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Chapter 1: Homotopy

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Homotopical Topology

Part of the book series: Graduate Texts in Mathematics ((GTM,volume 273))

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Abstract

Let X and Y be topological spaces. Continuous maps f, g: X → Y are called homotopic (f ∼ g) if there exists a family of maps h t : X → Y, t ∈ I such that (1) \(h_{0} = f,h_{1} = g\); (2) the map H: X × I → Y, H(x, t) = h t (x), is continuous.

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Notes

  1. 1.

    Condition (2) makes sense for \(n = -1\) and means that X is nonempty. Sometimes it is convenient to assume that (−1)-connected is the same as nonempty.

  2. 2.

    Sometimes, the terminology of the theory of coverings is based on a visual presentation of a covering, in which T lies “above” X and the projection p is vertical and directed down. This is reflected not only in terminology, but also in many pictures in this section.

  3. 3.

    This theorem is often called Seifert–Van Kampen Theorem.

  4. 4.

    “Borromeo” is not the name of a mathematician. It belongs to a family of Italian noblemen who had the picture of the link on their coat of arms.

  5. 5.

    To distinguish relative homotopy groups and spheroids from homotopy groups and spheroids considered before, we will sometimes call the latter absolute.

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Authors and Affiliations

  1. Department of Mathematics and Mechanics, Moscow State University, Moscow, Russia

    Anatoly Fomenko

  2. Department of Mathematics, University of California, Davis, CA, USA

    Dmitry Fuchs

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  1. Anatoly Fomenko
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Fomenko, A., Fuchs, D. (2016). Chapter 1: Homotopy. In: Homotopical Topology. Graduate Texts in Mathematics, vol 273. Springer, Cham. https://doi.org/10.1007/978-3-319-23488-5_1

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