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The antipodal map preserves orientation (is homotopic to the identity map) [2] when is odd, and reverses it when is even. Its degree is ( − 1 ) n + 1 . {\displaystyle (-1)^{n+1}.} If antipodal points are identified (considered equivalent), the sphere becomes a model of real projective space .
A homeomorphism is a special case of a homotopy equivalence, in which g ∘ f is equal to the identity map id X (not only homotopic to it), and f ∘ g is equal to id Y. [7]: 0:53:00 Therefore, if X and Y are homeomorphic then they are homotopy-equivalent, but the opposite is not true. Some examples:
For each dimensional k, the boundary maps d k : δD k → RP k−1 /RP k−2 is the map that collapses the equator on S k−1 and then identifies antipodal points. In odd (resp. even) dimensions, this has degree 0 (resp. 2): = + ().
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 ...
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.
The opposite is also true: If X has a hole with a d-dimensional boundary, then there is a d-dimensional sphere that is not homotopic to a constant map, so the d-th homotopy group of X is not trivial. In short, X has a hole with a d -dimensional boundary, if-and-only-if π d ( X ) ≇ 0 {\displaystyle \pi _{d}(X)\not \cong 0} .The homotopical ...
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. (However, there exist contractible spaces which do not strongly deformation retract to a point.) For any path-connected space Y, any two maps f,g: X → Y are homotopic.
For example, the category of (reasonable) topological spaces has a structure of a model category where a weak equivalence is a weak homotopy equivalence, a cofibration a certain retract and a fibration a Serre fibration. [20] Another example is the category of non-negatively graded chain complexes over a fixed base ring. [21