Homotopy invariance of loop concatenation

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Note: the homotopy in the title means homotopy

$\text{rel }\{0,1\}$

Statement

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

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

Then

$H:\ \ell_1*\ell_2\simeq\ell_1'*\ell_2'\ (\text{rel }\{0,1\})$ - $*$ is the operation of loop concatenation, an instance of path concatenation

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

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

Proof

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

Grade: A

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Notes

Jump up ↑ Recall that [[Omega(X,b)| $\Omega(X,b)$ is a subset of $C([0,1],X)$ , and $C(I,X)$ is the set of all paths in $X$ . $\Omega(X,b)$ is the set of all loop in $X$ based at $b\in X$ . That means if $\ell\in\Omega(X,b)$ that $\ell:I\rightarrow X$ is a path and $\ell(0)=\ell(1)=b$ .

Furthermore, $\Omega(X,b)$ is not just a set, it does have a group structure we can imbue on it, called the fundamental group. This page is actually an important step in the process.
Jump up ↑
Recall that means H1 is a homotopy between ℓ1 and ℓ2 that is relative to {0,1} or stationary on {0,1}. We may say ℓ1 is homotopic to ℓ2 (rel {0,1})
- In this case we have loop homotopies, an instance of end point preserving homotopy/homotopic maps

References

Jump up ↑ Introduction to Topological Manifolds - John M. Lee

[1] Jump up ↑ Recall that [[Omega(X,b)| $\Omega(X,b)$ is a subset of $C([0,1],X)$ , and $C(I,X)$ is the set of all paths in $X$ . $\Omega(X,b)$ is the set of all loop in $X$ based at $b\in X$ . That means if $\ell\in\Omega(X,b)$ that $\ell:I\rightarrow X$ is a path and $\ell(0)=\ell(1)=b$ .

Furthermore, $\Omega(X,b)$ is not just a set, it does have a group structure we can imbue on it, called the fundamental group. This page is actually an important step in the process.

[3] Jump up ↑ Recall that means $H_1$ is a homotopy between $\ell_1$ and $\ell_2$ that is relative to $\{0,1\}$ or stationary on $\{0,1\}$ . We may say $\ell_1$ is homotopic to $\ell_2$ (rel $\{0,1\}$ )
In this case we have loop homotopies, an instance of end point preserving homotopy/homotopic maps

[3] In this case we have loop homotopies, an instance of end point preserving homotopy/homotopic maps

[ITTMJML-2] Jump up ↑ Introduction to Topological Manifolds - John M. Lee

[Note 1]

[1]

[Note 2]

Homotopy invariance of loop concatenation

Contents

Statement

Proof

Notes

References

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