Singular Surfaces
H2 singularity by Tom
Banchoff
A surface in 3-space can intersect itself transversally. The points of
intersection are called double points. When 3 sheets intersect transversally
in a point, this is called a triple-point. Surfaces in 3-space with
transverse self-intersection, and a finite number of triple-points are
said to be stably immersed. Eg, the Boy surface (1900) (this is
an immersion of the projective plane in 3-space with a single triple-point).
(by J.
Scott Carter)
There are also some non-immersive points called pinch points (also
called Whitney umbrellas, Cayley umbrellas or cross-caps) that lie on the
closure of the double points.
Whitney umbrella from Geometry
Center
A singular surface in 3-space is a stably immersed surface except
at a finite number of pinch points.Eg, Steiner's roman surface (1795) (again,
this is the projective plane in 3-space with a single triple-point and
six pinch points).
Roman surface from Geometry
Center
Singular surfaces appear in many different contexts, for example, as the
image S of a stable mapping near a finite A-determined map-germ
from the plane into 3-space. This object plays an analogous role for mappings
as the Milnor fibre does for function germs. In particular, it is proved
that S is homotopy equivalent to a wedge of spheres and the number of spheres
is an upper bound for the codimension of the map-germ (v. D.
Mond, Springer Lect. Notes in Math. 1462).
The Thing