By Liviu Nicolaescu
This self-contained remedy of Morse concept specializes in purposes and is meant for a graduate direction on differential or algebraic topology. The ebook is split into 3 conceptually precise elements. the 1st half includes the principles of Morse thought. the second one half includes functions of Morse thought over the reals, whereas the final half describes the fundamentals and a few purposes of advanced Morse conception, a.k.a. Picard-Lefschetz theory.
This is the 1st textbook to incorporate themes comparable to Morse-Smale flows, Floer homology, min-max thought, second maps and equivariant cohomology, and intricate Morse idea. The exposition is more desirable 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 conception and with the differential and critical calculus on soft manifolds.
Some positive aspects of the second one version comprise further purposes, resembling Morse concept 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 routines, and diverse corrections from the 1st version.
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Additional info for An Invitation to Morse Theory (2nd Edition) (Universitext)
P/; 8u 2 U : The map i is an embedding. Set E WD E0 ˚ U and define F WU E ! u; e0 ˚ u / 7! p/, so that Fu jM D u, 8u 2 U . 21. 27. 25 may seem rather special, one can prove that any Morse function on a compact manifold is of the type hv . Indeed, let M be a compact smooth manifold, and f W M ! R be a Morse function on M . Fix an embedding ˚ W M ,! Rn . 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.
X0 /: 26 1 Morse Functions If both M0 and M1 are oriented, then we can give an alternate description of the Jacobian. M1 /. There exists a smooth function wF W M0 ! x0 /j; 8x0 2 M0 : Observe that if F W M0 ! M1 is a smooth map between compact smooth manifolds of identical dimensions, then Sard’s theorem implies that for almost every x1 2 M1 is a regular value of F . x1 / is a finite set. x1 / 2 Z 0 [ f1g its cardinality. We can now formulate the very special case of the coarea formula that we need in this section.
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.
An Invitation to Morse Theory (2nd Edition) (Universitext) by Liviu Nicolaescu