1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 3. THE LINEAR THEORY 21

subspace L is the image A(Lo) of Lo under the unitary transformation A that

takes 8 ~J. to Uj and lyJ · to Vj· Moreover, A(L 0 ) =Lo exactly when A belongs to

the orthogonal subgroup O(n) C U(n). D

Exercise 3.12. Show that the group Sp(2n) acts transitively on pairs of transver-

sally intersecting Lagrangians.

Lemma 3.13. 7r 1 (.C(n)) = Z.

Proof. The long exact homotopy sequence of the fibration O(n) ---+ U(n) ---+ .C(n)

contains the terms

7r1(0(n))---+ 7r1(U(n))---+ 7r1(.C(n))---+ ?ro(O(n)) = Z/2Z.

It is easy to check that the map

U(n)---+ S^1 : A r--t det(A)

induces an isomorphism on 7T 1 (where det denotes the determinant over C.) On the

other hand 7T 1 ( 0( n)) is generated by a loop in 0(2) and the inclusion 0(2) <---+ U(2)

takes its image in SU(2). Hence the map 7r1(0(n))---+ 7r 1 (U(n)) is trivial. D

It is not hard to check that a generating loop of 7r 1 (.C(n)) is

t r--t (e7ritR) EB R EB··· EB R c en, 0 :St :S l.

The Maslov index

There are several ways to use the structure of the Lagrangian Grassmannian
to get invariants. Typically the resulting invariants are called the "Maslov index".
Here is one way that is relevant when considering the Lagrangian intersection prob-
lem. Suppose we are given two Lagrangian submanifolds Qo, Q 1 in ( M, w) that
intersect transversally. For example Qo might be the zero section of the cotangent
bundle T*Q and Q 1 might be the graph of an exact 1-form df with nondegenerate
zeros. In this case one can assign an index to each transversal intersection point

x E Q 0 n Q 1 by using the usual Morse index for critical points of the function f.

Although this is not possible in the general situation, we will now explain how it is

possible to define a relative index of pairs x+, x _ of intersection points. If one has

chosen a homotopy class of connecting trajectories u, this index takes values in Z.

Here , a connecting trajectory means a map u : D^2 ---+ M such that

(3.1)

u(l) = x+, u(-l) = x_,

u( e7rit) E Q 0 when 0 :S t :S 1,

u( e7rit) E Q 1 when 1 :S t :S 2,

where D^2 is the unit disc in R^2 = C. Let us first see how to use this data to define
a closed loop L(t),O:::; t:::; 4, in .C(n). Note that u(TM) is a symplectic bundle
over the disc and so is symplectically trivial. Choose a trivialization </> : u (TM) ---+
D^2 x R^2 n. Then, define

L(t) = { (u:(Tu(e"" )Qo)), 0 :St :S 1,

<f>(u (Tu(e"i')Q1)), 2 :St :S 3.

For t E [1, 2] choose any path in .C(n) from L(l) to L(2). To complete the loop,

observe that by Exercise 3.12 there is A E Sp(n) such that

A(L(O)) = L(l), A(L(3)) = L(2).