LECTURE 5. MULTI-VALUED PERTURBATIONS 215Suppose first that x E Mi - int(Mi )· Then there exists a sequence xv E M - Mi
which converges to x. Since M(j) is open, we may assume without loss of generality
that Xv E M(j) for all v. Since I(j) is a finite set we may assume without loss of
generality that there exists an i' E I(j) - {i} such that Xv E M i' for all v. SinceMi' is closed, relative to M(j), this implies that x E M i'· Moreover, xis the limit
of a sequence in Mi' - (Mi n Mi'). Hence x E M i n Mi' - int Mi' (Min Mi').Conversely, suppose that x E MinMi' -intMi, (MinMi') for some i' E J(j)-{i}.
Then there exists a sequence Xv E Mi' - (Mi n Mi') which converges to x. Since
xv tJ. Mi for all v this implies x tJ. int(Mi )·
Step 3: Mreg is dense in M.It suffices to prove that int(Mi) is dense in Mi. To see this we write I(j) =
{ io' ... 'im} with io = i and prove by induction over e that the following holds for
e = 1, ... , m. For every x E Mi and every c > 0 there exists a y E Mi such that
(a) d(x, y) < c.
(b) For every k E {l, ... , f} either y tj. Mik or y E int Mi (Min Mik ).
First suppose f, = l. If x E int Mi (Mi n Mi 1 ) choose y = x. If x tj. int Mi (Mi n Mi, )
then there exists a sequence Xv E Mi - M i 1 converging to x and we can choose
y = xv for v sufficiently large. Now suppose that the assertion has been proved
with f, replaced by f, - 1 where f, 2: 2. Choose z E Mi such that d(x, z) < c/2 and,
fork E {l, ... ,f-1}, either z tJ. Mik or z E intMi(MinMiJ· If z E intMi(MinMie)
choose y = z. If z tJ. intMi (Min Mie) then there exists a sequence Zv E Mi - Mie
converging to z. There exists an N E N such that the following holds for v ;::: N
and k E { 1, ... , f, - 1}.
• d(z, zv) < c/2.
• If z E int Mi (Min Mik) then Zv E int Mi (Min Mik).
• If z tJ. Mik then Zv tJ. Mik·
Hence, for v;::: N, the pointy= Zv satisfies the conditions (a) and (b) above. This
completes the induction. Thus we have proved that every point x E M i can be
approximated by a sequence
Yv E n ((Mi - Mi' ) U intMi (Mi n Mi')) =int( M i)·
i'EJ(j)-{i}
The last equality follows from (iii) and Step 2. Hence int(Mi) is dense in Mi, as
claimed. This proves the lemma. D
Compact oriented branched 1-manifolds are also called train-tracks (see Fig-
ure 21). They were introduced by Thurston (with real weights) in connection with
the study of laminations on surfaces. A number of mathematicians generalized
these ideas and studied 3-manifolds by using branched 2-manifolds. The following
result about the ends of compact oriented branched 1-manifolds is the analogue ofthe observation that every compact 1-manifold has an even number of ends. It plays
the same role in the construction of rational invariants as the 1-manifold lemma
plays in differential topology (cf. Milnor [ 33]). Train tracks also play a crucial role
in the work of Fukaya-Ono [14].
Before stating the lemma we point out that every endpoint of a branched 1-