Triangulated Manifolds with Few Vertices: Combinatorial Manifolds

Abstract

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.

37 pages, 1 figure

References in corpus (6)

• Lectures on 0/1-polytopes
• Combinatorial 3-manifolds with 10 vertices
• Triangulated Manifolds with Few Vertices: Geometric 3-Manifolds
• Triangulated Manifolds with Few Vertices: Vertex-Transitive Triangulations I
• Triangulated Manifolds with Few Vertices: Centrally Symmetric Spheres and Products of Spheres
• Combinatorial triangulations of homology spheres

Cited by in corpus (27)

• Surface realization with the intersection edge functional
• Combinatorial 3-manifolds with 10 vertices
• Geometric Reasoning with polymake
• Generating irreducible triangulations of surfaces
• Triangulated Manifolds with Few Vertices: Vertex-Transitive Triangulations I
• Minimal Triangulations of Manifolds
• On stellated spheres, shellable balls, lower bounds and a combinatorial criterion for tightness
• Face enumeration - from spheres to manifolds
• On stellated spheres and a tightness criterion for combinatorial manifolds
• Topological field theory on r-spin surfaces and the Arf invariant
• Socles of Buchsbaum modules, complexes and posets
• Enriched homology and cohomology modules of simplicial complexes
• Isomorphism-free lexicographic enumeration of triangulated surfaces and 3-manifolds
• Equilibrium and equivariant triangulations of some small covers with minimum number of vertices
• 634 vertex-transitive and more than $10^{103}$ non-vertex-transitive 27-vertex triangulations of manifolds like the octonionic projective plane
• A characterization of tightly triangulated 3-manifolds
• A necessary condition for the tightness of odd-dimensional combinatorial manifolds
• Enumerative properties of triangulations of spherical bundles over S^1
• On Walkup's class $\mathcal{K}(d)$ and a minimal triangulation of $(S^3 \times \rotatebox{90}{\ltimes} S^1)^{\#3}$
• A minimal triangulation of complex projective plane admitting a chess colouring of four-dimensional simplices
• New examples and partial classification of 15-vertex triangulations of the quaternionic projective plane
• Applications of Klee's Dehn-Sommerville relations
• Triangulations of ${\mathbb{RP}}^n$ with few vertices
• $g$-vectors of manifolds with boundary
• Equilibrium triangulations of some quasitoric 4-manifolds
• Minimal triangulations of sphere bundles over the circle
• Tight and stacked triangulations of manifolds