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From arXiv

On sections of elliptic fibrations

We find a new relation among right-handed Dehn twists in the mapping class group of a

k

-holed torus for

4 \leq k \leq 9

. This relation induces an elliptic Lefschetz pencil structure on the four-manifold \cp

#(9-k)

\cpb

with

k

base points and twelve singular fibers. By blowing up the base points we get an elliptic Lefschetz fibration on the complex elliptic surface

E(1)=

\cp

#9

\cpb

\to S^2

with twelve singular fibers and

k

disjoint sections. More importantly we can locate these

k

sections in a Kirby diagram of the induced elliptic Lefschetz fibration. The

n

-th power of our relation gives an explicit description for

k

disjoint sections of the induced elliptic fibration on the complex elliptic surface

E(n) \to S^2

for

n \geq 1

Simplify

Published on April 24, 2006

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