Homotopy extension property

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In mathematics, in the area of algebraic topology, the homotopy extension property indicates which homotopies defined on a subspace can be extended to a homotopy defined on a larger space.

Definition[edit]

Let X\,\! be a topological space, and let A \subset X. We say that the pair (X,A)\,\! has the homotopy extension property if, given a homotopy f_t\colon A \rightarrow Y and a map F_0\colon X \rightarrow Y such that F_0 |_A = f_0, there exists an extension of F_0 to a homotopy F_t\colon X \rightarrow Y such that F_t|_A = f_t. [1]

That is, the pair (X,A)\,\! has the homotopy extension property if any map G\colon ((X\times \{0\}) \cup (A\times I)) \rightarrow Y can be extended to a map G'\colon X\times I \rightarrow Y (i.e. G\,\! and G'\,\! agree on their common domain).

If the pair has this property only for a certain codomain Y\,\!, we say that (X,A)\,\! has the homotopy extension property with respect to Y\,\!.

Visualisation[edit]

The homotopy extension property is depicted in the following diagram

Homotopy extension property.svg

If the above diagram (without the dashed map) commutes, which is equivalent to the conditions above, then there exists a map  \tilde{f} which makes the diagram commute. By currying, note that a map  \tilde{f} \colon X \to Y^I is the same as a map  \tilde{f} \colon X\times I \to Y .

Also compare this to the visualization of the homotopy lifting property.

Properties[edit]

  • If X\,\! is a cell complex and A\,\! is a subcomplex of X\,\!, then the pair (X,A)\,\! has the homotopy extension property.
  • A pair (X,A)\,\! has the homotopy extension property if and only if (X\times \{0\} \cup A\times I) is a retract of X\times I.

Other[edit]

If \mathbf{\mathit{(X,A)}} has the homotopy extension property, 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.

See also[edit]

References[edit]

  1. ^ A. Dold, Lectures on Algebraic Topology, pp. 84, Springer ISBN 3-540-58660-1