By Liviu Nicolaescu

This self-contained therapy of Morse thought specializes in functions and is meant for a graduate path on differential or algebraic topology. The publication is split into 3 conceptually distinctive elements. the 1st half comprises the principles of Morse idea. the second one half includes functions of Morse idea over the reals, whereas the final half describes the fundamentals and a few functions of advanced Morse concept, a.k.a. Picard-Lefschetz theory.

This is the 1st textbook to incorporate themes reminiscent of Morse-Smale flows, Floer homology, min-max conception, second maps and equivariant cohomology, and intricate Morse thought. The exposition is more suitable with examples, difficulties, and illustrations, and may be of curiosity to graduate scholars in addition to researchers. The reader is predicted to have a few familiarity with cohomology thought and with the differential and vital calculus on delicate manifolds.

Some positive factors of the second one variation comprise extra purposes, similar to Morse idea and the curvature of knots, the cohomology of the moduli area of planar polygons, and the Duistermaat-Heckman formulation. the second one version additionally encompasses a new bankruptcy on Morse-Smale flows and Whitney stratifications, many new workouts, and numerous corrections from the 1st variation.

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J ' J+l D S j €1 ' S cu . )|2d? )|2d§ . (X'1(x))a(\"1(x)) = xa(X^(x)) = xa(x, j) . 7. ) is constant for some n . If there is point spectrum, then the corresponding eigenspace is infinite dimensional. 7. »-J Chapters 3 and 4 are devoted to the analytic properties of the family of operators case. Lp . In chapter 3 we will show that The analysis is quite simple. parabolic case L- is holomorphic on Le is entire in the elliptic In chapter 4 we will prove that in the (0,2TT) . The singularities at 0 and 2TT are not removable.

2d? )|2d§ . (X'1(x))a(\"1(x)) = xa(X^(x)) = xa(x, j) . 7. ) is constant for some n . If there is point spectrum, then the corresponding eigenspace is infinite dimensional. 7. »-J Chapters 3 and 4 are devoted to the analytic properties of the family of operators case. Lp . In chapter 3 we will show that The analysis is quite simple. parabolic case L- is holomorphic on Le is entire in the elliptic In chapter 4 we will prove that in the (0,2TT) . The singularities at 0 and 2TT are not removable.

In the sequel we will refer to these as classes of type I, II and III respectively. 10) : T h(kT ) 2 2 k=1| kT T/2 i t y T / 2 e e c o s k ? -kTT/2 - i H T / 2 | -e e - 2 ikn? 2 T h(kT ) -e V gT gT r v-i „ kT /2 ikT(j> /2 -kT /2 -ike)) /2 k= 1 r m n •» i n ' n i^ mn mn IgT i gT e gT e _e gT gT e • J2 Observe that the first sum on the right hand side is independent of ? The { T : g € T } appearing in this sum are the lengths of the closed 3 geodesies on E /T . Each such length is represented by infinitely many 3 distinct closed geodesies on E fT .