Proper Morse functions on 3-dimensional manifolds
No. 171, 1975
( Topology of 3-dimensional manifolds)
India ink on paper, 31.5x44 cm.
In this long and indeterminate space, bells hang along lines, suspended from an upward infinity. It is a very deep space, reaching way beyond the visual horizon, far back toward some infinite point. The bells dangle, but in this case they are still, hovering motionless in a breezeless space filled only with a very distant light, perhaps emanating from some cluster of ancient stars. Here again, the mathematical idea of critical points comes into play. Think of each bell as a critical point, in fact a saddle point, of a Morse function on a three-dimensional manifold. Collectively, the critical points are distributed on the manifold in a fairly chaotic way. Yet, it is still possible to deform each Morse function smoothly so that it becomes a proper Morse function, which means that all of the critical points of the same index are located on a single level surface of the function. In the case of three-dimensional manifolds, the idea is that first come the minima, then saddle points of index 1, then saddle points of index 2, and finally the function's maxima. In this image, we see the level of the function where all of the critical points of index 2 are concentrated. But there is a condition—in three—dimensional Euclidean space ( in which this image is cast) it is impossible to show exactly how a two-dimensional separatrix disk will drop downward, the one-dimensional separatrix segment will climb upward, while at the same time keeping the central segment orthogonal to the disk. To do so, we must work in four-dimensional space.
