91,411 research outputs found
Edge Clique Cover of Claw-free Graphs
The smallest number of cliques, covering all edges of a graph , is
called the (edge) clique cover number of and is denoted by . It
is an easy observation that for every line graph with vertices,
. G. Chen et al. [Discrete Math. 219 (2000), no. 1--3, 17--26;
MR1761707] extended this observation to all quasi-line graphs and questioned if
the same assertion holds for all claw-free graphs. In this paper, using the
celebrated structure theorem of claw-free graphs due to Chudnovsky and Seymour,
we give an affirmative answer to this question for all claw-free graphs with
independence number at least three. In particular, we prove that if is a
connected claw-free graph on vertices with , then and equality holds if and only if is either the graph of
icosahedron, or the complement of a graph on vertices called twister or
the power of the cycle , for .Comment: 74 pages, 4 figure
The Lambrechts-Stanley Model of Configuration Spaces
We prove the validity over of a commutative differential graded
algebra model of configuration spaces for simply connected closed smooth
manifolds, answering a conjecture of Lambrechts--Stanley. We get as a result
that the real homotopy type of such configuration spaces only depends on the
real homotopy type of the manifold. We moreover prove, if the dimension of the
manifold is at least , that our model is compatible with the action of the
Fulton--MacPherson operad (weakly equivalent to the little disks operad) when
the manifold is framed. We use this more precise result to get a complex
computing factorization homology of framed manifolds. Our proofs use the same
ideas as Kontsevich's proof of the formality of the little disks operads.Comment: 61 pages. To appear in Inventiones Mathematica
A model for configuration spaces of points
The configuration space of points on a -dimensional smooth framed manifold
may be compactified so as to admit a right action over the framed little
-disks operad. We construct a real combinatorial model for these modules,
for compact smooth manifolds without boundary
Configuration Spaces of Manifolds with Boundary
We study ordered configuration spaces of compact manifolds with boundary. We
show that for a large class of such manifolds, the real homotopy type of the
configuration spaces only depends on the real homotopy type of the pair
consisting of the manifold and its boundary. We moreover describe explicit real
models of these configuration spaces using three different approaches. We do
this by adapting previous constructions for configuration spaces of closed
manifolds which relied on Kontsevich's proof of the formality of the little
disks operads. We also prove that our models are compatible with the richer
structure of configuration spaces, respectively a module over the Swiss-Cheese
operad, a module over the associative algebra of configurations in a collar
around the boundary of the manifold, and a module over the little disks operad.Comment: 107 page
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