570 research outputs found
Embeddings and immersions of tropical curves
We construct immersions of trivalent abstract tropical curves in the
Euclidean plane and embeddings of all abstract tropical curves in higher
dimensional Euclidean space. Since not all curves have an embedding in the
plane, we define the tropical crossing number of an abstract tropical curve to
be the minimum number of self-intersections, counted with multiplicity, over
all its immersions in the plane. We show that the tropical crossing number is
at most quadratic in the number of edges and this bound is sharp. For curves of
genus up to two, we systematically compute the crossing number. Finally, we use
our immersed tropical curves to construct totally faithful nodal algebraic
curves via lifting results of Mikhalkin and Shustin.Comment: 23 pages, 14 figures, final submitted versio
Configuration spaces and Vassiliev classes in any dimension
The real cohomology of the space of imbeddings of S^1 into R^n, n>3, is
studied by using configuration space integrals. Nontrivial classes are
explicitly constructed. As a by-product, we prove the nontriviality of certain
cycles of imbeddings obtained by blowing up transversal double points in
immersions. These cohomology classes generalize in a nontrivial way the
Vassiliev knot invariants. Other nontrivial classes are constructed by
considering the restriction of classes defined on the corresponding spaces of
immersions.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-39.abs.htm
Cubulations, immersions, mappability and a problem of Habegger
The aim of this paper (inspired from a problem of Habegger) is to describe
the set of cubical decompositions of compact manifolds mod out by a set of
combinatorial moves analogous to the bistellar moves considered by Pachner,
which we call bubble moves. One constructs a surjection from this set onto the
the bordism group of codimension one immersions in the manifold. The connected
sums of manifolds and immersions induce multiplicative structures which are
respected by this surjection. We prove that those cubulations which map
combinatorially into the standard decomposition of for large enough
(called mappable), are equivalent. Finally we classify the cubulations of
the 2-sphere.Comment: Revised version, Ann.Sci.Ecole Norm. Sup. (to appear
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