Homotopy invariance of loop concatenation
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]
Contents
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]}, then^{Corollary 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
- [ilmath]H:\ \ell_1*\ell_2\simeq\ell_1'*\ell_2'\ (\text{rel }\{0,1\})[/ilmath] - [ilmath]*[/ilmath] is the operation of loop concatenation, an instance of path concatenation
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
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Notes
- ↑ Recall that [[Omega(X,b)|[ilmath]\Omega(X,b)[/ilmath] is a subset of [ilmath]C([0,1],X)[/ilmath], and [ilmath]C(I,X)[/ilmath] is the set of all paths in [ilmath]X[/ilmath]. [ilmath]\Omega(X,b)[/ilmath] is the set of all loop in [ilmath]X[/ilmath] based at [ilmath]b\in X[/ilmath]. That means if [ilmath]\ell\in\Omega(X,b)[/ilmath] that [ilmath]\ell:I\rightarrow X[/ilmath] is a path and [ilmath]\ell(0)=\ell(1)=b[/ilmath].
Furthermore, [ilmath]\Omega(X,b)[/ilmath] 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. - ↑ Recall that means [ilmath]H_1[/ilmath] is a homotopy between [ilmath]\ell_1[/ilmath] and [ilmath]\ell_2[/ilmath] that is relative to [ilmath]\{0,1\} [/ilmath] or stationary on [ilmath]\{0,1\} [/ilmath]. We may say [ilmath]\ell_1[/ilmath] is homotopic to [ilmath]\ell_2[/ilmath] (rel [ilmath]\{0,1\} [/ilmath])
- In this case we have loop homotopies, an instance of end point preserving homotopy/homotopic maps
References
- Stub pages
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- Theorems
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- Algebraic Topology Theorems
- Algebraic Topology Theorems, lemmas and corollaries
- Algebraic Topology
- Homotopy Theory Theorems
- Homotopy Theory Theorems, lemmas and corollaries
- Homotopy Theory
- Topology Theorems
- Topology Theorems, lemmas and corollaries
- Topology