1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

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D. SALAMON, FLOER HOMOLOGY

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Figure 21. A compact oriented branched I-manifold with BM C Mreg

the outward unit normal vector. This vector together with the orientation of Mi

determines a sign

Ci -{ - +l,

-1,

if Vi is positively oriented,

if vi is negatively oriented.

Now consider the oriented sum of the weights Ai and define

(48)

iEJ

xEM;

Note that this number is not equal to >.(x) unless the ci are all equal to l. We leave

it to the reader to prove that p(x) is independent of the choice of j with x E M(j).

Lemma 5.11. Let (M, >.) be a compact oriented branched I-manifold with bound-

ary. For each x E 8M let p(x) E Q be defined by (48). Then

I: p(x) = o.
xEoM
Proof. The proof is in three steps. The first step shows that M can be covered
by sets Mi with endpoints in Mreg· The second step proves that (M, >.) admits the
structure of a rational cycle. The last step proves the lemma.

Step 1: We may assume without loss of generality that, for every i, the interval

cpi(Mi) is one of (-1, 1), [O, 1), or (-1, O], and that 'Pi-I extends smoothly to the

closure of this interval. Moreover, we may assume that the endpoints of Mi are

regular, i.e.

for every i EI with ±1 E cl(cpi(Mi )).

For every x E M(j) choose a number cx > 0 such that

• For every i E J(j), cx is a regular value of Bx o 'Pi-I·

• cx tf_ Bx(M - Mreg)·

• infM-M(j) Bx > cx and infM, Bx > cx for every i E J(j) with x tf. Mi·

To see that such a number cx exists, note first that the set cpi(Mi - int(Mi)) has

empty interior and is closed relative to cpi(Mi)· Hence the image of this set under

Bx o 'Pi -I is of the first category in the sense of Baire, i.e. is a countable union of