PLANAR LACES

Let S be a connected open subset of 2-sphere S² which is identified with the extended plane R² ∪ {∞}. We assume that S contains the n segments {1, 2, …, n} × [-1, 1]. An n-lace ℓ (in S) is the union ℓ₁ ∪ … ∪ ℓ_n of disjoint simple arcs in S such that ∂ ℓ_i = {(i, 1), (π (i),-1)}, i = 1, …, n, for some permutation π of {1, 2, …, n}.

In this paper we will do present mapping class group description of n-laces, and 1-lace in 1-punctured plane, and 2-laces in S². We will also describe the isotropy subgroup of the trivial lace in .