206 D. SALAMON, FLOER HOMOLOGY
Let us now fix a smooth time dependent Hamiltonian Ht = Ht+I : M ___. JR and
consider a sequence
uv E M(x-,x+;H,J)
of Floer connecting orbits, i.e. solutions of (7) and (8). Let us suppose that the
index of each uv is one, i.e.
μ(uv, H) = 1 = 7/H(x-) - 7/H(x+) + 2rE(uv).
This formula shows that the energy E( uv) is independent of v and hence there
exists a subsequence which converges modulo bubbling (Proposition 3.3). Let us
consider the simplest nontrivial limit configuration with a single J-holomorphic
sphere v : 82 --. M bubbling off, and the sequence of connecting orbits splitting
into a connected sum u#v where u E M(x-, x+; H, J) (see Figure 19).
x x
vu
Figure 19. Bubbling for Floer's connecting orbitsThe homotopy class of the limit configuration u#v must agree with that of u v
for large v, and hence we obtainμ(u, H) + 2c1(v) = μ(uv, H) = 1.
It follows from the Floer-Gromov compactness theorem that the image of v inter-
sects the image of u. Suppose, for example, that μ( u, H) = 2k + 1 and c 1 ( v) = -k.
Then the set of points lying on simple spheres of Chern number -k is the image
of the evaluation map ev : M^5 (-k; J) x 82 / G --. M and hence has codimension
2k +4 for a generic J. On the other hand the moduli space M(x-, x+; H, J)/JR has
dimension 2k near u and hence the points on connecting orbits in M(x-, x+; H, J)
near u form a subset of M of dimension 2k + 2. Comparing this with the above
statement about codimension 2k + 4, we find that, for a generic pair (J, H), the
connecting orbits of index 2k + 1 and the simple J-holomorphic spheres of Chern
number -k will never meet. Hence the above bubbling cannot occur, generically,
provided that vis a simple J-holomorphic sphere. Similar arguments work for arbi-
trarily complicated bubble trees, again under the assumption that the bubble tree
is simple.
Unfortunately, this argument breaks down completely in the case where v is
a multiply covered sphere with negative Chern number. In such a case the actualdimension of the moduli space M(-k; J) will be much bigger than the virtual
dimension, predicted by the index theorem. As a result, we can no longer argue
that the image of the connecting orbit u must be disjoint from the image of v. This
problem can be resolved by means of a perturbation which destroys the multiply
covered J-holomorphic spheres with negative Chern number. The construction of
such perturbations will be explained in the next four sections.