Linear homotopy: Difference between revisions
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==Definition== | ==Definition== | ||
Suppose <math>U</math> is a subset of | Suppose <math>U</math> is a subset of a (possibly infinite-dimensional) Euclidean space and <math>f,g:X \to U</math> are continuous maps. Suppose further that for every <math>x \in X</math>, the line segment joining <math>f(x)</math> to <math>g(x)</math> lies completely inside <math>U</math>. The '''linear homotopy''' between <math>f</math> and <math>g</math> is a [[homotopy]] defined as follows: | ||
<math>x \mapsto (1-t) f(x) +tg(x)</math> | <math>x \mapsto (1-t) f(x) +tg(x)</math> | ||
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A composite of several linear homotopies is termed a [[piecewise linear homotopy]]. If there exists a piecewise linear homotopy between two functions <math>f,g:X \to U</math> then we say that <math>f</math> and <math>g</math> are [[piecewise linearly homotopic maps]]. | A composite of several linear homotopies is termed a [[piecewise linear homotopy]]. If there exists a piecewise linear homotopy between two functions <math>f,g:X \to U</math> then we say that <math>f</math> and <math>g</math> are [[piecewise linearly homotopic maps]]. | ||
{{further|[[Linear homotopy theorem]]}} | |||
==Facts== | |||
One nice thing about linear homotopies is that they do not unnecessarily move points. In other words, if <math>f(x) = g(x)</math> for some point <math>x</math>, the linear homotopy from <math>f</math> to <math>g</math> fixes <math>x</math> at every point. Linear homotopies are thus useful for showing that given retracts are deformation retracts. | |||
Linear homotopies are commonly seen in the following kinds of sets: | |||
* [[Convex subset of Euclidean space]]: Any two functions to such a set are linearly homotopic | |||
* [[Star-like subset of Euclidean space]]: Any two functions to such a set are homotopic via a composite of at most two linear homotopies | |||
* Compact retract of open subset of Euclidean space |
Latest revision as of 02:56, 9 November 2010
Definition
Suppose is a subset of a (possibly infinite-dimensional) Euclidean space and are continuous maps. Suppose further that for every , the line segment joining to lies completely inside . The linear homotopy between and is a homotopy defined as follows:
where the computation on the right side is in . Essentially we are moving from to along a straight line.
A composite of several linear homotopies is termed a piecewise linear homotopy. If there exists a piecewise linear homotopy between two functions then we say that and are piecewise linearly homotopic maps.
Further information: Linear homotopy theorem
Facts
One nice thing about linear homotopies is that they do not unnecessarily move points. In other words, if for some point , the linear homotopy from to fixes at every point. Linear homotopies are thus useful for showing that given retracts are deformation retracts.
Linear homotopies are commonly seen in the following kinds of sets:
- Convex subset of Euclidean space: Any two functions to such a set are linearly homotopic
- Star-like subset of Euclidean space: Any two functions to such a set are homotopic via a composite of at most two linear homotopies
- Compact retract of open subset of Euclidean space