Homotopy

In mathematics, the topological concept of homotopy formalizes the natural concept of “continuous deformation” of an object towards another.

Homotopy between functions

One gives oneself two topological spaces X \, \! and Y \, \! . Two continuous functions f, \, G \: \, X \ rightarrow Y \, \! is known as homotopic (in Y \, \! ) if there exists a continuous application H \: \, X \ times \ rightarrow Y \, \! such as:

  • \ forall X \ in X, \, H (X, 0) = F (X) \, \!

  • \ forall X \ in X, \, H (X, 1) = G (X) \, \!

In other words, according to the values of the parameter t \, \! , the function H \, \! passes continuously from f \, \! (for t=0 \, \! ) with g \, \! (for t=1 \, \! ). Each value of the parameter t \, \! corresponds to a function:

h_t \: \, X \ rightarrow Y, \, X \ mapsto H (X, T) \, \!

“located between f \, \! and g \, \! ”.

Another manner of seeing it is that for each x \ in X \, \! , the function H \, \! defines a way \ gamma_x \, \! connecting f (X) \, \! with g (X) \, \! :

\ gamma_x \: \, \ rightarrow Y, \, T \ mapsto H (X, T) \, \!

Example 1 : One takes X = \ R \, \! , Y = \ R \, \! , f (X) = 1 \, \! and g (X) = -1 \, \! . Then f \, \! and g \, \! is homotopic in Y \, \! via the function continues:

H (X, T) = 1 - 2t \, \!

(to be noted that in this example nothing depends on the variable x \, \! what is exceptional…).

NB : Homotopic ranking “ in Y \, \! ” can prove very important; indeed in the preceding example if one replaces Y = \ R \, \! by the subspace Y' = \ R^* \, \! , f \, \! and g \, \! is always with values in Y' \, \! but they is not homotopic in Y' \, \! , because there does not exist continuous function connecting -1 \, \! with 1 \, \! in \ R^* \, \! (see the Theorem of the intermediate values).

Example 2 : One takes X = \, \! , Y = \ mathbb {C} \, \! , f (X) = e^ {2i \ pi X} \, \! and g (X) = 0 \, \! . f \, \! describes a circle of radius unit around the origin; g \, \! remains in the beginning. Then f \, \! and g \, \! is homotopic via the function continues:

H (X, T) = (1-t) e^ {2i \ pi X} \, \!

(for each value of t \, \! the function h_t (X) =H (X, T) \, \! describes a circle of radius 1-t \, \! around the origin).

The homotopy of the functions is a Relation of equivalence on the unit \ mathcal {C} (X, Y) \, \! of the continuous applications of X \, \! towards Y \, \! . One of the first applications of homotopy is the definition of the simple Connexité via the homotopy of the laces.

Homotopic equivalence between topological spaces

The definition of homotopy between two spaces can appear abstract, but it corresponds to the very simple idea of continuous deformation.

Being given two topological spaces E \, \! and F \, \! , one says that they are homotopiquement equivalent (or “in the same way standard of homotopy”) if and only if there exist two continuous applications f \: \, E \ rightarrow F \, \! and g \: \, F \ rightarrow E \, \! such as:

  • g \ circ F \, \! is homotopic with id_E \, \! identity of E \, \! ;
  • f \ circ G \, \! is homotopic with id_F \, \! identity of F \, \! .

One will more often speak about homotopic equivalence between two parts of topological spaces.

Two topological spaces homeomorphic are homotopiquement equivalent but the reciprocal one is false in general, as show it the following examples.

Examples:

  • a Circle, a ellipse are homotopiquement equivalent to \ mathbb {C} ^* \, \! i.e. a private plan of a point.
  • a segment \, \! , a disc closed or a swell closed are homotopiquement equivalent between them, and homotopiquement equivalent to a point.

Homotopic equivalence is a Relation of equivalence between topological spaces. Various important properties in algebraic Topologie are preserved by homotopic equivalence, among which: the simple Connexity, the Connexity by arcs, the groups of homology and Cohomologie…

Isotopy

isotopy is a refinement of homotopy; if two continuous applications f \: \, X \ rightarrow Y \, \! and g \: \, X \ rightarrow Y \, \! is Homéomorphisme S one can want to pass from f \, \! with g \, \! , not only continuously but in more by homeomorphisms.

It will thus be said that f \, \! and g \, \! is isotopes if and only if there exists a continuous application H \: \, X \ times \ rightarrow Y \, \! such as:

  • \ forall X \ in X, \, H (X, 0) = F (X) \, \!
  • \ forall X \ in X, \, H (X, 1) = G (X) \, \!
  • for all t \ in \, \! the partial application h_t \, \! is a homeomorphism.

The function h_t \, \! is defined by \ forall X \ in X, \, h_t (X) = H (X, T) \, \! .

The concept of isotopy is in particular important in Théorie of the nodes: two nodes are considered identical if they are homotopic, i.e. if one can deform one to obtain the other without the “cord” tearing or penetrates oneself.

See too

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