Homotopy concatenation

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Definition

Let [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath] be topological spaces. Let [ilmath]H_1:X\times I\rightarrow Y[/ilmath] and [ilmath]H_2:X\times I\rightarrow Y[/ilmath] be homotopies from [ilmath]X[/ilmath] to [ilmath]Y[/ilmath]. Suppose:

• [ilmath]\forall x\in X[H_1(x,1)=H_2(x,0)][/ilmath] - that the final stage of [ilmath]H_1[/ilmath] is the same as the initial stage of [ilmath]H_2[/ilmath]

Then we may concatenate [ilmath]H_1[/ilmath] and [ilmath]H_2[/ilmath] to form:

• [ilmath]H_1*H_2:X\times I\rightarrow Y[/ilmath] given by: [ilmath]H_1*H_2:(x,t)\mapsto\left\{\begin{array}{lr}H_1(x,2t)&\text{for }t\in[0,\frac{1}{2}]\\ H_2(x,2t-1)&\text{for }t\in[\frac{1}{2},1]\end{array}\right.[/ilmath]
  • Notice [ilmath]t=\frac{1}{2}[/ilmath] is in both parts of the domain, this is a nod to the pasting lemma

Claim 1: the concatenation homotopy is actually a homotopy

Proof of claims

Claim 1

We must show that [ilmath]H_1*H_2[/ilmath] is actually a homotopy. All that means showing really is that it is continuous. This is a quick application of the pasting lemma and using the fact that [ilmath]H_1[/ilmath] and [ilmath]H_2[/ilmath] are themselves continuous.

Grade: C

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I've basically done it. See note above

See also

• Path concatenation (topology)
  • Loop concatenation
• Fundamental group