolds as properly embedded co-dimension 1 submanifolds. Focus is given to the case of dimension 3 , where slicings are (discrete) normal surfaces. For the cases of 2 -neighborly 3 -manifolds as well as quadrangulated slicings, lower bounds on the number of quadrilaterals of slicings depending on its genus g are presented. These are shown to be sharp for infinitely many values of g . Furthermore, we classify slicings of combinatorial 3 -manifolds which are weakly neighborly polyhedral maps.