By Percival Frost

This obtainable therapy covers orders of small amounts, types of parabolic curves at an enormous distance, kinds of curves locally of the foundation, and types of branches whose tangents on the foundation are the coordinate axes. extra themes contain asymptotes, analytical triangle, singular issues, extra. 1960 variation.

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First we shall define a good collection of complexes on which one can define Morse functions, and then give the definitions of ascending and descending links. 1 (Affine cell complexes). A finite-dimensional cell-complex X is said to be an affine cell-complex if it is equipped with the following structure. An integer m dim(X) is given, and for each cell e of X we are given a convex polyhedral cell Ce ⊂ Rm and a characteristic function χe : Ce → e such that the restriction of χe to any face of Ce is a characteristic function of another cell, possibly precomposed by a partial affine homeomorphism (that is, the restriction of an affine homeomorphism) of Rm .

This is a recursive condition to check, because the links of vertices are piecewise spherical complexes of lower dimension. In order to see that these links are CAT(1), one needs to verify two things: 1. Check that the links are locally CAT(1), via a repeated application of the link condition to these piecewise spherical complexes. 2. Check that the links do not contain any geodesic loops of length less than 2π. In practice it is very hard to implement the link condition. The hard part occurs in checking part 2 above; namely, in showing that an arbitrary piecewise 34 Chapter 2.

This is a very fundamental viewpoint to which we shall return repeatedly in later sections. We summarize the ideas here. 1. We are given a real-valued Morse function f on a contractible cell complex X. The contractibility of X is often a consequence of some type of non-positive curvature, such as the fact that X might be CAT(0) or CAT(−1). 2. Suppose that there is a group G acting cocompactly by deck transformations on the complex X. Suppose that there is an epimorphism ϕ : G → Z. Let Z act cocompactly by translations on R, and suppose that the map f : X → R is ϕ-equivariant with respect to the action of G on X and of Z on R.

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