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Given two topological spaces X and Y, a homotopy equivalence between X and Y is a pair of continuous maps f : X → Y and g : Y → X, such that g ∘ f is homotopic to the identity map id X and f ∘ g is homotopic to id Y. If such a pair exists, then X and Y are said to be homotopy equivalent, or of the same homotopy type.
Similarly, since singular homology is a homotopy invariant, the reduced homology groups of a contractible space are all trivial. For a nonempty topological space X the following are all equivalent: X is contractible (i.e. the identity map is null-homotopic). X is homotopy equivalent to a one-point space. X deformation retracts onto a point.
Finally, the "true" homotopy category of pointed spaces is obtained from the category Top * by inverting the pointed maps that are weak homotopy equivalences. For pointed spaces X and Y, [X,Y ] may denote the set of morphisms from X to Y in either version of the homotopy category of pointed spaces, depending on the context.
These homotopy classes form a group, called the n-th homotopy group, (), of the given space X with base point. Topological spaces with differing homotopy groups are never homeomorphic , but topological spaces that are not homeomorphic can have the same homotopy groups.
We now show that if f and g are homotopically equivalent, then f * = g *. From this follows that if f is a homotopy equivalence, then f * is an isomorphism. Let F : X × [0, 1] → Y be a homotopy that takes f to g. On the level of chains, define a homomorphism : + ()
When X, Y are pointed spaces, the maps are required to ... Two spaces are said to be homotopy equivalent if there is a homotopy equivalence between them.
A homotopy pullback (or homotopy fiber-product) is the dual concept of a homotopy pushout. It satisfies the universal property of a pullback up to homotopy. [ citation needed ] Concretely, given f : X → Z {\displaystyle f:X\to Z} and g : Y → Z {\displaystyle g:Y\to Z} , it can be constructed as
The Whitehead theorem states that a weak homotopy equivalence from one CW complex to another is a homotopy equivalence. (That is, the map f: X → Y has a homotopy inverse g: Y → X, which is not at all clear from the assumptions.) This implies the same conclusion for spaces X and Y that are homotopy equivalent to CW complexes.