By Liviu Nicolaescu

This self-contained therapy of Morse idea makes a speciality of purposes and is meant for a graduate path on differential or algebraic topology. The booklet is split into 3 conceptually targeted elements. the 1st half comprises the principles of Morse conception. the second one half includes purposes of Morse idea 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 resembling Morse-Smale flows, Floer homology, min-max concept, second maps and equivariant cohomology, and complicated Morse conception. The exposition is greater with examples, difficulties, and illustrations, and should be of curiosity to graduate scholars in addition to researchers. The reader is anticipated to have a few familiarity with cohomology concept and with the differential and necessary calculus on gentle manifolds.

Some gains of the second one version contain additional functions, akin to Morse concept and the curvature of knots, the cohomology of the moduli area of planar polygons, and the Duistermaat-Heckman formulation. the second one variation additionally incorporates a new bankruptcy on Morse-Smale flows and Whitney stratifications, many new routines, and numerous corrections from the 1st variation.

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**Additional resources for An Invitation to Morse Theory (2nd Edition) (Universitext)**

**Example text**

Tx M , which E. In particular, For every x 2 M we have a smooth map @x f W ! Tx M; 7! 18. (a) We say that the family F W E ! R is sufficiently large relative to the submanifold M ,! E if the following hold: • dim dim M . • For every x 2 M , the point 0 2 Tx M is a regular value for @x f . 19. If F W submanifold M ,! E. 7! x/ 2 is surjective. E ! R is large, then it is sufficiently large relative to any Proof. From the equality @x f D Px @x F , we deduce that @x f is a submersion as a composition of two submersions.

This is known as the canonical framing1 of the knot. It defines a diffeomorphism between a tubular neighborhood U of the knot and the solid torus D2 S 1 . The canonical framing traces the curve ` D `K D f1g S1 @D2 S 1: The curve ` is called the longitude of the knot, while the boundary @D2 f1g of a fiber of the normal disk bundle defines a curve called the meridian of the knot and is denoted by D K . Any other framing of the normal bundle will trace a curve ' on @U Š @D2 S 1 isotopic inside U to the axis K D f0g S 1 of the solid torus U .

The difference between the number of positive crossings and the number of negative crossings. i / ; i . 1) i D1 Set G D GK , where K is the (left-handed) trefoil knot. In this case all the crossings in the diagram depicted in Fig. K/ D 3. Kbb / D x2 1 x3 1 x1 1 ; i . 4) For x 2 G we denote by Tx W G ! G the conjugation g 7! 3/ 1 . , x3 commutes with c D x2 1 x3 1 x1 1 . 2) that G has the presentation G D ha; bj aba D babi: Consider the group H D h x; yj x 3 D y 2 i: We have a map H ! G; x 7! ab; y 7!