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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 .
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.
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.
For example, the cohomology ring of a path-connected H-space with finitely generated and free cohomology groups is a Hopf algebra. [9] Also, one can define the Pontryagin product on the homology groups of an H-space. [10] The fundamental group of an H-space is abelian. To see this, let X be an H-space with identity e and let f and g be loops at e.
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.
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.
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 ...
Let and be closed and aspherical topological manifolds, and let : be a homotopy equivalence.The Borel conjecture states that the map is homotopic to a homeomorphism.Since aspherical manifolds with isomorphic fundamental groups are homotopy equivalent, the Borel conjecture implies that aspherical closed manifolds are determined, up to homeomorphism, by their fundamental groups.