19,172 research outputs found
Spanning surfaces in 3-graphs
We prove a topological extension of Dirac's theorem suggested by Gowers in
2005: for any connected, closed surface , we show that any
two-dimensional simplicial complex on vertices in which each pair of
vertices belongs to at least facets contains a homeomorph of
spanning all the vertices. This result is asymptotically sharp,
and implies in particular that any 3-uniform hypergraph on vertices with
minimum codegree exceeding contains a spanning triangulation of the
-sphere.Comment: 33 pages, 6 figure
Computing trisections of 4-manifolds
Algorithms that decompose a manifold into simple pieces reveal the geometric
and topological structure of the manifold, showing how complicated structures
are constructed from simple building blocks. This note describes a way to
algorithmically construct a trisection, which describes a -dimensional
manifold as a union of three -dimensional handlebodies. The complexity of
the -manifold is captured in a collection of curves on a surface, which
guide the gluing of the handelbodies. The algorithm begins with a description
of a manifold as a union of pentachora, or -dimensional simplices. It
transforms this description into a trisection. This results in the first
explicit complexity bounds for the trisection genus of a -manifold in terms
of the number of pentachora (-simplices) in a triangulation.Comment: 15 pages, 9 figure
Colouring the Triangles Determined by a Point Set
Let P be a set of n points in general position in the plane. We study the
chromatic number of the intersection graph of the open triangles determined by
P. It is known that this chromatic number is at least n^3/27+O(n^2), and if P
is in convex position, the answer is n^3/24+O(n^2). We prove that for arbitrary
P, the chromatic number is at most n^3/19.259+O(n^2)
Special Lagrangian torus fibrations of complete intersection Calabi-Yau manifolds: a geometric conjecture
For complete intersection Calabi-Yau manifolds in toric varieties, Gross and
Haase-Zharkov have given a conjectural combinatorial description of the special
Lagrangian torus fibrations whose existence was predicted by Strominger, Yau
and Zaslow. We present a geometric version of this construction, generalizing
an earlier conjecture of the first author.Comment: 23 pagers, 10 figure
- …