by Thomas Banchoff and Floris Takens

This paper addresses the question: Under what conditions
is there an immerstion or an embedding *f*: M^2 -> E^3
of a closed surface M^2 into Euclidean 3-space such that there is
a linear function on E^3, *z*:E^3 -> **R**, so that the
composition *zf*: M^2 -> **R** has exactly three critical
points, one of which may be degenerate? This problem for a
smooth embedding *f* was proposed by H. Hopf
in [6, p. 92]. A three critical point (3cp) smooth embedding
is only possible in the case of a 2-sphere embedded as a "shoe
surface" (see below).

When smooth immersions or polyhedral embeddings or immersions are considered, the results differ considerably. The following results are proven in this paper.

**Theorem 1.** *There is a smooth *3cp* immersion of
the torus but there is no smooth *3cp* immersion for any
orientable surface of a genus greater than one.*

**Theorem 2.** *There is a smooth *3cp* immersion of
any nonorientable surface. *

These immersions can be approximated by polyhedral 3cp immersions but in the polyhedral case we have an entirely different phenomenon:

**Theorem 3.** *For any orientable surface except the sphere
there is a polyhedral *3cp* embedding into E^3.*

These problems bear resemblance to the factorization
problems studied by Haefliger in [3], where he answered
the question of when an "excellent" mapping
*k*: M^2 -> E^2 could be expressed as the composition
of an immersion *f*: M^2 -> E^3 and the orthogonal projection
*p*: E^3 -> E^2. Since for any surface M^2 there exist
*h*: M^2 -> **R** with exactly three critical points, our
work may be thought of as an attempt to factorize some such
*h* as an immersion or embedding *f*: M^2 -> E^3
followed by an orthogonal projection into a line.

Section 5 gives examples of 3cp polyhedral embeddings which have been used as motivation for a study of the extensions of the idea of the degree of the Gauss spherical mapping to polyhedral embeddings in [1].

In Section 1 of this paper deals with arbitrary functions *h*
with 3cp on surfaces. In Section 2 we examine those
functions of the form *h = zf* and the rotation number
results (which we need in Section 3 to prove Theorem 1)
are derived. Section 4 gives the proof of Theorem 2, and discusses
smooth 3cp immersions of nonorientable surfaces.
The
last section, which contains the proof of Theorem
3, deals with polyhedral embeddings and can be read directly
after Section 1.

The following illustration (drawn by Cassidy Curtis) shows an immersion of the torus in three space that has exactly three critical points.