Mathematical Proceedings of the Cambridge Philosophical Society · 1964 · 281 citations · 2 references
Integral GeometryGeometric Group TheoryLie GroupGlobal GeometryGeometryHomeotopy GroupEducationTopological GroupsSpace XHomeomorphism HTopological Propertyλ XLie TheoryFinite SetTopological Invariant
The homeotopy group Λ_X consists of homeomorphisms of a space X modulo those isotopic to the identity, and for closed oriented 2‑manifolds this study focuses on piecewise‑linear maps, a restriction that simplifies arguments and introduces the notion of twists about simple closed curves. The paper aims to define and analyze the homeotopy group of a closed oriented 2‑manifold by restricting to piecewise‑linear homeomorphisms and isotopies. To construct elements of the group, each simple closed curve c is associated with a homeomorphism obtained by embedding a parametrized annulus A into X, applying a boundary‑fixed map H on A, and extending it by the identity outside the embedded neighbourhood.
The homeotopy group Λ x 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 Λ x 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: A → A by H is then fixed on the boundary of A . If now e : A → X 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 X − eA to a homeomorphism h:X → X . Any piecewise linear homeomorphism h c 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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