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Property they Homotopy extension property

Posted in Uncategorized by admin on the December 30th, 2006

In mathematics, in the area of algebraic topology, the homotopy extension property indicates when a homotopy can be extended to another one, so that the original homotopy is simply the restriction of the extended homotopy.

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Definition

Given <math>A \subset X </math>, we say that the pair <math>\mathbf{\mathit{(A,X)}}</math> has the homotopy extension property with respect to <math>\mathbf{\mathit{Y}}</math> if the following holds:

Given any continuous <math>f: X \to Y</math>, <math>g: A \to Y</math> for which there is a homotopy <math>G: A \times I \to Y</math> of <math>\mathbf{\mathit{f}}</math> and <math>\mathbf{\mathit{g}}</math>, we can extend this to a homotopy <math>F: X \times I \to Y</math> of <math>\mathbf{\mathit{f}}</math> and some <math>\mathbf{\mathit{g’}}</math>, where <math>g’ : X \to Y</math> and <math>g’\mid A = g</math>.


Other

If <math>\mathbf{\mathit{(A,X)}}</math> has the homotopy extension property independent of <math>\mathbf{\mathit{Y}}</math>, then the simple inclusion map <math>i: A \to X</math> is a cofibration.

In fact, if you consider any cofibration <math>i: Y \to Z</math>, then we have that <math>\mathbf{\mathit{Y}}</math> is homeomorphic to its image under <math>\mathbf{\mathit{i}}</math>. This implies that any cofibration can be treated as an inclusion map, and therefore it can be treated as having the homotopy extension property.


See also

  • Homotopy lifting property


References

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