Homotopy class

Definition

The relation of paths being end-point-preserving homotopic is an Equivalence relation

That is $\alpha\simeq\beta\text{ rel}\{0,1\}$ where $\alpha$ and $\beta$ are paths from $a$ to $b$ (which are not necessarily distinct as it may be a loop) is an equivalence relation, which is to say:

• Reflexive: $\alpha\simeq\alpha\text{ rel}\{0,1\}$
• Symmetric: $\alpha\simeq\beta\text{ rel}\{0,1\}\implies \beta\simeq\alpha\text{ rel}\{0,1\}$
• Transitive: $\alpha\simeq\beta\text{ rel}\{0,1\}\wedge\beta\simeq\gamma\text{ rel}\{0,1\}\implies \alpha\simeq\gamma\text{ rel}\{0,1\}$

The equivalence class of $\alpha$ is denoted (as is usual) by $[\alpha]$

Important properties

$\alpha,\ \beta$ and $\gamma$ denote paths

• For a continuous map $p:[0,1]\rightarrow[0,1]$ with $p(0)=0$ and $p(1)=1$ we have:

  $[\alpha\circ p]=[\alpha]$ - that is any reparametrisation of $\alpha$ is homotopic to $\alpha$

• $$[\alpha_1]=[\alpha_2]\wedge[\beta_1]=[\beta_2]\implies[\alpha_1\beta_1]=[\alpha_2\beta_2]$$
  • This allows us to define multiplication
• $$(\alpha\beta)\gamma\simeq\alpha(\beta\gamma)\text{ rel}\{0,1\}$$ or $$([\alpha][\beta])[\gamma]=[\alpha]([\beta][\gamma])$$
  • This allows us to define associativity
• Where $a$ is the constant loop at $a$ (ie $a(t)=a\ \forall t\in[0,1]$) we have

  $$a\alpha\simeq\alpha\simeq\alpha b\text{ rel}\{0,1\}$$ or $$[a][\alpha]=[\alpha]=[\alpha][b]$$

• if $\alpha^{-1}$ is the reverse path of $\alpha$, literally $\alpha^{-1}(t)=\alpha(1-t)$ then
  • $[\alpha_0]=[\alpha_1]\implies[\alpha_0^{-1}]=[\alpha_1^{-1}]$
    • we can now define the inverse, $[\alpha^{-1}]=[\alpha]^{-1}$
• $[\alpha][\alpha]^{-1}=[a]$

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TODO: Proofs for all of these p117

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See also

• Fundamental group