Morse theory

In Chapter 5, we have studied Morse theory on a symplectic manifold X given by an action of a compact torus T. As noted there, when X is a Kahler manifold, the gradient flow is given by the associated holomorphic action of the complexification T of T. Hence, the stable and the unstable manifolds can be expressed purely in terms of the group action. 

Now we shall calculate the Poincare polynomial. In order to apply the general argument of Morse theory, we need to find an appropriate iSCinvariant Kahler metric. Though we believe that there exists a natural Kahler metric on -we do not know how to construct... 

In this chapter we shall calculate the Poincare polynomial of (C2). This was first accomplished by Ellingsrud and Strpmme [14]. They have used the Bialynicki-Birula decomposition associated with the natural torus action on (C2), and then compute the Poincare polynomial using the Weil conjecture. Our approach is essentially the same, but we use Morse theory instead of the Weil conjecture. 

With this purpose in mind, we have developed a new specific Morse-type theory of integrable Hamiltonian systems which is distinct from the ordinary Morse theory and from the Bott theory of functions with degenerate singularities [172]. 

A Short List of the Basic Data From the Classical Morse Theory... 

An important stage in the construction of the ordinary Morse theory is the well-known Morse lemma. It asserts that in some open neighbourhood of a nondegenerate critical point Xq of a Morse function /, there always exist local regular coordinates yi,..., ym such that the function / is written in the form... 

Proof The analogue to this Lemma in the ordinary Morse theory is well known, but in our case the proof is more delicate, since here we deal with the integral f (and not merely with a smooth function), and therefore we should essentially use the conditons of Theorem 2.1.3. An arbitrary smooth perturbation of an integral... 

Fomenko, A. T. The Morse theory of integrable Hamiltonian systems. Dokl. Akad. Nauk SSSR, 287 (1986) No. 5, 1071-1075. 

Morse Theory, Betti Numbers, and Boron Nanotori... 

The classical reference for Theorem 9.6 is the book Stratified Morse Theory of Goresky and MacPherson, [GoM88]. 

Beyond the encoding of all allowed collapsing orders as the set of linear extensions of the universal object U (P, M), viewing the posets with small fibers as the central notion of the combinatorial part of discrete Morse theory is also invaluable for the structural explanation of a standard way to construct acyclic matchings as unions of acyclic matchings on fibers of a poset map. 

The Main Theorem of Discrete Morse Theory for CW Complexes... 

Main theorem of discrete Morse theory for CW complexes)... 

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