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A lantern lemma. (English) Zbl 1022.57001
Let \(S\) be an orientable surface and \(\text{Mod}(S)\) the mapping class group of \(S\) (i.e., the group of orientation preserving self-homeomorphisms of \(S\), modulo isotopy). In [Invent. Math. 135, 523-633 (2002; Zbl 0978.57014)], N. V. Ivanov and J. D. McCarty show the existence of an interesting interplay between the algebraic and topological aspects of \(\text{Mod}(S)\), by proving the algebraic characterization of topological reflectiveness, disjointness relation and braid relation between Dehn twists (as far as these topological relations are concerned, see, for example, N. V. Ivanov [Mapping class groups (North-Holland, Amsterdam) (2002; Zbl 1002.57001)]).
The present paper faces – and positively solves – the problem of the existence of a similar characterization of the lantern relation, which is a relation between Dehn twists about curves which lie on a sphere with four punctures (see M. Dehn [Papers on group theory and topology (Springer-Verlag, New York) (1987)] and D. L. Johnson [Proc. Amer. Math. Soc. 75, 119-125 (1979; Zbl 0407.57003)]). Furthermore, the author makes use of results proven by A. Ishida [Proc. Japan Acad. Ser. A Math. Sci. 72, 240-241 (1996; Zbl 0883.57016)] and H. Hamidi-Tehrani [Algebr. Geom. Topol. 2, 1155-1178 (2002; Zbl 1023.57001)] in order to obtain an analogue characterization of another relation involving multitwists, i.e., the 2-chain relation.
Note that the paper cited last independently yields similar results.

57M07 Topological methods in group theory
20F38 Other groups related to topology or analysis
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
57M99 General low-dimensional topology
Full Text: DOI EMIS EuDML arXiv
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