Homotopy invariance of path concatenation

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Statement

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Let [ilmath]p_1,p_2,p_1',p_2'\in[/ilmath][ilmath]C([0,1],X)[/ilmath] be given. Suppose [ilmath]H_1:\ p_1\simeq p_1'\ (\text{rel }\{0,1\})[/ilmath] and [ilmath]H_2:\ p_2\simeq p_2'\ (\text{rel }\{0,1\})[/ilmath] are end point preserving homotopies (where [ilmath]H_1,H_2:[0,1]\times [0,1]\rightarrow X[/ilmath] are the specific homotopies of the paths) then:

• [ilmath]H:p_1*p_2\simeq p_1'*p_2'\ (\text{rel }\{0,1\})[/ilmath] where
  • [ilmath]p_1*p_2[/ilmath] denotes path concatenation, explicitly:
    • [ilmath]p_1*p_2:[0,1]\rightarrow X[/ilmath] by [ilmath]p_1*p_2:t\mapsto\left\{\begin{array}{lr}p_1(2t)&\text{for }t\in[0,\frac{1}{2}]\\p_2(2t-1)&\text{for }t\in[\frac{1}{2},1]\end{array}\right. [/ilmath]
      • Note that the fact [ilmath]t=\frac{1}{2}[/ilmath] is in both parts is a nod towards the use of the pasting lemma
  • [ilmath]H:=H_1*H_2[/ilmath] - the homotopy concatenation, explicitly:
    • [ilmath]H:[0,1]\times[0,1]\rightarrow X[/ilmath] by [ilmath]H:(s,t)\mapsto\left\{\begin{array}{lr}H_1(s,2t)&\text{for }t\in[0,\frac{1}{2}]\\ H_2(s,2t-1)&\text{for }t\in[\frac{1}{2},1]\end{array}\right.[/ilmath]
      • Note that the fact [ilmath]t=\frac{1}{2}[/ilmath] is in both parts is a nod towards the use of the pasting lemma

Proof

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It's basically already done. All we have to show is that the homotopy concatenation, [ilmath]H[/ilmath], fits the requirements

See also

• The fundamental group

References