Let \(X\) be a topological space and let \(x_0\in X\). Define the product of homotopy classes of loops \([\alpha]_{x_0}\) based at \(x_0\) and verify in detail that this product is associative.
Solution
All homotopies of loops below are taken relative to the endpoints.
1. For loops\[ \alpha, \beta:I\to X \]based at \(x_0\), define their concatenation by\[ (\alpha\mathbin{\cdot}\beta)(s) = \begin{cases} \alpha(2s),&0\le s\le\frac12,\\ \beta(2s-1),&\frac12\le s\le1. \end{cases} \]Then define the product of homotopy classes by\[ [\alpha]_{x_0}[\beta]_{x_0} = [\alpha\mathbin{\cdot}\beta]_{x_0}. \]
Since \[ \alpha(1)=x_0=\beta(0), \] the two formulas for \(\alpha\mathbin{\cdot}\beta\) agree at \(s=1/2\). The pasting lemma therefore makes the concatenation continuous. Moreover, \[ (\alpha\mathbin{\cdot}\beta)(0)=x_0 \qquad\text{and}\qquad (\alpha\mathbin{\cdot}\beta)(1)=x_0, \] so it is again a loop based at \(x_0\).
2. The product in 1 is well-defined on homotopy classes.
Suppose \[ \alpha\simeq\alpha' \qquad\text{and}\qquad \beta\simeq\beta' \] through endpoint-fixed homotopies \[ H:I\times I\to X, \qquad K:I\times I\to X. \] Define \[ L:I\times I\to X \] by \[ L(s,t) = \begin{cases} H(2s,t),&0\le s\le\frac12,\\ K(2s-1,t),&\frac12\le s\le1. \end{cases} \] At \(s=1/2\), the first formula is \[ H(1,t)=x_0 \] and the second is \[ K(0,t)=x_0, \] so \(L\) is continuous by the pasting lemma. Also \[ L(-,0)=\alpha\mathbin{\cdot}\beta, \qquad L(-,1)=\alpha'\mathbin{\cdot}\beta', \] and \[ L(0,t)=L(1,t)=x_0. \] Thus \[ \alpha\mathbin{\cdot}\beta \simeq \alpha'\mathbin{\cdot}\beta', \] so the class of the concatenation depends only on \([\alpha]_{x_0}\) and \([\beta]_{x_0}\).
3. For three based loops \(\alpha,\beta,\gamma\), define a single traversal\[ P:[0,3]\to X \]by\[ P(u) = \begin{cases} \alpha(u),&0\le u\le1,\\ \beta(u-1),&1\le u\le2,\\ \gamma(u-2),&2\le u\le3. \end{cases} \]Then \(P\) is continuous.
At the two joining parameters, \[ \alpha(1)=\beta(0)=x_0 \] and \[ \beta(1)=\gamma(0)=x_0. \] Hence the three formulas agree on their overlaps, and the pasting lemma gives continuity.
4. The two parenthesized concatenations are reparametrizations of \(P\):\[ (\alpha\mathbin{\cdot}\beta)\mathbin{\cdot}\gamma =P\circ r_L, \]where\[ r_L(s) = \begin{cases} 4s,&0\le s\le\frac12,\\ 2s+1,&\frac12\le s\le1, \end{cases} \]and\[ \alpha\mathbin{\cdot}(\beta\mathbin{\cdot}\gamma) =P\circ r_R, \]where\[ r_R(s) = \begin{cases} 2s,&0\le s\le\frac12,\\ 4s-1,&\frac12\le s\le1. \end{cases} \]
For the left parenthesization, expanding the definition of concatenation gives \[ ((\alpha\mathbin{\cdot}\beta)\mathbin{\cdot}\gamma)(s) = \begin{cases} \alpha(4s),&0\le s\le\frac14,\\ \beta(4s-1),&\frac14\le s\le\frac12,\\ \gamma(2s-1),&\frac12\le s\le1. \end{cases} \] This is precisely \(P(r_L(s))\).
Similarly, \[ (\alpha\mathbin{\cdot}(\beta\mathbin{\cdot}\gamma))(s) = \begin{cases} \alpha(2s),&0\le s\le\frac12,\\ \beta(4s-2),&\frac12\le s\le\frac34,\\ \gamma(4s-3),&\frac34\le s\le1, \end{cases} \] which is \(P(r_R(s))\).
5. The two parenthesizations in 4 are homotopic relative to the endpoints.
For \(t\in I\), set \[ r_t(s)=(1-t)r_L(s)+t r_R(s). \] Both \(r_L\) and \(r_R\) are continuous and nondecreasing maps from \(I\) to \([0,3]\) with \[ r_L(0)=r_R(0)=0, \qquad r_L(1)=r_R(1)=3. \] Hence each \(r_t\) is continuous, takes values in \([0,3]\), and has the same endpoint values.
Define \[ F:I\times I\to X, \qquad F(s,t)=P(r_t(s)). \] This is continuous. By 4, \[ F(-,0)=(\alpha\mathbin{\cdot}\beta)\mathbin{\cdot}\gamma \] and \[ F(-,1)=\alpha\mathbin{\cdot}(\beta\mathbin{\cdot}\gamma). \] Finally, \[ F(0,t)=P(0)=x_0, \qquad F(1,t)=P(3)=x_0, \] so the endpoints remain fixed throughout the homotopy.
6. The product of based loop homotopy classes is associative:\[ \boxed{([\alpha]_{x_0}[\beta]_{x_0})[\gamma]_{x_0} =[\alpha]_{x_0}([\beta]_{x_0}[\gamma]_{x_0}).} \]
By the definition in 1, the left side is represented by \[ (\alpha\mathbin{\cdot}\beta)\mathbin{\cdot}\gamma \] and the right side by \[ \alpha\mathbin{\cdot}(\beta\mathbin{\cdot}\gamma). \] These loops are endpoint-fixed homotopic by 5, so they determine the same homotopy class.