Surfaces in Three-Manifolds and Non-Singular Equations in Groups.

John R. Stallings

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The present paper is the notes of a mini-course addressed mainly to non-experts. Its purpose is to provide a first approach to the theory of mapping class groups of non-orientable surfaces.

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Singular projections of generic 2-dim surfaces in ℝ³ with singular boundary to 2-space are studied. The case of projections of surfaces with nonsingular boundary has been treated by Bruce and Giblin. The aim of this paper is to generalise these results to the simplest singular case where the boundary of the surface consists of two transversally intersecting lines. Local models for germs of generic singular projections of corank ≤ 1 and codimension ≤ 3 are given. We also present geometrical...