Vertices: Faces: a = (-2,0,0) abk bgk bgj bij gfj b = (0,0,0) adl akl cdl flm clm c = (1,0,0) hij ehi bci cim fim d = (0,1,0) efi hkl hjl fjl ghk e = (-2,-1,2) f = (1,-1,2) abe bef bcf cfg g = (1,1,2) cgh cdh adh aeh h = (0,3,2) i = (-3/8,0,1/2) j = (1/2,1/4,1) k = (-1/4,7/12,7/6) l = (0,3/4,7/6) m = (1/4,0,1/2)A view of this surface is given below. In this picture, the

To see that this is really the projective plane with one handle, we
compute the Euler characteristic:
since there are 13 vertices, 42 edges, and 28 faces, the Euler
characteristic is *V *- *E *+ *F *= -1, as desired.
Another way to see this is from the triangulation shown below, which
shows the real projective plane (left) with two disks removed (grey)
together with the tube that connects the two holes (right). The
dotted line represents the self-intersection.

To see that it is an immersion, we need to check that the star of each vertex is embedded . This can be seen in the triangulation above, since the double locus does not contain any vertex. As expected, its image is the wedge of three circles, their common point forming the required triple point in the immersion .

To check that it is tight, one must show that all the edges of the convex hull are contained in the surface itself, and that any vertex that is not a vertex of the convex hull of the surface lies inside the convex hull of its neighbors .

To check these conditions, note that the convex hull of the surface is
formed by 7 vertices (*a*, *c*, *d*, *e*,
*f*, *g*, *h*), and the last 8 faces listed above
contain all the edges of the convex hull. Of the remaining 6
vertices, two lie on the straight line segment between two neighbors
(*b* lies on the segment *ac*, *k* lies on the segment
*ag*), two lie within a triangle formed by three neighbors
(*i* lies within triangle *beh*, and *j* within
*bfg*), and two lie inside tetrahedra formed by four neighbors
(*l* lies within *acfh*, and *m* lies within
*cfil*).

The
level sets
for the
height function
in the direction of the *z*-axis closely resemble those
originally given by Kuiper in
[**K2**],
where he describes a tight real projective plane with two handles.

The situation for smooth versions of the projective plane with a handle is markedly different; in fact, no tight immersion is possible in the smooth case . This polyhedral example is significant in that it represents one of only a handful of low-dimensional examples where the smooth and polyhedral theories differ in a significant way. The circumstances that provide for the difference in this specific case still deserve investigation, and should be a source of insight into both areas of study.

* 7/19/94 dpvc@geom.umn.edu -- *

*The Geometry Center*