Homotopy concatenation

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Definition

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

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

Then we may concatenate $H_1$ and $H_2$ to form:

• $H_1*H_2:X\times I\rightarrow Y$ given by: $H_1*H_2:(x,t)\mapsto\left\{\begin{array}{lr}H_1(x,2t)&\text{for }t\in[0,\frac{1}{2}]\\ H_2(x,2t-1)&\text{for }t\in[\frac{1}{2},1]\end{array}\right.$
  • Notice $t=\frac{1}{2}$ 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 $H_1*H_2$ 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 $H_1$ and $H_2$ are themselves continuous.

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I've basically done it. See note above

See also

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