On `lacets' and their manifolds
Henry Crapo
C.A.M.S., E.H.E.S.S, Paris
October 21,
4:15pm
refreshments at 3:45pm
2338
ABSTRACT

We study the class of embeddings of a simple closed curve
in closed 2manifolds such that
{\parindent=24pt\parskip=3pt
\item{(1)} there are only transversal double points,
\item{(2)} each residual region of the manifold is a disc, and
\item{(3)} this set of discs is twocolorable, thus furnishing
the vertices and faces of a
combinatorial map (`carte').
}
Such embeddings are characterized up to homotopy by two
combinatorial structures on the set $E$ of crossing points:
{\parindent=24pt\parskip=3pt
\item{(1)} The double occurrence word $\Delta$, or `combinatorial
lacet', giving the order of crossing points along the curve, and
\item{(2)} a bipartition $(K,L)$ of the set $E$, classifying
the type of reentry at each crossing.}
\vskip0pt
This combinatorial information gives rise to a unique map
having the lacet as its diagonal.
Given a combinatorial lacet, on what 2manifolds can it be represented?
Starting from a problem and partial response by Gauss (1840),
and using transformations of vector spaces over $GF(2)$, Rosenstiehl (1976)
and Lins, Richter \& Shank (1987) characterized lacets
representable in the Euclidean
and projective planes, respectively. We show how these
characterizations extend naturally to the case of arbitrary 2manifolds,
and give, in particular, simple procedures for determining whether a lacet is
representable on the torus or on the Klein bottle.
This is joint work with Pierre Rosenstiehl.

Speaker's Contact Info: Henry.Crapo(atsign)inria.fr
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