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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Showing that the natural map $X_0\cup_f X_1\to X_0\cup_F (X_1\times I)$ is a topological embedding

Let $X_0,X_1$ be topological spaces, $A\subset X_1$, and $f:A\to X_0$, $F:A\times I\to X_0$ continuous maps with $F(a,0)=f(a)$. Consider two adjunction spaces $X_0\cup_f X_1$ and $X_0\cup_F (X_1\times ...
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Does the pair $(S^n, \{p_1,p_2, \cdots, p_k\})$ have HEP?

Let $p_1, p_2, \cdots, p_k$ be $k $-distinct points on $S^n.$ Then does the pair $(S^n , \{p_1,p_2, \cdots , p_k\})$ have homotopy extension property? I am actually trying to show that $(S^n , \{p_1,...
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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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Induced homotopy extension property for adjunction spaces.

Let the pair $(X,A)$ have homotopy extension property. Let $B$ be a closed subset of $A$ and consider the map $\varphi : B \longrightarrow Y.$ Then can we say that the pair of adjunction spaces $\left ...
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Homotopy extension property of the mapping cylinder.

I am mimicking the argument of Hatcher here from example $0.15$ of page no. 15. What the author says is as follows $:$ To verify the homotopy extension property, notice first that $I \times I$ ...
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Non-example of a homotopy extension property.

Consider the space $$A = \left \{0,1,\frac 1 2, \frac 1 3, \cdots \right \} \subseteq I\ (= [0,1]).$$ Then $(I,A)$ doesn't have homotopy extension property. Our instructor elaborated it in the ...
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Homotopy extension property of pushout.

Consider the pushout diagram in the category Top $:$ $$\require{AMScd} \begin{CD} A @>{f}>{\text {inclusion}}> B\\ @V{g}VV @VV{i_1}V\\ Y @>{i_2}>{}> X\end{CD}$$ Suppose that $(B,A)$ ...
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Question on equivalent ways of describing homotopy extension property.

I am following a lecture note in algebraic topology. There I came across a property which is known as Homotopy Extension Property or in short HEP. The definition is given as follows $:$ Definition $:$ ...
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When does a homotopy exist?

Let $X\coprod X$ be the disjoint union of $X$ with itself and let there be a commutative square $$\require{AMScd} \begin{CD} X \coprod X @>{f,g}>> A \\ @V{(i_0, i_1)}VV @VV{p}V \\ X \times ...
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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 ...
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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 ...
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Prove that it is NOT $F_{0} \simeq_{S_{1} \cup 2 S_{1}} F_{1}$ for $F(x, t)= e^{2\pi t i (|x|-1)} x$

Let $X= \{ x\in R^{2} : 1\leq |x| \leq 2 \}$ be a topological space on $R^{2}$ $A= \partial X$ or $A= S^{1} \cup 2S^{1}$ and Let function $F(x, t) $ be: I=[0,1] $F: X\times I \rightarrow X$ $F(...
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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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Proving that two formulations of the Homotopy Extension Property via diagrams, are equivalent.

Suppose we are working with a collection of topological spaces for which there are a product functor $F:Set\to Set:X\to X\times I$ and an exponential functor $G:Set\to Set: X\to X^I$ such that $F\...
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$(M_f, X \cup Y)$ has the homotopy extension property

Let $X,Y$ be spaces and $f:X \to Y$ a continuous map. I want to show that $(M_f, X \cup Y)$ has the homotopy extension property. In the proof of Whitehead's theorem (Theorem 4.5 in Hatcher's ...
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All finite path connected CW complexes are homotopy equivalent to a CW complex with only one 0-cell.

Statement: if $X$ is a finite path connected CW-complex, then $X$ is homotopy equivalent to a CW-complex with only one $0$-cell. Thoughts: we can use the theorem that for a contractable subspace $A \...
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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 ...
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Question in Hatcher's Algebraic Topology Exercise 0.27

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 \...
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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 \...
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Properties of covering spaces replacing base points by contractible subspaces

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 ...
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