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unilogo Universität Stuttgart
Fakultät 8: Fachbereich Mathematik
Lehrstuhl für Differentialgeometrie

Prof. Wolfgang Kühnel


Triangulations with few vertices

Determine the minimum number of vertices for a triangulation of a given manifold, or find "good" triangulations with a reasonable number of vertices and/or with high symmetry. Find general lower bounds for the possible numbers of vertices. A basic inequality was given in
  • U. Brehm and W. Kühnel, Combinatorial manifolds with few vertices, Topology 26, 465-473 (1987).

Upper and Lower bound conjectures for combinatorial manifolds

There is the following CONJECTURE: For any 2k-dimensional combinatorial manifold M with n vertices the following inequality holds: ${{n-k-2} \choose {k+1}} \geq (-1)^k {{2k+1} \choose {k+1}}(\chi(M) - 2)$ with equality if and only if the triangulation is $(k+1)$-neighborly. For a first approach toward this conjecture see
  • I. Novik, Upper bound theorems for homology manifolds, Israel J. Math. 108, 45-82 (1998)
  • I. Novik, On face numbers of manifolds with symmetry, Advances Math. 192, 183-208 (2005)
For the positive answer to the conjecture see
  • I. Novik and Ed Swartz, Socles of Buchsbaum modules, complexes and posets, Advances Math. 222, 2059-2084 (2009).
A similar conjecture for the case of centrally-symmetric triangulations were first studied in the following work by my former student
  • E. Sparla, An upper and a lower bound theorem for combinatorial 4-manifolds, Discrete Comp. Geom. 19, 575-593 (1998)
  • E. Sparla, A new lower bound theorem for combinatorial 2k-manifolds, Graphs and Combinatorics 15, 109-125 (1999)

Tight polyhedral embeddings

There is the following conjecture: For any tight polyhedral embedding of a (k-1)-connected 2k-dimensional manifold M into Euclidean d-space (not in any hyperplane) the following inequality holds: ${{d-k-1} \choose {k+1}} \leq (-1)^k {{2k+1} \choose {k+1}}(\chi(M) - 2)$ with equality if and only if the image is a (k-1)-neighborly subcomplex of the d-dimensional simplex. For partial results see
  • W. Kühnel, Tight polyhedral submanifolds and tight triangulations, Lecture Notes in mathematics 1612, 122 pages, Springer 1995
More recently new tight polyhedral embeddings of 4-manifolds were investigated, see
  • M. Casella and W. Kühnel, A triangulated K3 surface with the minimum number of vertices, Topology 40, 753-772 (2001)
  • W. Kühnel and F. H. Lutz, A census of tight triangulations, Periodica Math. Hungarica 39, 161-183 (1999)
  • W. Kühnel, Tight embeddings of simply connected 4-manifolds, Documenta Math. 9, 401-412 (2004) (electronic)
For tight polyhedral submanifolds of higher-dimensional octahedra see
  • W. Kühnel, Centrally-symmetric tight surfaces and graph embeddings, Beiträge zur Algebra und Geometrie 37, 347-354 (1996)
  • F. Effenberger and W. Kühnel, Hamiltonian submanifolds of regular polytopes, Discrete Comput. Geometry 43, 242-262 (2010)
There is a close connection with the subject "Upper and lower bound theorem", because k-Hamiltonian 2k-manifolds in the boundary complex of a cross polytope belong to both categories, at least in the centrally-symmetric case. Further progress was made in
  • F. Effenberger, Stacked polytopes and tight triangulations of manifolds, J. Comb. Theory, Ser. A 118, No. 6, 1843-1862 (2011)
  • F. Effenberger, Hamiltonian submanifolds of regular polytopes, Doctoral Thesis, Stuttgart 2010

Combinatorial manifolds satisfying complementarity, and triangulated projective planes

The problem arises from three different points of view:
  • Find triangulations of 2k-manifolds with 3k+3 vertices. This is the minimum number for any manifold which is not a sphere, see
    • U. Brehm and W. Kühnel, Combinatorial manifolds with few vertices, Topology 26, 465-473 (1987)
  • Find tight polyhedral embeddings of the projective planes over the real, complex numbers, quaternions, or octonions in the same codimension as the classical Veronese-type embeddings of them. Halfway between two antipodal copies of them we find polyhedral analogous of the Cartan isoparametric hypersurfaces in spheres (which are tubes around the Veronese-type embeddings), see
    • T.F. Banchoff and W. Kühnel, Tight polyhedral models of isoparametric families, and PL--taut submanifolds,
      Advances in Geometry 7, 613--639 (2007)
  • Find triangulated (pseudo-)manifolds satisfying the following combinatorial complementarity condition:
    • A subset $A \subset V$ of the set of vertices V spans a simplex if and only if the complement $V \setminus A$ does not.
    This can be regarded as a kind of a combinatorial duality condition. Among other implications, it leads to a triangulated sphere as the deleted join (the so-called Bier-sphere after work of Thomas Bier).
So far the only known examples are the 6-vertex RP2, the 9-vertex CP2 and a number of 15-vertex triangulations of an 8-manifold, presumably HP2. The case of a 27-vertex triangulation of the octonion plane is still open, see
  • U. Brehm and W. Kühnel, 15-vertex triangulations of an 8-manifold, Math. Annalen 294, 167-193 (1992)

Triangulations of the d-dimensional torus

There is the following conjecture: Any combinatorial triangulation of the d-dimensional torus must have at least 2d+3 vertices. Examples attaining this bound are available, see
  • W. Kühnel and G. Lassmann, Combinatorial d-tori with a large symmetry group, Discrete Comp. Geom. 3, 169-176 (1988)
  • G. Dartois and A. Grigis, Separating maps of the lattice E8 and triangulations of the eight-dimensional torus, Discrete Comp. Geom. 23, 555-567 (2000)
In the case of lattice triangulations the conjecture is proved in
  • U. Brehm and W. Kühnel, Lattice triangulations of E3 and of the 3-torus , Israel J. Math. 189, 97-133 (2012).

Manifolds associated with abstract regular polytopes

Abstract regular polytopes are natural generalizations of the Platonic solids and the other classical regular polytopes. The requirement is that the automorphism group acts transitively on flags. So in particular one can transform any point to any other and then, while keeping the point fixed, any edge to any other and so on. For a monograph on the subject see
  • P. McMullen and E. Schulte, Abstract Regular Polytopes, 570 pages, Cambridge University Press 2001
Among the abstract regular polytopes there are 4-polytopes with toroidal facets. When transforming these abstract toroidal facets into solid tori, a 3-manifold can be associated with the polytope, see
  • U. Brehm, W. Kühnel and E. Schulte, Manifold structures on abstract regular polytopes, Aequationes math. 49, 12-35 (1995)
The higher-dimensional case is still work in progress.