Abstract
The Epstein–Baer theory of curve isotopies is basic to the remarkable theorem that homotopic homeomorphisms of surfaces are isotopic. The groundbreaking work of R. Baer was carried out on closed, orientable surfaces and extended by D. B. A. Epstein to arbitrary surfaces, compact or not, with or without boundary and orientable or not. We give a new method of deducing the theorem about homotopic homeomorphisms from the results about homotopic curves via the hyperbolic geometry of surfaces. This works on all but 13 surfaces where ad hoc proofs are needed.
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References
Baer, R.: Isotopien von Kurven auf orientierbaren, geshlossenen Fächen. Journal für die Reine und Angewandte Mathematik 159, 101–116 (1928)
Benedetti, R., Petronio, C.: Lectures on Hyperbolic Geometry. Springer, Berlin (1991)
Bleiler, S.A., Casson, A.J.: Automorphisms of Surfaces after Nielsen and Thurston. Cambridge Univ. Press, Cambridge (1988)
Epstein, D.B.A.: Curves on \(2\)-manifolds and isotopies. Acta Math. 115, 83–107 (1966)
Handel, M., Thurston, W.: New proofs of some results of Nielson. Adv. Math. 56, 173–191 (1985)
Massey, W.S.: Algebraic Topology: An Introduction, Harcourt. Brace & World, New York, NY (1967)
Ratcliffe, J.G.: Foundations of Hyperbolic Manifolds. Springer, New York (1994)
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Cantwell, J., Conlon, L. Hyperbolic geometry and homotopic homeomorphisms of surfaces. Geom Dedicata 177, 27–42 (2015). https://doi.org/10.1007/s10711-014-9975-1
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DOI: https://doi.org/10.1007/s10711-014-9975-1