Difference between revisions of "Homotopy concatenation"

Latest revision as of 05:09, 6 November 2016

Definition

Let $(X,\mathcal{ J })$ and $(Y,\mathcal{ K })$ be topological spaces. Let $H_1:X\times I\rightarrow Y$ and $H_2:X\times I\rightarrow Y$^{[Note 1]} 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^[1] $H_1$ 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.

See also

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

Notes

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

References

1. Jump up ↑ Topology and Geometry - Glen E. Bredon