Abstract. On a compact oriented surface of genus g with n ≥ 1 boundary components, δ1, δ2, . . . , δn, we consider positive factorizations of the boundary multitwist t δ 1 t δ 2 · · · t δn , where t δ i is the positive Dehn twist about the boundary δi. We prove that for g ≥ 3, the boundary multitwist t δ 1 t δ 2 can be written as a product of arbitrarily large number of positive Dehn twists about nonseparating simple closed curves, extending a recent result of Baykur and Van Horn-Morris, who proved this result for g ≥ 8. This fact has immediate corollaries on the Euler characteristics of the Stein fillings of contact three manifolds.
We give an elementary proof of a relation, first discovered in its full generality by Korkmaz, in the mapping class group of a closed orientable surface. Our proof uses only the well-known relations between Dehn twists.
Let T denote a binding component of an open book ( , φ) compatible with a closed contact 3-manifold (M, ξ). We describe an explicit open book ( , φ ) compatible with (M, ζ ), where ζ is the contact structure obtained from ξ by performing a full Lutz twist along T . Here, ( , φ ) is obtained from ( , φ) by a local modification near the binding.
We study perfect discrete Morse functions on closed, connected, oriented n-dimensional manifolds. We show how to compose such functions on connected sums of manifolds of arbitrary dimensions and how to decompose them on connected sums of closed oriented surfaces.
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