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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:
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 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 ...
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 ...
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
For example, the homotopy pushout encountered above always maps to the ordinary pushout. This map is not typically a weak equivalence, for example the join is not weakly equivalent to the pushout of X 0 ← X 0 × X 1 → X 1 {\displaystyle X_{0}\leftarrow X_{0}\times X_{1}\rightarrow X_{1}} , which is a point.
Indeed, both above composites are homotopic, for example, to the loop that traverses all three loops ,, with triple speed. The set of based loops up to homotopy, equipped with the above operation therefore does turn π 1 ( X , x 0 ) {\displaystyle \pi _{1}(X,x_{0})} into a group.