6 SELMA ALTINOK AND MOHAN BHUPAL
components of Fφ(i),k for i = 2, . . . , n and k = 1, . . . , (dφ(i), di). By the observation
above, it is easy to check that this part is given by
II1 =
n
i=2
(mφ(i), mi)(dφ(i), di)
dφ(i)di
1
dφ(i)
(dφ(i), di)
di
(dφ(i), di)
(dφ(i), di)
=
n
i=2
(mφ(i), mi)
dφ(i)di
(dφ(i), di)
.
The second part arises as a consequence of the possibility that the number of
connected components of Fi,k and Fφ(i),k may both be greater than one for some i
and each k = 1, . . . , (dφ(i), di). It is not difficult to check that this part is given by
II2 =
n
i=2
dφ(i)
(dφ(i), di)
1
di
(dφ(i), di)
1 (dφ(i), di)
=
n
i=2
dφ(i)di
(dφ(i), di)
dφ(i) di + (dφ(i), di) .
Finally, the third part corresponds to the nonlocal increase in genus when we attach
the part of Σ corresponding to Ei to the part of Σ corresponding to E1, . . . , Ei−1
for i = 2, . . . , n. If d1 = 1, this is given by summing
dφ(i)
(dφ(i), di)
1 (dφ(i), di) = dφ(i) (dφ(i), di)
over i = 2, . . . , n. In general, this is given by
II3 =
n
i=2
(
dφ(i) (dφ(i), di)
)
(d1 1)
Therefore the total second contribution II = II1 + II2 + II3 to the page-genus of
OB(Y ) is given by
II =
n
i=2
(
(mφ(i), mi) di
)
(d1 1).
Putting the first and second contributions I and II together we now find that
the page-genus of OB(Y ) is given by
genus(OB(Y )) = 1 +
n
i=1
(vi + ki 2)mi ki
2
,
which is equivalent to the formula in the statement of the lemma.
In the following lemma, note that OB(Z(T )) is an open book decomposition
of a manifold other than MX ; however, we may still use Lemma 3.1 formally to
compute genus(OB(Z(T ))).
Lemma 3.2. Let Y =
∑n
i=1
miEi be an element of E + and T be a Tjurina
component for Y . Then
genus(OB(Y + Z(T ))) = genus(OB(Y )) + genus(OB(Z(T ))) 1
+ {Non-Tjurina components intersecting T }.
6
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