Morse Homology by Matthias Schwarz

By Matthias Schwarz

This ebook offers a hyperlink among smooth research and topology. dependent upon classical Morse thought it develops the finite dimensional analogue of Floer homology which, within the fresh years, has come to play an important function in geometry. Morse homology clearly arises from the gradient dynamical method linked to a Morse functionality. The underlying chain advanced, already thought of by way of Thom, Smale, Milnor and Witten, analogously types the elemental element of Floer's homology thought. this idea of relative Morse concept together with Conley's continuation precept lends itself to an axiomatic homology functor. the current technique consistenly employs analytic equipment in strict analogy with the development of Floers homology teams. that could be a calculus for definite nonlinear Fredholm operators on Banach manifolds which listed here are curve areas and in which the answer units shape the focal moduli areas. The e-book bargains a scientific and finished presentation of the research of those moduli areas. All theorems inside this analytic time table comprising Fredholm concept, regularity and compactness effects, gluing and orientation research, including their proofs and pre-requisite fabric, are tested right here intimately. This exposition hence brings a methodological perception into present-day research.

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This bundle map F: H ~ 0 - L together with the identification Fx = Fwx thus has the explicit representation = [(gt + V7 f) oexpupe] (t), V72 expv_)C)-I. [(gt + V7 f) expv_ p C] (t), V72exPu)O-I. { 0 Fup(e)(t), { FV_p(C)(t), with eE H 1 ,2 Up' CE H 1 ,2 V_ p ' t ~-1 t ~ 1 t ~ -1 t ~ 1 ~r(-00,-11 == ~1(-00,-11 til "'1[1,00) - t = "'1[1,00) ' , and we obtain the identity Fwx (0) = Fo(w x ) Moreover, the fibre derivative D 2 F(X)(0) = DFwx (0) : H x - Lx (~#x()(t) a at + X(t) . 65). #0 as it is given by equation The crucial point of the construction of the gluing map can be found in the process which deduces the linearization of the trajectory gluing from this lin­ ear version of the approximate gluing #0.

Hm}J (};Pn' ~n 1£2 n Xn (+ + l'1m n ,8 'Y-p n ,~n \IL2 Xn lim ((};#Xn 'Y, ~n)£2Xn n 0, as ~n E L;n The proof of the assertion follows. L --=-. L2 82 CHAPTER 2. 5. GLUING 83 (un#~m vn)*TM «Pm,n. h that the isomorphism G satisfies the estimate IIGx~lll,2,x ~ C K ,l 11~llo,2,x for all X E K X [p2' (0), ~ E L~ . Proof. The decisive feature within this proof is the uniformness with respect to the broken trajectories (u, v) E K. An equivalent statement of the assertion via (= G x . ~ E L~ is given by: 1I(1I1,2,x ~ CK,l IID x .

44 yields the pointwise convergence for approriate sub­ sequences. It is worthwhile to mention that at this point we crucially relied on the coercivity of the asymptotically constant homotopy. 60) "y = due to the auxiliary proposition. Thus, each critical point X a E Crit fa can be connected to only finitely many critical points xf3 E Crit ff3 via h af3 -trajecto­ ries. In fact, this will be the reason that the homotopy morphism f3 a , which will be dealt with in Chapter 4, is indeed well-defined.

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