By Liviu Nicolaescu

This self-contained remedy of Morse conception specializes in purposes and is meant for a graduate path on differential or algebraic topology. The e-book is split into 3 conceptually detailed elements. the 1st half includes the principles of Morse concept. the second one half contains purposes of Morse idea over the reals, whereas the final half describes the fundamentals and a few functions of advanced Morse idea, a.k.a. Picard-Lefschetz theory.

This is the 1st textbook to incorporate issues resembling Morse-Smale flows, Floer homology, min-max conception, second maps and equivariant cohomology, and intricate Morse concept. 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 necessary calculus on tender manifolds.

Some positive aspects of the second one variation comprise extra functions, similar to Morse conception and the curvature of knots, the cohomology of the moduli house of planar polygons, and the Duistermaat-Heckman formulation. the second one variation 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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**Additional info for An Invitation to Morse Theory (2nd Edition) (Universitext)**

**Sample text**

The identity mapping of E' into E is continuous. For, the identity mapping of E' into E' is obviously continuous. 15. 2. 1. Let E l' E 2 , . . , En be topological spaces. We are going to define a natural topology on E = El X E2 X ... x En. •. X Un, where Vi is an open subset of E j • Let us call open set in E any union of elementary open sets. 1. First of all, E = El X E2 X ... x En 0 and = 0 X E2 x En X ... are open sets, even elementary open sets. Axiom (ii) is obvious. Finally, let A, B be open subsets of E and let us show that A n B is an open subset of E.

There exists an open subset Vi of Ei such that Xi E Vi C Vi. Then and U 1 x ... inE. X Vn is open in E, thus VI x ... 2. Finite Products of Topological Spaces Let V be a neighborhood of x in E. There exists an open subset V of E such that x EVe V. The set V is the union of elementary open sets, therefore x belongs to one of these sets, say VI X ••• X Vn' where Vi is an open subset of E i . Then XiEVi, thus Vi is a neighborhood of Xi in E i , and VI x ... X Vn cV. 5. Theorem. Let E = EI X ... x En be a product of topological spaces.

J,,(t», where /; is a mapping of T into E i . The following conditions are equivalent: ~ (i) f is continuous; (ii) fl' ... ,f,. are continuous. Indeed, for every to E T, the conditions lim f(t) = f(to), lim j;(t) = j;(t o) for i = 1, ... 6. 8. Corollary. •. x En be a product of topological spaces. The canonical projections of E onto E l , ... , En are continuous. Let fbe the identity mapping ofE. It is continuous. Now, it is the mapping x H (f1(X), ... , fn{x», where ft, ... , fn are the canonical projections of E onto E t , ...