Top Sets and Top Cycles:

The study of top sets and top cycles is critical to an understanding of tight immersions.

A fundamental property of top sets is that a top set of a tight immersion is itself a tight immersion [K4]. A surprising result is that a top set need not be an embedding, as shown below.

This is the topset for a three-handled torus: the torus itself is formed by taking the boundary of the figure (black line) cross an interval and pasting a copy of the top set at each end of the tube so formed. The outer ring forms a torus, and the two cross pieces form two additional handles.

In general, the interesting top sets are those containing the top cycles. For smooth surfaces, distinct top sets must be disjoint, but for polyhedral surfaces, different top sets can share points or even edges, so the top cycles need not be distinct.

A top cycle has the following important properties:

It is planar.
It is embedded.
It is convex.
It does not bound a region of the surface (that is, it is a non-trivial 1-cycle).
It has an orientable neighborhood.

To see that the last is true, note that, in the smooth case, since the remainder of the surface lies on one side of the support plane for the top cycle, the surface is tangent to the plane, so for each point of the top cycle, the surface is locally the graph of a function with respect to the plane; thus in a small enough neighborhood of the top cycle, the projection of the surface onto the plane is an embedding. Since a Möbius band can not be embedded in the plane, the neighborhood of the top cycle must be orientable.

Another way to see this is to consider the supporting plane for the top cycle, and imagine pushing the plane down (through the surface). As it moves, it intersects the surface in two curves, one on each side of the top cycle. These curves trace out a neighborhood that is an orientable strip (since it has two boundary curves). This holds for both smooth and polyhedral surfaces.

The Number of Top Cycles:

Cecil and Ryan [CR1] showed that the number of top cycles of a tight immersion (of a connected surface other than a sphere) is always at least 2 and no more than 2 - X(M), where X(M) is the Euler characteristic of the surface.

The lower bound is easy to see (using the decomposition of the surface into M+ and M- regions ): if a tight immersion of a connected surface has no top cycles, then it is necessarily a sphere, since in this case the image of the M+ region is all of the convex envelope (and there is no M- region). So a tight immersion of any surface other than the sphere must have at least one top cycle. On the other hand, it can't have just one, since such a top cycle would bound a region (namely, the region M+), so there must be at least two top cycles.

The upper bound comes from the fact that the Euler characteristic is tied to the genus of the surface, and the genus limits the number of tubes that can be used to attach the interior M- region to the outer M+ region along the top cycles.

Tightness and polar height functions

Tightness and the convex hull

Tightness and its consequences

7/22/94 dpvc@geom.umn.edu -- The Geometry Center