Homotopy Equivalence

Algebraic Topology · Axiom Academy

Understanding when two spaces are homotopy equivalent and the concept of deformation retracts Before defining homotopy equivalence of spaces, we need the concept of homotopy between continuous maps. The homotopy H provides a continuous family of maps H_t(x) = H(x,t) that deforms f into g . 2. Homotopy Equivalence of Spaces Two spaces are homotopy equivalent if there exist maps going both directions that compose to give homotopic identity maps. g ∘ f ≃ id_X (homotopic to the identity on X ) f ∘ g ≃ id_Y (homotopic to the identity on Y ) Homotopy equivalence is weaker than homeomorphism but preserves all algebraic topological invariants (like the fundamental group). A deformation retract is a particularly nice type of homotopy equivalence where one space sits inside another. r(a) = a for all a ∈ A (fixes points in A ) The inclusion i : A → X followed by r gives r ∘ i ≃ id_A The composition i ∘ r ≃ id_X through a homotopy that fixes A Intuitively, X can be continuously "shrunk" onto A while keeping A fixed. Here are fundamental examples of homotopy equivalences that appear throughout algebraic topology: R n ≃ point : Euclidean space is contractible, homotopy equivalent to a single point S n − pt ≃ R n : An n -sphere with one point removed is homotopy equivalent to n -dimensional Euclidean space Cylinder ≃ Circle: A cylinder deformation retracts onto its central circle Mobius strip ≃ Circle: The Mobius strip deformation retracts onto its core circle

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