1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 1 Symplectic Fixed Points and Morse Theory 1.1. The Arnold conjecture Let ( M, w) be a compact symplectic manifold. The ...

148 D. SALAMON, FLOER HOMOLOGY where Hi(M, Q) denotes the singular homology of M with rational coefficients. In contrast, the Le ...

LECTURE 1. SYMPLECTIC FIXED POINTS AND MORSE THEORY 149 1.2. The monotonicity condition An almost complex structure J on TM is c ...

150 D. SALAMON, FLOER HOMOLOGY Definition 1.6. Let ( M, w) be a compact symplectic manifold. Then the minimal Chern number of (M ...

LECTURE l. SYMPLECTIC FIXED POINTS AND MORSE THEORY 151 of points in M whose gradient lines connect y to x (the space of connect ...

152 D. SALAMON, FLOER HOMOLOGY the free abelian group generated by the critical points of f. This complex is graded by the Morse ...

LECTURE 1. SYMPLECTIC FIXED POINTS AND MORSE THEORY 153 of the Morse-Smale-Witten homology class [l::i m i xi] E HM* ( M, f ; Z) ...

154 D. SALAMON, FLOER HOMOLOGY Figure 3. A Morse-Smale gradient flow on the 2-torus Figure 4. A Morse-Smale gradient flow on the ...

LECTURE 1. SYMPLECTIC FIXED POINTS AND MORSE THEORY 155 Exercise 1.19. Prove that the I-form iI! His closed. Hint: Consider a 2- ...

156 D. SALAMON, FLOER HOMOLOGY The construction of the Floer homology groups relies on a careful analysis of the gradient flow l ...

LECTURE 1. SYMPLECTIC FIXED POINTS AND MORSE THEORY 157 converges to zero uniformly as s -t ±oo.^3 Hence OtU - Xt(u) converges t ...

158 D. SALAMON, FLOER HOMOLOGY The meaning of this result is not only that, by accident, M(x-, x+) is a smooth manifold, but tha ...

2.1. Fredholm operators LECTURE 2 Fredholm Theory Let X and Y be Banach spaces. A bounded linear operator D : X ----t Y is calle ...

160 D. SALAMON, FLOER HOMOLOGY and, since df(x) is onto, the dimension of M agrees with the index off. 2.2. The linearized opera ...

LECTURE 2. FREDHOLM THEORY 161 The limit matrices are symmetric and hence, modulo some compact perturbation, we may as well assu ...

162 D. SALAMON, FLOER HOMOLOGY denotes the Laplace operator. Once this is established, the proof of Lemma 2.3 is an easy exercis ...

LECTURE 2. FREDHOLM THEORY 163 for ( E W^1 ·P([-l, 2] x S^1 ). Moreover, if ( E W^1 •^2 and D( E W 1 ~;,1', then ( E w,k+l,p Joe ...

164 D. SALAMON, FLOER HOMOLOGY for every ( E C 0 (JR x S^1 , JR^2 n). Since C 0 is dense in W^1 ,P, this estimate continues to h ...

LECTURE 2. FREDHOLM THEORY 165 Here D* = -8 8 + Jo8t + 8 denotes the formal adjoint operator and the last equality holds since D ...

166 D. SALAMON, FLOER HOMOLOGY where sign S is the signature (the number of positive minus the number of negative eigenvalues). ...