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A finite set of generators for the homeotopy group of a 2-manifold

  • W. B. R. Lickorish (a1)

The homeotopy group Λx of a space X is the group of all homeomorphisms of X to itself, modulo the subgroup of those homeomorphisms that are isotopic to the identity. In this paper X will be taken to be a closed oriented 2-manifold, together with a polyhedral structure, and the definition of Λx is then restricted to the consideration of piecewise-linear homeomorphisms and isotopies. Although this restriction to the polyhedral category is not really essential to what follows, it does tend to simplify some of the arguments. In (2) a homeomorphism of X was associated with every simple closed (polyhedral) curve c in X in the following way. First, let A be an annulus in the Euclidean plane parametrized by (r, θ) where 1 ≤ r ≤ 2 and θ is a real number mod 2 π. We define a homeomorphism H: AA by

H is then fixed on the boundary of A. If now e: AX is an orientation-preserving embedding, and eA is a neighbourhood of c in X, then eHe−1|eA can be extended by the identity on XeA to a homeomorphism h:XX. Any piecewise linear homeomorphism hc which is isotopic to h will be called a twist about c or, if c is not specified, just a twist.

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(1)Goeritz, L.Die Abbildungen der Brezelflächen und der Vollbrezel vom Geschlecht 2. Abh. Math. Sem. Univ. Hamburg, 9 (1933), 244259.
(2)Lickorish, W. B. R.A representation of orientable combinatorial 3-manifolds. Ann. of Math. (2), 76 (1962), 531540.
(3)Lickorish, W. B. R.Homeomorphisms of non-orientable two-manifolds. Proc. Cambridge Philos. Soc. 59 (1963), 307317.
(4)Nielsen, J.Die Struktur periodischer Transformationen von Flächen. Mat.-Fys. Medd. Danske Vid. Selsk. 15 (1937), 127.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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