30 D. MCDUFF, INTRODUCTION TO SYMPLECTIC TOPOLOGYcan often prove they exist for all other J.^2 That is exactly what we need here. Here
is an outline of how this works. For more details see [MSl] as well as the Park
City lectures by Salamon.Fredholm theoryLet M(A, .:!) be the space of all pairs (u, J), where u : (S^2 ,j) ___, (M, J) is J-
holomorphic, u*([S^2 ]) = A E H 2 (M), and J E J(w). One shows that a suitable
completion of M(A, J) is a Banach manifold and that the projection
7f : M(A, J) ___, J
is Fredholm of index2(c1 (A)+ n),
where c 1 = c 1 (TM, J). In this situation one can apply an infinite dimensional
version of Sard's theorem (due to Smale) that states that there is a set .:lreg of
second category in J consisting of regular values of 7f. Moreover by the implicit
function theorem for Banach manifolds the inverse image of a regular value is a
smooth manifold of dimension equal to the index of the Fredholm operator. Thus
one finds that for almost every J
7f-^1 (J) = M(A, J)is a smooth manifold of dimension 2(c 1 (A) + n). Moreover, by a transversality
theorem for paths, given any two elements Jo, J 1 E .:lreg there is a path Jt. 0:::; t:::; 1,
such that the union
W = UtM(A,Jt) = 7f-^1 (UtJt)
is a smooth (and also oriented) manifold with boundary8W = M(A, Ji) u - (M(A, Jo)).
It follows that the evaluation map
eVJ: M(A, J) Xe S^2 ~ M, (u, z) f-+ u( z),
is independent of the choice of (regular) J up to oriented bordism.^3 (Here G =
PSL(2, C) is the reparametrization group and has dimension 6.) In particular, ifwe could ensure that everything is compact and if we arrange that ev maps between
manifolds of the same dimension then the degree of this map would be independent
of J.
In the case of the nonsqueezing theorem we are interested in looking at curvesin the class A = [S^2 x pt] in the cylinder S^2 x r^2 n-^2. Thus C1 (A) = 2 since the
normal bundle to S^2 x pt is trivial (as a complex vector bundle with the induced
structure from Jsplit· ) Thusdim(M(A, J) xe S^2 ) =2(c 1 (A) + n) - 6 + 2
=4 + 2n - 6 + 2 = 2n = dim(M).
(^2) An existence theory for J-holomorphic curves had to wait until the r ecent work of Donaldson
and Taubes.
(^3) Two m a ps, ei : Mi --+ X for i = 1, 2, a re said to b e oriented bordant if there is an oriented
manifold W with boundary 8W = M1 U (-M2) and a map e : W --+ X that restricts to ei on
the boundary component M i · Often the compactness that is needed to get any results from this
notion is built into the definition. For example, if all m a nifolds M1, M2, W are compact and if
M1, M 2 have no boundary then bordant m aps ei represent the same homology class.