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| <math>{\mathbb K}=\Q</math> or <math>{\mathbb K}=\Z_2</math>, with a variable <math>T</math>, | | <math>{\mathbb K}=\Q</math> or <math>{\mathbb K}=\Z_2</math>, with a variable <math>T</math>, |
| <center> | | <center> |
− | <math>\Lambda := \Lambda_{\mathbb K} := \left\{ \sum_{i=0}^\infty a_i T^{\lambda_i} \,\left|\, \R\ni \lambda_i \underset {i\to\infty}{\to} \infty , a_i\in \mathbb{K} \right.\right\}</math> | + | <math>\textstyle |
| + | \Lambda := \Lambda_{\mathbb K} := \left\{ \sum_{i=0}^\infty a_i T^{\lambda_i} \,\left|\, \R\ni \lambda_i \underset {i\to\infty}{\to} \infty , a_i\in \mathbb{K} \right.\right\}</math> |
| </center> | | </center> |
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| Now for two Lagrangians <math>L_0, L_1 \subset M</math> that are transverse (i.e. <math> T_x L_0 \oplus T_x L_1 = T_x M \;\text{for each}\; x\in L_0\cap L_1</math>) and hence have a finite set of intersection points, the natural choice of morphism space is the Floer chain complex | | Now for two Lagrangians <math>L_0, L_1 \subset M</math> that are transverse (i.e. <math> T_x L_0 \oplus T_x L_1 = T_x M \;\text{for each}\; x\in L_0\cap L_1</math>) and hence have a finite set of intersection points, the natural choice of morphism space is the Floer chain complex |
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− | <math>\text{Hom}(L_0,L_1) := \sum_{x\in L_0\cap L_1} \Lambda \, x </math>. | + | <math>\textstyle |
| + | \text{Hom}(L_0,L_1) := \sum_{x\in L_0\cap L_1} \Lambda \, x </math>. |
| </center> | | </center> |
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| When considering the isomorphism space <math>\text{Hom}(L,L)</math> of a fixed Lagrangian <math>L_0 = L_1=L</math>, then the Hamiltonian symplectomorphism <math>\phi</math> can be obtained by lifting a Morse function <math>f: L \to \R</math> to a Hamiltonian function in a Lagrangian neighborhood <math> T^*L \subset M</math>. After extending it suitably, the intersection points can be identified with the critical points, <math>\phi(L)\cap L = \text{Crit} f</math>. Thus in this case we define the morphism space by the Morse chain complex | | When considering the isomorphism space <math>\text{Hom}(L,L)</math> of a fixed Lagrangian <math>L_0 = L_1=L</math>, then the Hamiltonian symplectomorphism <math>\phi</math> can be obtained by lifting a Morse function <math>f: L \to \R</math> to a Hamiltonian function in a Lagrangian neighborhood <math> T^*L \subset M</math>. After extending it suitably, the intersection points can be identified with the critical points, <math>\phi(L)\cap L = \text{Crit} f</math>. Thus in this case we define the morphism space by the Morse chain complex |
− | <center><math>\text{Hom}(L,L) := \sum_{x\in \text{Crit} f} \Lambda \, x </math>.</center> | + | <center><math>\textstyle |
| + | \text{Hom}(L,L) := \sum_{x\in \text{Crit} f} \Lambda \, x </math>.</center> |
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| Each of the morphism spaces is a vector space over the Novikov field <math>\Lambda=\Lambda_{\mathbb K}</math>. | | Each of the morphism spaces is a vector space over the Novikov field <math>\Lambda=\Lambda_{\mathbb K}</math>. |
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| To simplify notation, we will in the following use the universal construction of these vector spaces as | | To simplify notation, we will in the following use the universal construction of these vector spaces as |
| <center> | | <center> |