1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 5. MULTI-VALUED PERTURBATIONS

Perturbations and marked points

Consider the space

B = B(A, k) = MapA(8^2 , M) x ((8^2 )k - ~k)

219

whose elements are tuples ( u, z) = ( u, z 1 , ... , Zk), where u : 82 ---+ M is a smooth

map representing the homology class A E H 2 (M,'ll) and z 1 , ... ,zk are pairwise

distinct points in 82. Denote by E. ---+ B vector bundle with fibers

E.u,z = {TJ E n°•^1 (8^2 ,u*TM): T) = 0 near Zi for i = 1, ... ,k}.

We shall consider multi-valued perturbations r : B ---+ 2£ which satisfy all the

axioms of Section 5.2. The "equivariance" axiom now takes the form

(49)

The definition of E. guarantees that each solution (u, z 1 , ... , zk) of the equation

(50)

is an unperturbed J-holomorphic curve in some neighbourhood of the marked

points.

Stable Maps

Fix a tree T and consider the space B(A; T, k) of all stable maps

(u, z) = ( {uo:}o:ET, {zo:,e}o:e,e, {ai, zi}i=l, ... ,k),

modelled over the tree T , which satisfy

Here the u°' : 82 ---+ M are arbitrary smooth functions, J-holomorphic or not, and

the stability condition takes the form #Zo: 2: 3 whenever u°'*[8^2 ] = 0 E H 2 (M,'ll).

The space B = B(A; T , k) is an infinite dimensional manifold whose tangent space
T(u,z)B consists of tuples

( {~o:}o:ET, {(o:,B}o:E,B, {(i}i=l, ... ,k)

with~°' E C^00 (8^2 ,uo:*TM), (o:,e E TZa(38^2 , (i E Tz; 82 , which satisfy

~o:(zo:,e) + duo:(zo:,e)(o:,e = ~13(z130:) + du13(z130:)({30:·

For a tree T the perturbed equations will take the form

(51)

where

Z°' = {zo:f3 : aE{J} U {zi : ai =a}

(as in Section 4.2). We claim that r and J can be chosen such that the space of

solutions of (51) is an oriented branched manifold of the predicted dimension. The
details are as in [31] and are carried out in [22].
The space
B(T,k) = LJB(A; T ,k)
A