An Invitation to Morse Theory (2nd Edition) (Universitext) by Liviu Nicolaescu

By Liviu Nicolaescu

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

This is the 1st textbook to incorporate subject matters equivalent to Morse-Smale flows, Floer homology, min-max concept, second maps and equivariant cohomology, and complicated Morse conception. The exposition is more advantageous 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 thought and with the differential and essential calculus on soft manifolds.

Some beneficial properties of the second one variation comprise extra functions, corresponding to Morse thought 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 numerous corrections from the 1st variation.

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For each y ∈ Y , there are y , δy > 0 such that B y (x) ∩ Bδy (y) = ∅. Since {Bδy (y) : y ∈ Y } is an open cover for Y , there are y1 , . . , y1 ∈ Y such that Y ⊂ Bδy1 (y1 ) ∪ · · · ∪ Bδyn (yn ). Letting := min{ y1 , . . , yn }, we obtain that B (x) ∩ Y ⊂ B (x) ∩ Bδy1 (y1 ) ∪ · · · ∪ Bδyn (yn ) = ∅ and thus B (x) ⊂ X \ Y . Since x ∈ X \ Y was arbitrary, this means that X \ Y is open. 5. Let (K, dK ) be a compact metric space, let (Y, dY ) be any metric space, and let f : K → Y be continuous. Then f (K) is compact.

This proves (i). For (ii), let (yn )∞ n=1 be a sequence in Y that converges to y ∈ X. Since (yn )∞ converges in X, it is a Cauchy sequence in X and thus in Y . Since Y n=1 is complete, there is y ∈ Y with y = limn→∞ yn . If (yn )∞ n=1 converges to y in Y , it does so in X. Uniqueness of the limit yields that y = y. Hence, y lies in Y . 5 thus yields that Y is closed in X. 6. Let (X, dX ) and (Y, dY ) be metric spaces. We define C(X, Y ) := {f : X → Y : f is continuous} and Cb (X, Y ) := B(X, Y ) ∩ C(X, Y ).

Let Y be the subspace of B(N, F) consisting of those sequences tending to zero. Show that Y is separable. 6. Let (X, d) be a metric space, and let Y be a subspace of X. Show that U ⊂ Y is open in Y if and only if there is V ⊂ X that is open in X such that U = Y ∩ V . 3 Convergence and Continuity The notion of convergence in Rn carries over to metric spaces almost verbatim. 1. Let (X, d) be a metric space. A sequence (xn )∞ n=1 in X is said to converge to x ∈ X if, for each > 0, there is n ∈ N such that d(xn , x) < for all n ≥ n .

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