By Graves L. (ed.)

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**Example text**

M. Let X be the set of common zeros for the functions hi. If for each x E X the vectors D hi, i = 1, ... , m, are linearly independent then X is a -manifold of dimension n - m. 5. Rn. Let X EX and denote by¢; a coordinate system for X around x. Then the derivative of ¢;- 1 at cj;(x): is well-defined. The image of IRk under D¢(x)¢- 1 is called: the tangent space (TxX) of X at x. Rn from 0 to x. It is easily proved that the concept of tangent space is independent of the incidental choice of the coordinate system (cf.

F- 1((-oo, -E]) uD; is the identity map, for all t E [0, 1]. f- 1 ([ -E, c]) is moved -at constant speed- along the bounded interval joining (v, w) to (v, s,r · w) E j- 1 ( -E) U D:; see Fig. 3(b) for the case n = 2, k = 1). D 48 MORSE THEORY (WITHOUT CONSTRAINTS) With respect to a suitably chosen local coordinate system around a nondegen-function will take the form of the function erate critical point, an arbitrary fin the preceding example (cf. 2). Therefore results of the type as stated in the next theorem may be expected (the crucial point in the proof being the reduction from the global to the local point of view).

2 is called a homotopy equivalence and a map as h : Y ~ X in this definition is called a homotopy inverse of g. 3 (see Fig. 2). R 3 . Wedefineg: X~ Y;g(x) = (O,x)and h : Y ~ X; h(t, x) = :r. Obviously both g and hare continuous mappings. dy, with F2: [0, 1] x Y ~ Y; F 2(T, (t, x)) = (T · t, :r:) 30 MORSE THEORY (WITHOUT CONSTRAINTS) As a consequence we have: X :::::: Y. 2 Obviously, two homeomorphic topological spaces are also homotopy equivalent. The converse of this statement however is not true.