{TALK 
        {"October 21}
        {"On `lacets' and their manifolds}
        {"Henry Crapo}
        {"C.A.M.S., E.H.E.S.S, Paris}
        {"Henry.Crapo@inria.fr}
        {"We study the class of embeddings of a simple closed curve 
in closed 2-manifolds 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 two-colorable, 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 2-manifolds 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 2-manifolds, 
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.
}