

SELECTED TOPICS OF MY RESEARCH IN COMBINATORIAL TOPOLOGY AT PRESENT AND IN THE PAST:
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, 465473 (1987).
Upper and Lower bound conjectures for combinatorial manifolds
There is the following CONJECTURE:
For any 2kdimensional combinatorial manifold M with n vertices
the following inequality holds:
${{nk2} \choose {k+1}} \geq (1)^k {{2k+1} \choose {k+1}}(\chi(M)  2)$
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, 4582 (1998)
 I. Novik, On face numbers of manifolds with symmetry,
Advances Math. 192, 183208 (2005)
For the positive answer to the conjecture see
 I. Novik and Ed Swartz, Socles of Buchsbaum modules, complexes
and posets,
Advances Math. 222, 20592084 (2009).
A similar conjecture for the case of centrallysymmetric 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, 575593 (1998)
 E. Sparla, A new lower bound theorem for combinatorial $2k$manifolds,
Graphs and Combinatorics 15, 109125 (1999)
Tight polyhedral embeddings
There is the following conjecture:
For any tight polyhedral embedding of a (k1)connected
2kdimensional manifold M into Euclidean dspace (not in any hyperplane)
the following inequality holds:
${{dk1} \choose {k+1}} \leq (1)^k {{2k+1} \choose {k+1}}(\chi(M)  2)$
with equality if and only if the image is a (k1)neighborly
subcomplex of the ddimensional 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 4manifolds were investigated,
see
 M. Casella and W. Kühnel, A triangulated K3 surface
with the minimum number of vertices, Topology 40, 753772 (2001)
 W. Kühnel and F. H. Lutz, A census of tight triangulations,
Periodica Math. Hungarica 39, 161183 (1999)
 W. Kühnel, Tight embeddings of simply connected
$4$manifolds, Documenta Math. 9, 401412 (2004)
(electronic)
For tight polyhedral submanifolds of higherdimensional octahedra see
 W. Kühnel, Centrallysymmetric tight surfaces and graph embeddings,
Beiträge zur Algebra und Geometrie 37, 347354 (1996)
 F. Effenberger and W. Kühnel, Hamiltonian submanifolds
of regular polytopes, Discrete Comput. Geometry 43, 242262 (2010)
There is a close connection with the subject "Upper and lower bound
theorem",
because kHamiltonian 2kmanifolds in the boundary complex of a cross
polytope belong to both categories, at least in
the centrallysymmetric case. Further progress was made in
 F. Effenberger, Stacked polytopes and tight triangulations of manifolds,
J. Comb. Theory, Ser. A 118, No. 6, 18431862 (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 2kmanifolds 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, 465473 (1987)
 Find tight polyhedral embeddings of the projective planes
over the real, complex numbers, quaternions, or octonions
in the same codimension as the classical Veronesetype 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 Veronesetype embeddings), see
 T.F. Banchoff and W. Kühnel, Tight polyhedral models of
isoparametric families, and PLtaut submanifolds,
Advances in Geometry 7, 613639 (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 socalled Biersphere after work of Thomas Bier).
So far the only known examples are the 6vertex
$RP2$,
the 9vertex $CP2$ and
a number of 15vertex triangulations of an 8manifold, presumably
$HP2$.
The case of a 27vertex triangulation of the octonion plane is still open, see
 U. Brehm and W. Kühnel, $15$vertex triangulations of an $8$manifold,
Math. Annalen 294, 167193 (1992)
Triangulations of the ddimensional torus
There is the following conjecture:
Any combinatorial triangulation of the ddimensional 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, 169176 (1988)
 G. Dartois and A. Grigis, Separating maps of the lattice
$E$_{8}
and triangulations of the eightdimensional torus,
Discrete Comp. Geom. 23, 555567 (2000)
In the case of lattice triangulations the conjecture is proved in
 U. Brehm and W. Kühnel, Lattice triangulations of
$E3$ and of the 3torus ,
Israel J. Math. 189, 97133 (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 4polytopes
with toroidal facets.
When transforming these abstract toroidal facets into solid tori,
a 3manifold can be associated with the polytope, see
 U. Brehm, W. Kühnel and E. Schulte, Manifold structures
on abstract regular polytopes, Aequationes math. 49, 1235 (1995)
The higherdimensional case is still work in progress.

