This paper gives a combinatorial description of spin and spin^c-structures on triangulated PL-manifolds of arbitrary dimension. These formulations of spin and spin^c-structures are established primarily for the purpose of aiding in computations. The novelty of the approach is we rely heavily on the naturality of binary symmetric groups to avoid lengthy explicit constructions of smoothings of PL-manifolds.
In this survey on combinatorial properties of triangulated manifolds we discuss various lower bounds on the number of vertices of simplicial and combinatorial manifolds. Moreover, we give a list of all known examples of vertex-minimal triangulations.
Combinatorial Ricci flow on an ideally triangulated compact 3-manifold with boundary was introduced by Luo as a 3-dimensional analog of Chow-Luo's combinatorial Ricci flow on a triangulated surface and conjectured to find algorithmically the complete hyperbolic metric on the compact 3-manifold with totally geodesic boundary. In this paper, we prove Luo's conjecture affirmatively by extending the combinatorial Ricci flow through the singularities of the flow if the ideally triangulated compact 3-manifold with boundary admits such a metric.
We give a sufficient condition for an open 3-manifold to admit a decomposition along properly embedded open annuli and tori, generalizing the toric splitting of Jaco-Shalen and Johannson.
In this paper, we describe geometrical constructions to obtain triangulations of connected sums of closed orientable triangulated 3-manifolds. Using these constructions, we show that it takes time polynomial in the number of tetrahedra to check if a closed orientable 3-manifold, equipped with a minimal triangulation, is reducible or not. This result can easily be generalized to compact orientable 3-manifolds with non-empty boundary.
It is a theorem of Casson and Rivin that the complete hyperbolic metric on a cusp end ideal triangulated 3-manifold maximizes volume in the space of all positive angle structures. We show that the conclusion still holds if some of the tetrahedra in the complete metric are flat.
Although Kirby and Siebenmann showed that there are manifolds that do not admit PL structures, the possibility remained that all manifolds could be triangulated. In the late seventies Galewski and Stern and independently Matumoto showed that non-triangulable manifolds exist in all dimensions > 4 if and only if homology 3-spheres with certain properties do not exist. In 2013 Manolescu showed that, indeed, there were no such homology 3-spheres and hence, non-triangulable manifolds exist in each dimension >4. It follows from work of Freedman in 1982 that there are 4-manifolds that cannot be trian…
Tight triangulated manifolds are generalisations of neighborly triangulations of closed surfaces and are interesting objects in Combinatorial Topology. Tight triangulated manifolds are conjectured to be minimal. Except few, all the known tight triangulated manifolds are stacked. It is known that locally stacked tight triangulated manifolds are strongly minimal. Except for three infinite series and neighborly surfaces, very few tight triangulated manifolds are known. From some recent works, we know more on tight triangulation. In this article, we present a survey on the works done on tight tria…
We present new computational methods for proving diffeomorphy of triangulated 4-manifolds, including algorithms and topological software that can for the first time effectively handle the complexities that arise in dimension four and be used for large scale experiments.
We solve some computational problems for triangulated closed three-dimensional manifolds using groups of simplicial homology and cohomology modulo 2. Two efficient algorithms for computing the intersection numbers of 1- and 2-dimensional cycles are developed. By means of these algorithms it is possible to construct a basis of cohomology group from the homology group of two cycles of complementary dimensions.
In this paper, we introduce two discrete curvature flows, which are called $α$-flows on two and three dimensional triangulated manifolds. For triangulated surface $M$, we introduce a new normalization of combinatorial Ricci flow (first introduced by Bennett Chow and Feng Luo \cite{CL1}), aiming at evolving $α$ order discrete Gauss curvature to a constant. When $αχ(M)\leq0$, we prove that the convergence of the flow is equivalent to the existence of constant $α$-curvature metric. We further get a necessary and sufficient combinatorial-topological-metric condition, which is a generalization of T…
Based in the isomorphism between Lie algebroid cohomology and piecewise smooth cohomology, it is proved that the Rham cohomology of a locally trivial Lie groupoid $G$ on a smooth manifold $M$ is isomorphic to the piecewise Rham cohomology of $G$, in which $G$ and $M$ are manifolds without boundary and $M$ is smoothly triangulated by a finite simplicial complex $K$ such that, for each simplex $Δ$ of $K$, the inverse images of $Δ$ by the source and target mappings of $G$ are transverses submanifolds in the ambient space $G$. As a consequence, it is shown that the piecewise de Rham cohomology of …
We propose a finite dimensional variational principle on triangulated 3-manifolds so that its critical points are related to solutions to Thurston's gluing equation and Haken's normal surface equation. The action functional is the volume. This is a generalization of an earlier program by Casson and Rivin for compact 3-manifolds with torus boundary. Combining the result in this paper and the work of Futer-Guéritaud, Segerman-Tillmann and Luo-Tillmann, we obtain a new finite dimensional variational formulation of the Poncare-conjecture. This provides a step toward a new proof the Poincaré conjec…
An unusual formula for the Euler characteristics of even dimensional triangulated manifolds is deduced from the generalized Dehn-Sommerville equations.
We describe an algorithm for the enumeration of (candidates of) vertex-transitive combinatorial $d$-manifolds. With an implementation of our algorithm, we determine, up to combinatorial equivalence, all combinatorial manifolds with a vertex-transitive automorphism group on $n\leq 13$ vertices. With the exception of actions of groups of small order, the enumeration is extended to 14 and 15 vertices.
Principal circle bundle over a PL polyhedron can be triangulated and thus obtains combinatorics. The triangulation is assembled from triangulated circle bundles over simplices. To every triangulated circle bundle over a simplex we associate a necklace (in combinatorial sense). We express rational local formulas for all powers of first Chern class in the terms of mathematical expectations of parities of the associated necklaces. This rational parity is a combinatorial isomorphism invariant of triangulated circle bundle over simplex, measuring mixing by triangulation of the circular graphs over …
We explicitly construct small triangulations for a number of well-known 3-dimensional manifolds and give a brief outline of some aspects of the underlying theory of 3-manifolds and its historical development.
The present paper follows the computational approach to 3-manifold classification via edge-coloured graphs, already performed by several authors with respect to orientable 3-manifolds up to 28 coloured tetrahedra, non-orientable 3-manifolds up to 26 coloured tetrahedra, genus two 3-manifolds up to 34 coloured tetrahedra: in fact, by automatic generation and analysis of suitable edge-coloured graphs, called crystallizations, we obtain a catalogue of all orientable 3-manifolds admitting coloured triangulations with 30 tetrahedra. These manifolds are unambiguously identified via JSJ decomposition…
We give a brief summary of some of our work and our joint work with Stephan Tillmann on solving Thurston's equation and Haken equation on triangulated 3-manifolds in this paper. Several conjectures on the existence of solutions to Thurston's equation and Haken equation are made. Resolutions of these conjecture will lead to a new proof of the Poincaré conjecture without using the Ricci flow. We approach these conjectures by a finite dimensional variational principle so that its critical points are related to solutions to Thurston's gluing equation and Haken's normal surface equation. The action…
We construct 2^{Ω(n^{5/4})} combinatorial types of triangulated 3-spheres on n vertices. Since by a result of Goodman and Pollack (1986) there are no more than 2^{O(n log n)} combinatorial types of simplicial 4-polytopes, this proves that asymptotically, there are far more combinatorial types of triangulated 3-spheres than of simplicial 4-polytopes on n vertices. This complements results of Kalai (1988), who had proved a similar statement about d-spheres and (d+1)-polytopes for fixed d >= 4.
Here we are fixing an output of a trivial calculation based on Konsevich's differential 2-form for the Chern class of polygon bundle. As a result an interesting combinatorics and arithmetics jumps right out of a jukebox. The calculation gives very simple rational combinatorial characteristics (we call it "curvature") of a triangulated $S^1$ bundle over a 2-simplex, which is a local combinatorial formula for the first Chern class. The curvature is expressed in terms of cyclic word in 3-character alphabet associated to the bundle. From the point of view of simplicial combinatorics the word is a …
We prove two results on stacked triangulated manifolds in this paper: (a) every stacked triangulation of a connected manifold with or without boundary is obtained from a simplex or the boundary of a simplex by certain combinatorial operations; (b) in dimension $d \geq 4$, if $Δ$ is a tight connected closed homology $d$-manifold whose $i$th homology vanishes for $1 < i < d-1$, then $Δ$ is a stacked triangulation of a manifold.These results give affirmative answers to questions posed by Novik and Swartz and by Effenberger.
In this paper, we introduce a framework of $(α,β)$-flows on triangulated manifolds with two and three dimensions, which unifies several discrete curvature flows previously defined in the literature.
For the purposes of this paper, Dehn surgery along a curve K in a 3-manifold M with slope r is `exceptional' if the resulting 3-manifold M_K(r) is reducible or a solid torus, or the core of the surgery solid torus has finite order in the fundamental group of M_K(r). We show that, providing the exterior of K is irreducible and atoroidal, and the distance between r and the meridian slope is more than one, and a homology condition is satisfied, then there is (up to ambient isotopy) only a finite number of such exceptional surgery curves in a given compact orientable 3-manifold M, with the boundar…
We show that solutions of Thurston equation on triangulated 3-manifolds in a commutative ring carry topological information. We also introduce a homogeneous Thurston equation and a commutative ring associated to triangulated 3-manifolds.
One of the apparent advantages of quantum computers over their classical counterparts is their ability to efficiently contract tensor networks. In this article, we study some implications of this fact in the case of topological tensor networks. The graph underlying these networks is given by the triangulation of a manifold, and the structure of the tensors ensures that the overall tensor is independent of the choice of internal triangulation. This leads to quantum algorithms for additively approximating certain invariants of triangulated manifolds. We discuss the details of this construction i…
This paper contains examples of closed aspherical manifolds obtained as a by-product of recent work by the author [arXiv:math.GR/0509490] on the relative strict hyperbolization of polyhedra. The following is proved. (I) Any closed aspherical triangulated n-manifold M^n with hyperbolic fundamental group is a retract of a closed aspherical triangulated (n+1)-manifold N^(n+1) with hyperbolic fundamental group. (II) If B_1,...,B_m are closed aspherical triangulated n-manifolds, then there is a closed aspherical triangulated manifold N of dimension n+1 such that N has nonzero simplicial volume, N r…
Geometric torsions are torsions of acyclic complexes of vector spaces which consist of differentials of geometric quantities assigned to the elements of a manifold triangulation. We use geometric torsions to construct invariants for a manifold with a triangulated boundary. These invariants can be naturally united in a vector, and a change of the boundary triangulation corresponds to a linear transformation of this vector. Moreover, when two manifolds are glued by their common boundary, these vectors undergo scalar multiplication, i.e., they work according to M. Atiyah's axioms for a topologica…
This article presents an improvement and extension of the heuristic first presented by Hougardy, Lutz, and Zelke in 2010 for realizing triangulated orientable surfaces with few vertices by a simplex-wise linear embedding. The improvement consists in the applicability to non-orientable surfaces (simplex-wise linear immersions) as well as symmetric realizations. With the help of our algorithm, numerous - often symmetric - realizations of non-orientable surfaces with the minimal number or few vertices have been obtained for the first time. These examples include the the Projective Plane with one …
For a field $\mathbb{F}$, the notion of $\mathbb{F}$-tightness of simplicial complexes was introduced by Kühnel. Kühnel and Lutz conjectured that any $\mathbb{F}$-tight triangulation of a closed manifold is the most economic of all possible triangulations of the manifold. The boundary of a triangle is the only $\mathbb{F}$-tight triangulation of a closed 1-manifold. A triangulation of a closed 2-manifold is $\mathbb{F}$-tight if and only if it is $\mathbb{F}$-orientable and neighbourly. In this paper we prove that a triangulation of a closed 3-manifold is $\mathbb{F}$-tight if and only if it i…
Walkup's class ${\cal K}(d)$ consists of the $d$-dimensional simplicial complexes all whose vertex links are stacked $(d-1)$-spheres. According to a result of Walkup, the face vector of any triangulated 4-manifold $X$ with Euler characteristic $χ$ satisfies $f_1 \geq 5f_0 - 15/2 χ$, with equality only for $X \in {\cal K}(4)$. Kühnel observed that this implies $f_0(f_0 - 11) \geq -15χ$, with equality only for 2-neighborly members of ${\cal K}(4)$. For $n = 6, 11$ and 15, there are triangulated 4-manifolds with $f_0=n$ and $f_0(f_0 - 11) = -15χ$. In this article, we present triangulated 4-manifo…
We give an explicit construction of vertex-transitive tight triangulations of $d$-manifolds for $d\geq 2$. More explicitly, for each $d\geq 2$, we construct two $(d^2+5d+5)$-vertex neighborly triangulated $d$-manifolds whose vertex-links are stacked spheres. The only other non-trivial series of such tight triangulated manifolds currently known is the series of non-simply connected triangulated $d$-manifolds with $2d+3$ vertices constructed by Kühnel. The manifolds we construct are strongly minimal. For $d\geq 3$, they are also tight neighborly as defined by Lutz, Sulanke and Swartz. Like Kühne…
In this paper, we introduce a new combinatorial curvature on triangulated surfaces with inversive distance circle packing metrics. Then we prove that this combinatorial curvature has global rigidity. To study the Yamabe problem of the new curvature, we introduce a combinatorial Ricci flow, along which the curvature evolves almost in the same way as that of scalar curvature along the surface Ricci flow obtained by Hamilton \cite{Ham1}. Then we study the long time behavior of the combinatorial Ricci flow and obtain that the existence of a constant curvature metric is equivalent to the convergenc…
A combinatorial formula for the Pontrjagin classes of a triangulated manifold is given. The main ingredients are oriented matroid theory and a modified formulation of Chern-Weil theory.
A triangulated $d$-manifold $K$, satisfies the inequality $\binom{f_0(K)-d-1}{2}\geq \binom{d+2}{2}β_1(K;\mathbb{Z}_2)$ for $d\geq 3$. The triangulated $d$-manifolds that meet the bound with equality are called {\em tight neighborly}. In this paper, we present tight neighborly triangulations of 4-manifolds on 15 vertices with $\mathbb{Z}_3$ as automorphism group. One such example wasconstructed by Bagchi and Datta in 2011. We show that there are exactly 12 such triangulations up to isomorphism, 10 of which are orientable.
This notes explores angle structures on ideally triangulated compact $3$-manifolds with high genus boundary. We show that the existence of angle structures implies the existence of a hyperbolic metric with totally geodesic boundary, and conversely each hyperbolic $3$-manifold with totally geodesic boundary has an ideal triangulation that admits angle structures.
We introduce the $k$-stellated spheres and consider the class ${\cal W}_k(d)$ of triangulated $d$-manifolds all whose vertex links are $k$-stellated, and its subclass ${\cal W}^{\ast}_k(d)$ consisting of the $(k+1)$-neighbourly members of ${\cal W}_k(d)$. 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 its 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-v…
Suppose $K$ is an unknot lying in the 1-skeleton of a triangulated 3-manifold with $t$ tetrahedra. Hass and Lagarias showed there is an upper bound, depending only on $t$, for the minimal number of elementary moves to untangle $K$. We give a simpler proof, utilizing a normal form for surfaces whose boundary is contained in the 1-skeleton of a triangulated 3-manifold. We also obtain a significantly better upper bound of $2^{120t+14}$ and improve the Hass--Lagarias upper bound on the number of Reidemeister moves needed to unknot to $2^{10^5 n}$, where $n$ is the crossing number.
We define a 2-normal surface to be one which intersects every 3-simplex of a triangulated 3-manifold in normal triangles and quadrilaterals, with one or two exceptions. The possible exceptions are a pair of octagons, a pair of unknotted tubes, an octagon and a tube, or a 12-gon. In this paper we use the theory of critical surfaces developed in earlier work to prove the existence of topologically interesting 2-normal surfaces. Our main results are (1) if a ball with normal boundary in a triangulated 3-manifold contains two almost normal 2-spheres then it contains a 2-normal 2-sphere and (2) in …
We show that for a representation of the fundamental group of a triangulated closed 3-manifold (not necessarily hyperbolic) into $\PSL$ so that any edge loop has non-trivial image under the representation, there exist uncountably many solutions to the hyperbolic gluing equation whose associated representations are conjugate to the given representation, and whose volumes are equal to the volume of the given representation.
A (complete) matching of the cells of a triangulated manifold can be thought as a combinatorial or discrete version of a nonsingular vector field. We give several methods for constructing such matchings.
Combinatorial Ricci flow on a cusped $3$-manifold is an analogue of Chow-Luo's combinatorial Ricci flow on surfaces and Luo's combinatorial Ricci flow on compact $3$-manifolds with boundary for finding complete hyperbolic metrics on cusped $3$-manifolds. Dual to Casson and Rivin's program of maximizing the volume of angle structures, combinatorial Ricci flow finds the complete hyperbolic metric on a cusped $3$-manifold by minimizing the co-volume of decorated hyperbolic polyhedral metrics. The combinatorial Ricci flow may develop singularities. We overcome this difficulty by extending the flow…
We consider triangulations of surfaces with edges painted three colors so that edges of each triangle have different colors. Such structures arise as Belyi data (or Grothendieck dessins d'enfant), on the other hand they enumerate pairs of permutations determined up to a common conjugation. The topic of these notes is links of such combinatorial structures with infinite symmetric groups and their representations.
We propose an approach to find constant curvature metrics on triangulated closed 3-manifolds using a finite dimensional variational method whose energy function is the volume. The concept of an angle structure on a tetrahedron and on a triangulated closed 3-manifold is introduced following the work of Casson, Murakami and Rivin. It is proved by A. Kitaev and the author that any closed 3-manifold has a triangulation supporting an angle structure. The moduli space of all angle structures on a triangulated 3-manifold is a bounded open convex polytope in a Euclidean space. The volume of an angle s…
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.
In this paper, we introduce a new combinatorial curvature on two and three dimensional triangulated manifolds, which transforms in the same way as that of the smooth scalar curvature under scaling of the metric and could be used to approximate the Gauss curvature on two dimensional manifolds. Then we use the flow method to study the corresponding constant curvature problem, which is called combinatorial Yamabe problem.
We define a new combinatorial class of triangulations of closed 3-manifolds, satisfying a weak version of 0-efficiency combined with a weak version of minimality, and study them using twisted squares. As an application, we obtain strong restrictions on the topology of a 3-manifold from the existence of non-smooth maxima of the volume function on the space of circle-valued angle structures.
We give a complete enumeration of all combinatorial 3-manifolds with 10 vertices: There are precisely 247882 triangulated 3-spheres with 10 vertices as well as 518 vertex-minimal triangulations of the sphere product $S^2\times S^1$ and 615 triangulations of the twisted sphere product $S^2_\times_S^1$. All the 3-spheres with up to 10 vertices are shellable, but there are 29 vertex-minimal non-shellable 3-balls with 9 vertices.
A combinatorial condition is obtained for when immersed or embedded incompressible surfaces in compact 3-manifolds with tori boundary components remain incompressible after Dehn surgery. A combinatorial characterisation of hierarchies is described. A new proof is given of the topological rigidity theorem of Hass and Scott for 3-manifolds containing immersed incompressible surfaces, as found in cubings of non-positive curvature.
We study a variation of Bagchi and Datta's $σ$-vector of a simplicial complex $C$, whose entries are defined as weighted averages of Betti numbers of induced subcomplexes of $C$. We show that these invariants satisfy an Alexander-Dehn-Sommerville type identity, and behave nicely under natural operations on triangulated manifolds and spheres such as connected sums and bistellar flips. In the language of commutative algebra, the invariants are weighted sums of graded Betti numbers of the Stanley-Reisner ring of $C$. This interpretation implies, by a result of Adiprasito, that the Billera-Lee sph…
In graph theory, as well as in 3-manifold topology, there exist several width-type parameters to describe how "simple" or "thin" a given graph or 3-manifold is. These parameters, such as pathwidth or treewidth for graphs, or the concept of thin position for 3-manifolds, play an important role when studying algorithmic problems; in particular, there is a variety of problems in computational 3-manifold topology - some of them known to be computationally hard in general - that become solvable in polynomial time as soon as the dual graph of the input triangulation has bounded treewidth. In view of…
We introduce the $k$-stellated spheres and compare and contrast them with $k$-stacked spheres. It is shown that for $d \geq 2k$, any $k$-stellated sphere of dimension $d$ bounds a unique and canonically defined $k$-stacked ball. In parallel, any $k$-stacked polytopal sphere of dimension $d\geq 2k$ bounds a unique and canonically defined $k$-stacked ball. We consider the class ${\cal W}_k(d)$ of combinatorial $d$-manifolds with $k$-stellated links. For $d\geq 2k+2$, any member of ${\cal W}_k(d)$ bounds a unique and canonically defined "$k$-stacked" $(d+1)$-manifold. We introduce the mu-vector o…
