Homotopy invariance of path concatenation

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Really not in the mood for this, done it anyway, check first and flesh out

Statement

TODO: This caption

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

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

Proof

Grade: A*

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The message provided is:

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

See also

• The fundamental group

References