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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 .
Since the Betti numbers of a 2-sphere are 1, 0, 1, 0, 0, ... the Lefschetz number (total trace on homology) of the identity mapping is 2. By integrating a vector field we get (at least a small part of) a one-parameter group of diffeomorphisms on the sphere; and all of the mappings in it are homotopic to the identity. Therefore, they all have ...
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:
In mathematics, specifically in topology, the mapping torus of a homeomorphism f of some topological space X to itself is a particular geometric construction with f.Take the cartesian product of X with a closed interval I, and glue the boundary components together by the static homeomorphism:
Antipodal In mathematics , the Borsuk–Ulam theorem states that every continuous function from an n -sphere into Euclidean n -space maps some pair of antipodal points to the same point. Here, two points on a sphere are called antipodal if they are in exactly opposite directions from the sphere's center.
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.
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, given a space , for each integer , let be the set of all maps from the n-simplex to . Then the sequence S n X {\displaystyle S_{n}X} of sets is a simplicial set. [ 22 ] Each simplicial set K = { K n } n ≥ 0 {\displaystyle K=\{K_{n}\}_{n\geq 0}} has a naturally associated chain complex and the homology of that chain complex is the ...