# 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]^{[Note 1]} 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*^{[1]} [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.

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I've basically done it. See note above

## See also

## Notes

- ↑ Where [ilmath]I:=[0,1]:=\{x\in\mathbb{R}\ \vert\ 0\le x\le 1\}\subset\mathbb{R}[/ilmath] - the closed unit interval