By Liviu Nicolaescu

This self-contained remedy of Morse concept makes a speciality of purposes and is meant for a graduate path on differential or algebraic topology. The ebook is split into 3 conceptually particular components. the 1st half comprises the principles of Morse idea. the second one half comprises functions of Morse conception over the reals, whereas the final half describes the fundamentals and a few purposes of complicated Morse thought, a.k.a. Picard-Lefschetz theory.

This is the 1st textbook to incorporate issues resembling Morse-Smale flows, Floer homology, min-max idea, second maps and equivariant cohomology, and complicated Morse idea. The exposition is superior with examples, difficulties, and illustrations, and may be of curiosity to graduate scholars in addition to researchers. The reader is anticipated to have a few familiarity with cohomology thought and with the differential and crucial calculus on soft manifolds.

Some good points of the second one version contain further functions, comparable to Morse idea and the curvature of knots, the cohomology of the moduli house of planar polygons, and the Duistermaat-Heckman formulation. the second one version additionally contains a new bankruptcy on Morse-Smale flows and Whitney stratifications, many new routines, and diverse corrections from the 1st variation.

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0/ D 0; xi D X uij y j H) x i 2 J' ; 8i: t u j Set ı WD ' Q, so that 't D ' tı. 16. For every g 2 m, there exists a smooth vector field Vt satisfying (Hg ) for any t 2 Œ0; 1. Moreover, if g 2 m2 we can find a solution Vt of (Hg ) satisfying the initial condition (I) as well. g Proof. We start with some simple observations. Observe that if Vt i is a solution of g g (Hgi ), i D 0; 1, and ui 2 E, then u0 Vt 0 C u1 Vt 1 is a solution of (Hu0 g0 Cu1 g1 ). Since every g 2 m can be written as a linear combination gD m X x i ui ; ui 2 E; i D1 it suffices to find solutions Vti of (Hx i ).

X 1 ; : : : ; x m / W U ! x j /2 : j> We have f jD D c We fix a smooth function Fig. 8). u C uC : W Œ0; 1/ ! 6) Now let (see Fig. t/ D 0g Ä 2": Define F W M ! 2 The Topology of Sublevel Sets 49 μ h ε t 2ε r Fig. u C 2uC /; while on M n D we have F D f . 8. The function F satisfies the following properties. q/; 8q 2 Crf nfpg: fF Ä ag, 8a 2 R, fF Ä c C ıg D ff Ä c C ıg, 8ı ". Proof. M n U /. , q D p. 0/ < c ". Clearly p is a nondegenerate critical point of F . q/.

We can then define another embedding ˚f W M ,! R Rn ; x 7! ˚f / he0 : (b) The Whitney embedding theorem states something stronger: any smooth manifold of dimension m can be embedded as a closed subset of an Euclidean space of dimension at most 2m C 1. We deduce that any smooth manifold admits exhaustive Morse functions. xn /jg ! 0 contains a subsequence convergent to a critical point of f . x/ 2 Tx M with respect to some fixed Riemannian metric on M . 28. A Morse function f W M ! q/. 3 It is possible that a Morse function f constructed in this corollary may be resonant.