Topology, Morse theory

Lexicographic shellability is an important tool for studying the topological properties of the order complexes of partially ordered sets. Although, as we shall see in Remark 12.4, discrete Morse theory is more powerful as a method, shellability may still be useful in concrete applications. We take a detailed look at this concept in this section. 

Morse, M. and Cairns, S.S. (1969) Critical Point Theory in Global Analysis and Differential Topology An Introduction, Academic Press, New York, London. 

It is well known that if a smooth function / with nondegenerate critical points, i.e., a Morse function, is given on a smooth manifold Q, then knowing these points and their indices allows us to say much about the topology of the manifold Q. It will be shown in the present chapter that an analogue of this theory exists also in the case where on a symplectic manifold a set of independent functions in involution is given, the number of which is equal to half the dimension of the manifold. 

Since each Bott integral is apparently tame, a trivial inclusion (H ) 3 (H) takes place. In other words, extending the class of considered integrals, we have also extended the class of three-dimensional manifolds which are isoenergy surfaces. From the point of view of three-dimensional topology and its applications to Hamiltonian mechanics, the question is of interest whether or not the classes H ) and [H) coincide. To say it differently, to what extent the assumption about the Bott character is essential in many theorems of the Morse-type theory developed in Chapter 2. 

The Euler-Poincare formula invokes the use of Betti numbers [10] which may be calculated as the count of the number of critical points, of various types, associated with the geometrical structure of nanotori. The theory of Morse flmctions [11] relates critical points to topological structure. We shall show, an alternating sum of Betti numbers defines the Euler characteristic of a torus to be zero. This connects the topology of a nanotorus, nanotube, and plan sheet, which have the same Euler characteristic. We show that for every possible carbon nanotorus there is a geometrical dual boron nanotorus. 

---