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The relative homotopy group $\pi_n(X, A)$ for a pair of topological spaces $(X, A)$ (with $A \subseteq X$) consists of homotopy classes of maps $(I^n, \partial I^n) \to (X, A)$, where $I^n$ is the $n$-dimensional unit cube and $\partial I^n$ is its boundary. This set admits a group structure for $n \geq 2$, but unlike the absolute homotopy groups $\pi_n(X)$ which are abelian for all $n \geq 2$, the relative versions $\pi_n(X, A)$ are abelian only for $n \geq 3$.
To see why $\pi_n(X, A)$ is abelian for $n \geq 3$, start by recalling how the group operation is defined. Represent elements of $\pi_n(X, A)$ by maps $\alpha, \beta: (I^n, \partial I^n, J_{n-1}) \to (X, A, x_0)$ (now in the pointed setting for precision, where $J_{n-1}$ is a distinguished face of the boundary and $x_0 \in A$ is the basepoint). The group operation is given by concatenation along one coordinate direction, say the first: the sum $[\beta] + [\alpha]$ (or $\beta * \alpha$) is the map that glues $\alpha$ and $\beta$ along the first coordinate $t_1$, specifically:
$(\beta * \alpha)(t_1, t_2, \dots, t_n) = \alpha(2t_1, t_2, \dots, t_n)$ for $0 \leq t_1 \leq 1/2$,
$(\beta * \alpha)(t_1, t_2, \dots, t_n) = \beta(2t_1 - 1, t_2, \dots, t_n)$ for $1/2 \leq t_1 \leq 1$.
This is well-defined up to homotopy (relative to the boundary), and the identity is the class of the constant map to $x_0$.
For $n \geq 3$, one can define additional pairings $\star_i$ for $1 \leq i \leq n-1$, which concatenate along the $i$-th coordinate instead:
$(\beta \star_i \alpha)(t_1, \dots, t_n) = \alpha(t_1, \dots, 2t_i, \dots, t_n)$ for $0 \leq t_i \leq 1/2$,
$(\beta \star_i \alpha)(t_1, \dots, t_n) = \beta(t_1, \dots, 2t_i - 1, \dots, t_n)$ for $1/2 \leq t_i \leq 1$.
These pairings all induce the same group structure on $\pi_n(X, A)$.
Moreover, by the Eckmann-Hilton argument (adapted to the relative setting), these operations distribute over each other in a way that forces commutativity. Specifically, consider two elements $[\alpha]$ and $[\beta]$. Using pairings along two different directions (say $\star_1$ and $\star_2$), one can construct a homotopy that interchanges the order of concatenation: the "horizontal" and "vertical" compositions (analogous to those in the absolute case) satisfy $ ([\delta] \star_1 [\gamma]) \star_2 ([\beta] \star_1 [\alpha]) = ([\delta] \star_2 [\beta]) \star_1 ([\gamma] \star_2 [\alpha]) $, and by filling in constant maps for $[\gamma]$ and $[\delta]$, this reduces to $[\beta] \star_1 [\alpha] = [\alpha] \star_1 [\beta]$ after showing the operations coincide. The extra dimension (since $n \geq 3$) provides "room" to perform this interchange via a continuous deformation, ensuring the group is abelian.
For $n=2$, there is effectively only one direction for concatenation (up to the relative boundary constraints), so the multiple pairings do not exist in the same way, and the Eckmann-Hilton argument does not apply to force commutativity—leading to examples where $\pi_2(X, A)$ is non-abelian (e.g., when $A$ is a wedge of circles and $X$ is chosen appropriately). |
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