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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.
The homotopy group functors assign to each path-connected topological space the group () of homotopy classes of continuous maps . Another construction on a space X {\displaystyle X} is the group of all self-homeomorphisms X → X {\displaystyle X\to X} , denoted H o m e o ( X ) . {\displaystyle {\rm {Homeo}}(X).}
Remarkably, Whitehead's theorem says that for CW complexes, a weak homotopy equivalence and a homotopy equivalence are the same thing. Another important result is the approximation theorem. First, the homotopy category of spaces is the category where an object is a space but a morphism is the homotopy class of a map. Then
Any two knots in 3-space are equivalent by regular homotopy, though not by isotopy. This curve has total curvature 6π, and turning number 3.. The Whitney–Graustein theorem classifies the regular homotopy classes of a circle into the plane; two immersions are regularly homotopic if and only if they have the same turning number – equivalently, total curvature; equivalently, if and only if ...
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces.The first and simplest homotopy group is the fundamental group, denoted (), which records information about loops in a space.
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.
In homotopy theory, a branch of algebraic topology, a Postnikov system (or Postnikov tower) is a way of decomposing a topological space by filtering its homotopy type.What this looks like is for a space there is a list of spaces {} where
For example, CP 2 and a fake projective plane with the same homotopy type are not homeomorphic but both have intersection form of (1). For n = 3, the h -cobordism theorem for smooth manifolds has not been proved and, due to the 3-dimensional Poincaré conjecture , is equivalent to the hard open question of whether the 4-sphere has non-standard ...