Homotopy extension property

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In algebraic topology, a field of mathematics, the Homotopy Extension Property is a very important concept. It arises from the definition of homotopy and continuous functions. Basically, the name is self-explanatory; the Homotopy Extension Property exists when one homotopy can be extended to another one, that is the original homotopy is simply the restriction of the extended homotopy.

[edit] Definition

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

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

[edit] Other

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

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