1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

s=-oo LECTURE 2. FREDHOLM THEORY 1 eigenvalues iof A(s) I s Figure 6. The spectral flow regular crossings the spectral flow is d ...

168 D. SALAMON, FLOER HOMOLOGY Proof of Theorem 2.2 (the index formula): Suppose, without loss of generality, that w(s, t) = w-( ...

LECTURE 2. FREDHOLM THEORY 169 Exercise 2.9. Let x E P(H) and u: B ____, M be as above. Prove that μH(x,A#u) = μH(x, u) - 2c1(A) ...

170 D. SALAMON, FLOER HOMOLOGY as the subset of all those Hamiltonians for which all the periodic solutions x E P(H) are nondege ...

LECTURE 2. FREDHOLM THEORY 171 e^68 f(s) is nonincreasing and this implies f(s) :::; e-o(s-sa) f(s 0 ) for s ;::: s 0. The argum ...

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LECTURE 3 Rational Floer Homology The goal of this lecture is to explain the definition of the Floer homology groups of a Hamilt ...

174 D. SALAMON, FLOER HOMOLOGY Lemma 3.2. For every IE .:J(M,w) there exists a constant n = n(M,w, I) > 0 such that E( v) ;:: ...

LECTURE 3. FLOER HOMOLOGY 175 E(v) ~ n. Now the subsequences (still denoted by u" and v") satisfy the following, for every (smal ...

176 D. SALAMON, FLOER HOMOLOGY + ---X 0 =X Figure 7. Limit behaviour for Floer's connecting orbits with bounded derivatives Proo ...

LECTURE 3. FLOER HOMOLOGY 177 detail in Floer-Hofer [12]. The rough idea is to prove first that the moduli spaces M(x-, x+) are ...

178 D. SALAMON, FLOER HOMOLOGY The Floer homology groups of a regular pair (H, J) are defined as the homology of the chain compl ...

LECTURE 3. FLOER HOMOLOGY 179 3.3. Floer's gluing theorem The goal of this section is to provide more details for the proof of T ...

180 D. SALAMON, FLOER HOMOLOGY Proposition 3.9. Suppose that HE Hreg, x,y,z E P(H), u E M(y,x;H,J), and v E M(z, y; H, J) .-Then ...

LECTURE 3. FLOER HOMOLOGY In particular, DuR 'T/u + DvR 'T/v = DR*'T/ and hence llDR'T/llw1.p < llDuR 'T/ullLP + llDvR *'T/vl ...

182 D. SALAMON, FLOER HOMOLOGY which, for a generic homotopy H°'f3, is a smooth manifold of dimension dim M(x°',xf3;H°'f3) = μHa ...

LECTURE 3. FLOER HOMOLOGY 183 for R > 0 sufficiently large. Then there exists an Ro > 0 such that, for every R > Ro, Ha ...

184 D. SALAMON, FLOER HOMOLOGY 0 1 Figure 10. The parametrized moduli space M^0 (y"'-, yf3; {H~^13 }) Proof of Theorem 3.6: Let ...

LECTURE 3. FLOER HOMOLOGY 185 u y u(Oc () • ) x Figure 11. The isomorphism between Morse homology and Floer homology μH(x, u). I ...

186 D. SALAMON, FLOER HOMOLOGY diverging to infinity. In other words, the moduli space M^1 (x, y; H, J) may not be a finite set ...