The Floer cohomology group $HF^{*}(L_{2},L_{1};\mathbb{C}\,)$ [FO3] is the cohomology of a cochain complex made from a copy of $\mathbb{C}\,$ for each intersection point of two graded Lagrangians in general position. The differential is defined by counting holomorphic strips, with boundary in the Lagrangians, running from one intersection point to another. It is a symplectic refinement of the topological intersection theory of $L_{1},\,L_{2}$, and as such is invariant only under hamiltonian deformations of the $L_{i}$. What is important to us is the grading of a particular transverse intersection point, as defined in [S2], [FO3].

While this can be defined completely topologically, it is most easily (and equivalently) defined via a complex structure. Again we work at the level of tangent spaces, pick local coordinates and, without loss of generality, take $L_{2}$ to have phase 0 and to be the $x$-axes: $L_{2}=\{y_{i}=0\}$. Write $L_{1}$ as

$$L_{1}=\{z_{i}=re^{i\alpha_{i}}\},$$

where the $\alpha_{i}$s are all in $(0,\pi)$. Then $\sum\alpha_{i}=\theta_{p}(L_{1})$ mod $\pi$, and the following integer is the definition of the Floer index of the point $p$:

$$\ind_{p}(L_{2},L_{1}):={1\over\pi}\left(\sum\alpha_{i}-\theta_{p}(L_{1})\right).$$

(3.4)

Notice therefore that $\ind_{p}(L_{2},L_{1})+\ind_{p}(L_{1},L_{2})=n$. Applying the definition (3.4) to the connect sums defined in the last section (for which $\theta_{p}(L_{1})=\sum\alpha_{i}-\pi$), we recover a result of Seidel [S2]:

$$L_{1}\#L_{2}\text{\emph{ exists as a graded connect sum if and only if }}\ind_{p}(L_{2},L_{1})=1.$$

(3.5)

(The only if part follows from the independence of gradings and the Floer index from the complex structure; we may therefore pick the complex structure locally to have the form of the local model above.) Given $L_{1}$ there is at most one choice of the grading on $L_{2}$ such that $\ind_{p}(L_{2},L_{1})=1$ at all intersection points $p$, so that $L_{1}\#L_{2}$ can be graded.

In fact connect sums $L_{1}\#L_{2}$ whose own Floer cohomology is well defined [FO3] should correspond to elements of $HF^{1}(L_{2},L_{1})$, mirror to the fact that extensions of sheaves $0\to E_{1}\to E\to E_{2}\to 0$ correspond to elements of Ext${}^{1}(E_{2},E_{1})$, as discussed in [Th].

We can also deal with the connect sums mentioned in [Th] which are relative versions of the above construction; $(n-r)$-dimensional connect sums carried out in a smooth family over an $r$-dimensional base. Then the same Floer index can be defined; there are now $r$ angles between the Lagrangians that are zero, and $(n-r)$ whose signs can be computed to get the Floer index. (Signs cannot change over the family since the intersection of the Lagrangians fibres over the base of the family with fibres of constant dimension; an angle going to zero would cause a fibre dimension to increase.)