TAILIEUCHUNG - A note on the generalized Matsumoto relation
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. | Turk J Math (2017) 41: 524 – 536 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article A note on the generalized Matsumoto relation 2 ˘ Elif DALYAN1 , Elif MEDETOGULLARI , Mehmetcik PAMUK3,∗ 1 Department of Mathematics, Hitit University, C ¸ orum, Turkey 2 Department of Mathematics, Atılım University, Ankara, Turkey 3 Department of Mathematics, Middle East Technical University, Ankara, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: 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. Key words: Mapping class groups, braid relation, chain relation 1. Introduction Our aim here is to give an alternative proof of Theorem of [3], given below. This theorem is a generalization of the Matsumoto relation in the mapping class group of a closed orientable surface of genus 2 obtained in [4], to the higher genus case. We will refer to this relation as the generalized Matsumoto relation. It gives a relation involving 2g + 4 (respectively 2g + 10) Dehn twists when the genus of the surface is even (respectively odd). Throughout the paper we denote the isotopy class of the right-handed Dehn twist about a simple closed curve c by the same letter c . We use functional notation, that is, for any two mapping classes f and g , the multiplication f g means that g is applied first. Let Σg denote a closed connected orientable surface of genus g. Theorem(Korkmaz). In the mapping class group of Σg , the following relations between right Dehn twists hold (see Figures 1 and 2): (i) (B0 B1 B2 · · · Bg σ)2 = 1 if g is even, (ii) (B0 B1 B2 · · · Bg a2 b2 )2 = 1 if g is odd. The above theorem is used to show that there are infinitely many pairwise nonhomeomorphic 4 -manifolds that admit .
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