By Liviu Nicolaescu
This self-contained therapy of Morse concept specializes in functions and is meant for a graduate direction on differential or algebraic topology. The publication is split into 3 conceptually precise elements. the 1st half comprises the rules of Morse thought. the second one half contains functions of Morse thought 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 subject matters equivalent to Morse-Smale flows, Floer homology, min-max conception, second maps and equivariant cohomology, and intricate Morse concept. 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 anticipated to have a few familiarity with cohomology conception and with the differential and essential calculus on gentle manifolds.
Some beneficial properties of the second one variation contain additional functions, similar 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 variation additionally incorporates 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)
In [M0], the number 1 cK D N K 2 was called the crookedness of the knot. v/ is the number of local minima of the Morse function hv . 15) Here are some interesting consequences. 34. For any knot K ,! E , we have TK 2 . Proof. 14) implies TK 2 . 35. If K is planar and convex, then TK D 2 . Proof. Since K is convex, any local minimum of a height function must be an absolute minimum. Thus, any Morse height function will have a unique local minimum. This implies that the crookedness of K is 1. 15). 36.
K; K 0 /. When we perform a p-surgery on K, we remove the solid torus U D D2 S 1 and we replace it with a new solid torus UO D S 1 D2 , so that in the new manifold the attaching curve Kp D ` C p will bound the disk f1g D2 UO . Let us look at a very simple yet fundamental example. z0 ; z1 / 2 C2 I jz0 j2 C jz1 j2 D 2 : 1 Its homotopy class is indeed independent of the choice of the Seifert surface X. 1 Surgery, Handle Attachment, and Cobordisms 37 K0 K1 Fig. z0 ; z1 / 2 S 3 I jzi j Ä 1g, i D 0; 1.
K 1/-dimensional sphere ˙ ,! @X embedded in @X with trivializable normal bundle T˙ @X . This normal bundle has rank m k D dim @X dim ˙. • A framing ' of the normal bundle T˙ @X . The framing defines a diffeomorphism from Dm k S k 1 to a tubular neighborhood N of ˙ in @X . Using this identification, we detect inside N a copy of @ Hk;m D ˙ Dm k . ˙; '/. 1 Surgery, Handle Attachment, and Cobordisms 43 Σ Fig. 5 Attaching a 2-handle of dimension three x+ x_ H X X+ X Fig. 6 Attaching a 1-handle of dimension two and smoothing the corners The manifold X C has corners, but they can be smoothed out (see Fig.
An Invitation to Morse Theory (2nd Edition) (Universitext) by Liviu Nicolaescu