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My textbook defines a space X to be contractible to $x_0 ∈ X$ with $x_0$ held fixed if there is a map $F:X×[0,1]→X$ such that

  • ${F(x,0)=x_0}$, $x∈X$
  • ${F(x,1)=x}$, $x∈X$
  • ${F(x_0,t)=x_0}$, $0≤t≤1$

Two paths $γ_0$ and $γ_1$ from $a$ to $b$ are said to be homotopic with endpoints fixed if there's a map $H:[0,1]×[0,1]→X$ such that

  • ${H(s,0)=γ_0(s)}$, $0≤s≤1$
  • ${H(s,1)=γ_1(s)}$, $0≤s≤1$
  • ${H(0,t)=a}$, $0≤t≤1$
  • ${H(1,t)=b}$, $0≤t≤1$

I was given the hint that I need to make some composite function to show that any loop at $x_0$ is homotopic to the constant loop at $x_0$, but I don't know how to proceed.

1 Answers1

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Being "contractible to $x_0 \in X$" is not really a standardized concept. In the sense of your textbook it means that $X$ is what I would denote as pointed contractible to $x_0 \in X$. See my answer to Is Armstrong saying that the comb space is not contractible? for definitions.

Anyway, taking the homotopy $F$ from your question, the map $H(s,t) = F(\gamma(s),t)$ has the desired properties:

  1. $H(s,0) = F(\gamma(s),1) = \gamma(s)$

  2. $H(s,1) = F(\gamma(s),0) = x_0$

  3. $H(0,t) = F(\gamma(0),t) = F(x_0,t) = x_0$

  4. $H(1,t) = F(\gamma(1),t) = F(x_0,t) = x_0$

Paul Frost
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