Triple Points of Immersions:

Banchoff showed [B2, B3] that the number of triple points in an immersion of a surface is congruent modulo 2 to the Euler characteristic of the surface (provided the immersion is in sufficiently general position). He does this in two different ways, one using singularities of projections and normal Euler classes for smooth surfaces, and one using modifications of surfaces by surgery near the double curves for smooth and polyhedral surfaces.

In the case of the real projective plane, with Euler characteristic equal to 1, any immersion must have at least one triple point. Furthermore, an immersion of the projective plane with any number of handles also has odd Euler characteristic, and so must have a triple point. In particular, any immersion of the real projective plane with one handle must have a triple point.

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8/10/94 -- The Geometry Center