By Liviu Nicolaescu
This self-contained remedy of Morse thought specializes in purposes and is meant for a graduate path on differential or algebraic topology. The booklet is split into 3 conceptually distinctive elements. the 1st half includes the principles of Morse thought. the second one half comprises purposes of Morse conception over the reals, whereas the final half describes the fundamentals and a few functions of complicated Morse thought, a.k.a. Picard-Lefschetz theory.
This is the 1st textbook to incorporate issues akin to Morse-Smale flows, Floer homology, min-max idea, second maps and equivariant cohomology, and complicated Morse idea. The exposition is greater 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 idea and with the differential and fundamental calculus on tender manifolds.
Some gains of the second one version contain further functions, akin to Morse conception and the curvature of knots, the cohomology of the moduli area 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 workouts, and numerous corrections from the 1st version.
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Extra resources for An Invitation to Morse Theory (2nd Edition) (Universitext)
Loosely speaking, if a point x is not very far from the stationary point 0 of the flow «t , then in one second it cannot travel very far along this flow. 0/ D 0, we deduce from Hadamard’s lemma that D ˇ f 2 m so that uˇ 2 m for all ˇ. t u Denote by J' the ideal in E generated by the germs at 0 of the partial derivatives @x i ', i D 1; : : : ; m. It is called the Jacobian ideal of ' at 0. Since 0 is a critical point of ', we have J' m. Because 0 is a nondegenerate critical point, we have an even stronger result.
W/ D 0, the differential D' W Tw W ! Rm is surjective. Set T1 WD T ; T2 D Tw W; V D Rm ; D1 W D ˚ W T1 ! V; D2 D Dw ˚ W T2 ! V: Note that D˚ D D1 C D2 , z D . D1 C D2 W T1 ˚ T2 ! V /: The lemma is then a consequence of the following linear algebra fact. Suppose T1 ; T2 ; and V are finite dimensional real vector spaces and Di W Ti ! V; i D 1; 2; are linear maps such that D1 C D2 W T1 ˚ T2 ! V is surjective and the restriction of the natural projection P W T1 ˚ T2 ! D1 C D2 / is surjective. Then D2 is surjective.
This algorithmic presentation is known as the Wirtinger presentation. We describe the special case of the (left-handed) trefoil knot depicted in Fig. A] for proofs. The Wirtinger algorithm goes as follows. • Choose an orientation of the knot and a basepoint situated off the plane of the diagram. Think of the basepoint as the location of the eyes of the reader. • The diagram of the knot consists of several disjoint arcs. Label them by a1 ; a2 ; : : : ; a ; in increasing cyclic order given by the above chosen orientation of the knot.
An Invitation to Morse Theory (2nd Edition) (Universitext) by Liviu Nicolaescu