580 research outputs found
Combinatorial Stokes formulas via minimal resolutions
We describe an explicit chain map from the standard resolution to the minimal
resolution for the finite cyclic group Z_k of order k. We then demonstrate how
such a chain map induces a "Z_k-combinatorial Stokes theorem", which in turn
implies "Dold's theorem" that there is no equivariant map from an n-connected
to an n-dimensional free Z_k-complex.
Thus we build a combinatorial access road to problems in combinatorics and
discrete geometry that have previously been treated with methods from
equivariant topology. The special case k=2 for this is classical; it involves
Tucker's (1949) combinatorial lemma which implies the Borsuk-Ulam theorem, its
proof via chain complexes by Lefschetz (1949), the combinatorial Stokes formula
of Fan (1967), and Meunier's work (2006).Comment: 18 page
The chromatic number of almost stable Kneser hypergraphs
Let be the set of -subsets of such that for all
, we have We define almost -stable Kneser hypergraph
to be the
-uniform hypergraph whose vertex set is and whose edges are the
-uples of disjoint elements of .
With the help of a -Tucker lemma, we prove that, for prime and for
any , the chromatic number of almost 2-stable Kneser hypergraphs
is equal
to the chromatic number of the usual Kneser hypergraphs ,
namely that it is equal to
Defining to be the number of prime divisors of , counted with
multiplicities, this result implies that the chromatic number of almost
-stable Kneser hypergraphs is equal to the
chromatic number of the usual Kneser hypergraphs for any
, namely that it is equal to $\lceil\frac{n-(k-1)r}{r-1}\rceil.
Residue currents with prescribed annihilator ideals
Given a coherent ideal sheaf we construct locally a vector-valued residue
current whose annihilator is precisely the given sheaf. In case is a
complete intersection, is just the classical Coleff-Herrera product. By
means of these currents we can extend various results, previously known for a
complete intersection, to general ideal sheaves. Combining with integral
formulas we obtain a residue version of the Ehrenpreis-Palamodov fundamental
principle
Polytopal complexes: maps, chain complexes and... necklaces
The notion of polytopal map between two polytopal complexes is defined.
Surprisingly, this definition is quite simple and extends naturally those of
simplicial and cubical maps. It is then possible to define an induced chain map
between the associated chain complexes. Finally, we use this new tool to give
the first combinatorial proof of the splitting necklace theorem of Alon. The
paper ends with open questions, such as the existence of Sperner's lemma for a
polytopal complex or the existence of a cubical approximation theorem.Comment: Presented at the TGGT 08 Conference, May 2008, Paris. The definition
of a polytopal map has been modifie
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