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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.
The category of topological spaces Top has topological spaces as objects and as morphisms the continuous maps between them. The older definition of the homotopy category hTop, called the naive homotopy category [1] for clarity in this article, has the same objects, and a morphism is a homotopy class of continuous maps.
Two maps , are called homotopic relative to A if they are homotopic by a basepoint-preserving homotopy : [,] such that, for each p in and t in [,], the element (,) is in A. Note that ordinary homotopy groups are recovered for the special case in which A = { x 0 } {\displaystyle A=\{x_{0}\}} is the singleton containing the base point.
A map is called a homotopy equivalence if there is another map such that and are both homotopic to the identities. Two spaces are said to be homotopy equivalent if there is a homotopy equivalence between them.
Two chain homotopic maps f and g induce the same maps on homology because (f − g) sends cycles to boundaries, which are zero in homology. In particular a homotopy equivalence is a quasi-isomorphism. (The converse is false in general.)
The null homotopic class acts as the identity of the group addition, and for X equal to S n (for positive n) — the homotopy groups of spheres — the groups are abelian and finitely generated. If for some i all maps are null homotopic, then the group π i consists of one element, and is called the trivial group.
The degree of a map is a homotopy invariant; moreover for continuous maps from the sphere to itself it is a complete homotopy invariant, i.e. two maps ,: are homotopic if and only if = (). In other words, degree is an isomorphism between [ S n , S n ] = π n S n {\displaystyle \left[S^{n},S^{n}\right]=\pi _{n}S^{n}} and Z {\displaystyle ...
A simply connected map (1-connected map) is one that is an isomorphism on path components (0th homotopy group) and onto the fundamental group (1st homotopy group). n -connectivity for spaces can in turn be defined in terms of n -connectivity of maps: a space X with basepoint x 0 is an n -connected space if and only if the inclusion of the ...