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Homotopic arcs are isotopic

Joseph Martin, Dale Rolfsen

1968
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Proceedings of the American Mathematical Society
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The purpose of this note is to show that if a and 8 are flat arcs in an re-manifold M («^3) and a and 8 are homotopic (with fixed endpoints), then there is an isotopy of M onto itself, leaving the endpoints of a and 8 fixed, which carries a onto 8. This result is actually an application of an unstated theorem in §6 of [2] and is of primary interest when w = 3, as theorems of this kind are well known for larger values of re. This same sort of technique has been used by James Kister to show that
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... ach arc in an re-manifold is isotopic, with fixed endpoints, to a flat arc. Of course, in this case, the isotopy is not ambient. An n-manifold is a topological space which may be covered by open sets, each of which is homeomorphic to £", Euclidean re-dimensional space. An arc in an w-manifold M is flat if there exists a closed neighborhood yl of a and a homeomorphism h of A onto Dn, the unit ball in E", which carries a onto a straight line interval. A homotopy of a space X in a space Y is a map/: XX7->F; we shall sometimes write ftix) for fix, t) or use another closed real interval to replace /= [0, l]. We say that/is/ixecf on a subset 5 of Xif/(| S=/t'| 5 whenever t, t' EI-ll ft is a homeomorphism for each t, we call/an isotopy, and if each ft is also surjective and /o is the identity, then / is an ambient isotopy of X. A path in a space Z is a map of / into Z. The paths o> and a/ are path homotopic in Z il there is a homotopy g of I in Z, fixed on {0, 1} such that co = go and w' =gi. If two arcs are images of / by homeomorphisms which are path homotopic, we shall also call the arcs path homotopic. It may easily be seen that two arcs a and 8 are path homotopic if and only if they have common endpoints and there exists a path X such that X maps [0, j] homeomorphically onto a, [|, l] homeomorphically onto 8, and X is path homotopic to a constant path. Theorem. Suppose that w=g3, M is an n-manifold, and a and 8 are path homotopic flat arcs in M with common endpoints p and q. Then there exists an ambient isotopy ht (0^/gl) of AI, fixed on p and q, such that hiia) = 8. Furthermore, if K is a closed set such that a and 8 are path homotopic in M -K then we may require that ht be fixed on K.

doi:10.1090/s0002-9939-1968-0232394-6
fatcat:66ggr6z345cwzopicmqfcqmspu