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

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