# Homotopy invariance of loop concatenation

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Stub page, chore to write, probably needs redoing
Marked as A* because I wrote it in such a bored mood, it needs to be checked ASAP
Note: the homotopy in the title means homotopy [ilmath]\text{rel }\{0,1\} [/ilmath]

## Statement

Here [ilmath]I:=[0,1]:=\{x\in\mathbb{R}\ \vert\ 0\le x\le 1\}\subset\mathbb{R}[/ilmath] will denote the closed unit interval. Let [ilmath]Top.[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath] be topological spaces, let [ilmath]b\in X[/ilmath] be given and let [ilmath]\ell_1,\ell_2,\ell_1',\ell_2'\in[/ilmath][ilmath]\Omega(X,b)[/ilmath][Note 1], thenCorollary to 7.10:[1] we have:

• If [ilmath]H_1:\ \ell_1\simeq\ell_1'\ (\text{rel }\{0,1\})[/ilmath] and [ilmath]H_2:\ \ell_2\simeq\ell_2'\ (\text{rel }\{0,1\})[/ilmath][Note 2]

Then

This can perhaps be better written symbolically using [ilmath][\ell][/ilmath] to denote the equivalence class of [ilmath]\ell[/ilmath] under (equivalence) relation of end point preseriving homotopy:

• [ilmath]\forall\ell_1,\ell_2,\ell_1',\ell_2'\in\Omega(X,b)[([\ell_1]=[\ell_1']\wedge[\ell_2]=[\ell_2'])\implies[\ell_1*\ell_2]=[\ell_1'*\ell_2']][/ilmath]

## Proof

The homotopy concatenation, [ilmath]H:=H_1*H_2[/ilmath], is easily shown to be the required homotopy. This is actually an instance of Homotopy invariance of path concatenation