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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 general, every manifold has the homotopy type of a CW complex; [3] in fact, Morse theory implies that a compact manifold has the homotopy type of a finite CW complex. [citation needed] Remarkably, Whitehead's theorem says that for CW complexes, a weak homotopy equivalence and a homotopy equivalence are the same thing.
For example, if we model our ∞-groupoids as Kan complexes (quasi-categories [3]), then the homotopy types of the geometric realizations of these sets give models for every homotopy type (perhaps in the weak form). It is conjectured that there are many different "equivalent" models for ∞-groupoids all which can be realized as homotopy types.
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 ...
Different from the numerical technique of homotopy continuation, the homotopy analysis method is an analytic approximation method as opposed to a discrete computational method. Further, the HAM uses the homotopy parameter only on a theoretical level to demonstrate that a nonlinear system may be split into an infinite set of linear systems which ...
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).}
an element of the stable -equivariant homotopy group of maps from to . Here "stable" means "stable under suspension", i.e. the direct limit over V {\displaystyle V} (or k {\displaystyle k} , if you will) of the ordinary, equivariant homotopy groups; and the Z 2 {\displaystyle \mathbb {Z} _{2}} -action is the trivial action on X {\displaystyle X ...
It began by saying "The homotopy λ-calculus is a hypothetical (at the moment) type system" and ended with "At the moment much of what I said above is at the level of conjectures. Even the definition of the model of TS in the homotopy category is non-trivial" referring to the complex coherence issues that were not resolved until 2009.