On a minimal factorization conjecture

• Published in: Topology and its applications Vol. 154; no. 15; pp. 2786 - 2794
• Main Authors: Cadavid, Carlos A., Vélez, Juan D.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01-08-2007
• Subjects: Deformation; Dehn twist; Elliptic fibration; Modular group; Monodromy; Singular fiber; Modular group; Dehn twist; Singular fiber; Deformation; 11F06; 32S30; Elliptic fibration; 14D05; 14B07; Monodromy
• ISSN: 0166-8641; 1879-3207
• DOI: 10.1016/j.topol.2007.06.003

Let
ϕ
:
S
→
D
be a proper holomorphic map from a connected complex surface
S onto the open unit disk
D
⊂
C
, with
0
∈
D
as its unique singular value, and having fiber genus
g
>
0
. Assume that in case
g
⩾
2
,
ϕ
:
S
→
D
admits a deformation
ϕ
′
:
S
′
→
D
whose singular fibers are all of simple Lefschetz type. It has been conjectured that the factorization of the monodromy
f
∈
M
g
around
ϕ
−1
(
0
)
in terms of right-handed Dehn twists induced by the monodromy of
ϕ
′
:
S
′
→
D
has the least number of factors among all possible factorizations of
f as a product of right-handed Dehn twists in the mapping class group (see [M. Ishizaka, One parameter families of Riemann surfaces and presentations of elements of mapping class group by Dehn twists, J. Math. Soc. Japan 58 (2) (2006) 585–594]). In this article, the validity of this conjecture is established for
g
=
1
.