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Topological Interactions Between Homotopy and Dehn Twist Varieties

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  • Susmit Bagchi

    (Department of Software Engineering (Earlier: Department of Aerospace and Software Engineering (Informatics)), IT Engineering College, Gyeongsang National University, Jinju 660-701, Republic of Korea)

Abstract

The topological Dehn twists have several applications in mathematical sciences as well as in physical sciences. The interplay between homotopy theory and Dehn twists exposes a rich set of properties. This paper generalizes the Dehn twists by proposing the notion of pre-twisted space, orientations of twists and the formation of pointed based space under a homeomorphic continuous function. It is shown that the Dehn twisted homotopy under non-retraction admits a left lifting property (LLP) through the local homeomorphism. The LLP extends the principles of Hurewicz fibration by avoiding pullback. Moreover, this paper illustrates that the Dehn twisted homotopy up to a base point in a based space can be formed by considering retraction. As a result, two disjoint continuous functions become point-wise continuous at the base point under retracted homotopy twists. Interestingly, the oriented Dehn twists of a pre-twisted space under homotopy retraction mutually commute in a contractible space.

Suggested Citation

  • Susmit Bagchi, 2024. "Topological Interactions Between Homotopy and Dehn Twist Varieties," Mathematics, MDPI, vol. 12(20), pages 1-12, October.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:20:p:3282-:d:1502364
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