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homotopy equivalence∗ matte† 2013-03-21 13:16:43 Definition Suppose that X and Y are topological spaces and f : X → Y is a continuous map. If there exists a continuous map g : Y → X such that f ◦ g ' idY (i.e. f ◦ g is homotopic to the identity mapping on Y ), and g ◦ f ' idX , then f is a homotopy equivalence. This homotopy equivalence is sometimes called strong homotopy equivalence to distinguish it from weak homotopy equivalence. If there exist a homotopy equivalence between the topological spaces X and Y , we say that X and Y are homotopy equivalent, or that X and Y are of the same homotopy type. We then write X ' Y . 0.0.1 Properties 1. Any homeomorphism f : X → Y is obviously a homotopy equivalence with g = f −1 . 2. For topological spaces, homotopy equivalence is an equivalence relation. 3. A topological space X is (by definition) contractible, if X is homotopy equivalent to a point, i.e., X ' {x0 }. References [1] A. Hatcher, Algebraic Topology, Cambridge University Press, 2002. Also available online. ∗ hHomotopyEquivalencei created: h2013-03-21i by: hmattei version: h31589i Privacy setting: h1i hDefinitioni h55P10i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1