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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.
In mathematics and more specifically in topology, a homeomorphism (from Greek roots meaning "similar shape", named by Henri Poincaré), [2] [3] also called topological isomorphism, or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous inverse function.
Homotopy groups are such a way of associating groups to topological spaces. A torus A sphere. That link between topology and groups lets mathematicians apply insights from group theory to topology. For example, if two topological objects have different homotopy groups, they cannot have the same topological structure—a fact that may be ...
A space X is (topologically) homogeneous if for every x and y in X there is a homeomorphism : such that () =. Intuitively speaking, this means that the space looks the same at every point. All topological groups are homogeneous. Finitely generated or Alexandrov.
A continuous map between two topological spaces induces a group homomorphism between the associated homotopy groups. In particular, if the map is a continuous bijection (a homeomorphism ), so that the two spaces have the same topology, then their i -th homotopy groups are isomorphic for all i .
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. That is, two continuous maps f : X → Y are considered the same in the naive homotopy category if one can be continuously deformed to the other.
A key concept in defining simplicial homology is the notion of an orientation of a simplex. By definition, an orientation of a k-simplex is given by an ordering of the vertices, written as (v 0,...,v k), with the rule that two orderings define the same orientation if and only if they differ by an even permutation.
A group homomorphism between topological groups is continuous if and only if it is continuous at some point. [4] An isomorphism of topological groups is a group isomorphism that is also a homeomorphism of the underlying topological spaces. This is stronger than simply requiring a continuous group isomorphism—the inverse must also be continuous.