By Liviu Nicolaescu
This self-contained remedy of Morse conception makes a speciality of functions and is meant for a graduate path on differential or algebraic topology. The booklet is split into 3 conceptually precise components. the 1st half comprises the principles of Morse conception. the second one half involves functions of Morse conception over the reals, whereas the final half describes the fundamentals and a few purposes of advanced Morse concept, 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 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 predicted to have a few familiarity with cohomology thought and with the differential and crucial calculus on delicate manifolds.
Some good points of the second one version contain extra functions, reminiscent of Morse conception 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 workouts, and numerous corrections from the 1st version.
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Additional resources for An Invitation to Morse Theory (2nd Edition) (Universitext)
Suppose T1 ; T2 ; and V are finite dimensional real vector spaces and Di W Ti ! V; i D 1; 2; are linear maps such that D1 C D2 W T1 ˚ T2 ! V is surjective and the restriction of the natural projection P W T1 ˚ T2 ! D1 C D2 / is surjective. Then D2 is surjective. Indeed, let v 2 V . t1 ; t2 / 2 T1 ˚T2 such that v D D1 t1 CD2 t2 . On the other hand, since P W K ! t1 ; t20 / 2 K. t2 t20 / H) v 2 Im D2 : t u Using the Morse–Sard–Federer theorem we deduce that the set M of critical values of W Z ! 1).
UO 0 / we can choose j0 . 0 / because j0 is surjective and ' 2 ker j0 . Mp / is generated by 0 ; ' with the relation 1 D j0 . O 0 / D jp . `0 /; jp . 0/ D j0 . Mp / Š Z=p. In fact, Mp is a lens space. 4 (Surgery on the trefoil knot). Suppose that K is a knot in S 3 . Choose a closed tubular neighborhood U of K. The canonical framing of K defines a diffeomorphism U D D2 S 1 . T / ! EK / the inclusion induced morphism. , a simple closed curve that intersects each meridian D @D2 fptg of the knot exactly once.
D1 C D2 / is surjective. Then D2 is surjective. Indeed, let v 2 V . t1 ; t2 / 2 T1 ˚T2 such that v D D1 t1 CD2 t2 . On the other hand, since P W K ! t1 ; t20 / 2 K. t2 t20 / H) v 2 Im D2 : t u Using the Morse–Sard–Federer theorem we deduce that the set M of critical values of W Z ! 1). Thus, for every 2 n M the function f W M ! R is a Morse function. This completes Step 1. 2 Existence of Morse Functions 21 Step 2. M is general. Mk /k 1 of M such that Mk is special 8k 1. We deduce from Step 1 that for every k 1 there exists a negligible set k such that for every 2 n k the restriction of f to Mk is a Morse function.