# Questions tagged [homotopy-extension-property]

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.

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### Does $(D^n,\mathring{D^n)}$ have the homotopy extension property?

We say that a pair $(X,A)$ has the homotopy extension property if every pair of maps $X \times \{0\} \rightarrow Y$ and $A \times I \rightarrow Y$ that agree on $A \times \{0\}$ can be extended to a ...
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### Does the pair $(CX, X)$ have HEP?

Let $X$ be a space and $CX$ be the cone of $X.$ Then does $(CX, X)$ have homotopy extension property? If so, how do I show that? Any help in this regard will be greatly appreciated. Thanks in advance.
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### Hatcher Algebraic Topology 0.28

Exercise 0.28 in Hatcher's Algebraic Topology states Show that if $(X_1,A)$ satisfies the homotopy extension property, then so does every pair $(X_0 \sqcup_f X_1, X_0)$ obtained by attaching $X_1$ ...
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### Show that either $X$ or $Z$ is homotopy equivalent to a point.

Prove or disprove the following statement: Suppose $X,Y,$ and $Z$ are simply connected $CW$ complexes and that $X \rightarrow Y \rightarrow Z$ is simultaneously a cofiber sequence and a fiber ...
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### Proof that higher homotopy groups of kan complexes are abelian using Eckmann-Hilton

I try to prove that higher homotopy groups of kan complexes are abelian using an Eckmann-Hilton argument. For the definitions I followed the book "Simplicial objects in algebraic Topology" by Peter ...
Hatcher's Exercise 0.27 is : $\mathbf{27.}$ Given a pair $(X,A)$ (this just means that $A$ is a subspace of a space $X$) and a homotopy equivalence $f:A \to B$, show that the natural map $X \to B \... 2 votes 0 answers 105 views ### Homotopy Extension and Deformation Retracts I am trying to solve Exercise 0.26 from Hatcher's Algebraic Topology in a manner different from the hint in the book. It says that if$(X,A)$satisfies the homotopy extension property, then$X \times \...
Let $(X,A)$ and $(Y,B)$ pairs satisfying the homotopy extension property (or a CW-pairs if necessary) with $A$ and $B$ contractible. Let $f:(X,A)\to (Y,B)$ a map of pairs and let \$p_X:\widetilde{X}\to ...