Download An Invitation to Morse Theory (2nd Edition) (Universitext) by Liviu Nicolaescu PDF
By Liviu Nicolaescu
This self-contained remedy of Morse conception specializes in functions and is meant for a graduate path on differential or algebraic topology. The publication is split into 3 conceptually particular components. the 1st half includes the rules of Morse thought. the second one half includes purposes of Morse concept over the reals, whereas the final half describes the fundamentals and a few purposes of advanced Morse idea, a.k.a. Picard-Lefschetz theory.
This is the 1st textbook to incorporate subject matters reminiscent of Morse-Smale flows, Floer homology, min-max thought, second maps and equivariant cohomology, and complicated Morse concept. The exposition is better with examples, difficulties, and illustrations, and may be of curiosity to graduate scholars in addition to researchers. The reader is predicted to have a few familiarity with cohomology thought and with the differential and quintessential calculus on gentle manifolds.
Some positive aspects of the second one version comprise additional purposes, comparable to Morse concept 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 variation.
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Additional resources for An Invitation to Morse Theory (2nd Edition) (Universitext)
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.
S/j. s/jds is called the total curvature of the knot, and it is denoted by TK . It measures how “twisted” is the curve K. Large TK signifies that K is very twisted. The above formula shows that if K is very twisted, then the height function hv will have lots of critical points on K. 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 .
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.