# 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.

*The polyhedral solution*

*Introduction*

* 8/10/94 dpvc@geom.umn.edu -- *

*The Geometry Center*