On the commutativity of the relative homotopy groups

Question

Can you explain to me why relative homotopy groups $\pi_{n}(X, A; x_0)$ are commutative for $n \geq 3$? It would be great if you will show me explicit homotopy.

Answer 1

It is easiest to see this using a model of relative homotopy groups $\pi_n(X,A,a)$ in terms of maps $f$ of an $n$-cube $I^n \to X$ which maps the face $\partial^-_1$ to $A$ and all other faces $\partial^\pm _i$ to the base point $a$. (Here $I^n$ consists of points $x =(x_1, \ldots, x_n) \in \mathbb R^n$ such that $0 \leqslant x_i \leqslant 1$ and the face $\partial^-_i$ has $x_i=0$, while $\partial ^+_i$ has $x_i=1$.) Given $i >1$ and two such maps $f,g$ such that $f(x)=g(y) $ if $x_i=1, y_i=0$ and $x_j=y_j $ for $j \ne i$ (this condition can be written $ \partial^+_i f= \partial^-_i g$) we can define $h=f+_ig: I^n \to X$ by

$$h(x) = f(x_1, \ldots, x_{i-1},2x_i ,x_{i+1}, \ldots, x_n) \; \text{if} \; x_i \leqslant 1/2,$$

$$h(x) = g(x_1, \ldots, x_{i-1},2x_i-1 ,x_{i+1}, \ldots, x_n) \; \text{if}\; x_i \geqslant 1/2. $$

(Compare the addition of paths.) If $n \geqslant 3$ and $i\ne j$ are $>1$ then we find the interchange law

$$(f+_i g)+_j (h+_ik) = (f+_jh)+_i (g+_j k) $$ whenever both sides are defined, and so both sides can be represented by the matrix

$$\begin{bmatrix} f& h \\ g & k \end{bmatrix} \; \;\; \begin{matrix} & \rightarrow &j\\\downarrow && \\ i & & \end{matrix} $$

where down is direction $i$ and across is direction $j$. This rule passes to homotopy classes, and we can now apply the standard interchange law argument to get the result asked for. The usual addition on relative homotopy groups is given by $+_2$ but for $n \geqslant 3$ we can also use $+_3$ and the argument says they agree, and are commutative.

It is not quite so easy to extract an explicit homotopy from this argument!

The above can be put in the explicit context of the cubical singular complex of a filtered space, and so obtain further useful properties of relative homotopy groups. See the book, with pdf available, Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids.

Later: on second thoughts, the homotopy required is the composite of the homotopies, in which $0$ stands for the constant map with value $a$:

$$\begin{bmatrix} f&g \end{bmatrix} \simeq \begin{bmatrix} f&0\\0&g \end{bmatrix} \simeq \begin{bmatrix} 0&f\\g&0 \end{bmatrix}\simeq \begin{bmatrix} g&f \end{bmatrix}. $$

Discussion of the use of matrix notation and of the use of the interchange law is on p. 148-149 of the above book.