# 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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### 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 ...
### If a homotopy can be extended to a neighborhood of a closed subspace of a normal space $X$, then it can be extended to all of $X$.
During some self-study, I came across the following problem in Spanier's Algebraic Topology: Statement: Suppose $X$ is a normal space, and $A$ is a closed subspace of $X$. Let $f\colon X \to Y$ be a ...
### any cofibration $i:A \to B$ is a homeomorphism onto its image (question regarding the inverse map)
I was recently working on a problem that introduced the homotopy extension property as a cofibration $i:A \to B$. Let's say we are given the commutative diagram: Now, if $i:A \to B$ is the inclusion ...