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Section 6.1 Relative Homotopy Groups

Throughout, \(I^n = [0,1]^n\) is the \(n\)-cube with coordinates \((s_1, \ldots, s_n)\text{,}\) and \(\partial I^n\) is its boundary, the set of points with some coordinate equal to \(0\) or \(1\text{.}\) The definition of relative homotopy groups singles out one face of the cube.

Definition 6.1.

Let \(n \geq 1\text{.}\) The distinguished face of \(I^n\) is the face
\begin{equation*} I^{n-1} = \{\,(s_1, \ldots, s_n) \in I^n \mid s_n = 0\,\}, \end{equation*}
which we identify with the \((n-1)\)-cube in the obvious way. We write
\begin{equation*} J^{n-1} = \overline{\partial I^n \setminus I^{n-1}} \end{equation*}
for the closure of the complement of the distinguished face in \(\partial I^n\text{,}\) i.e. the union of the remaining \(2n - 1\) faces. Thus \(\partial I^n = I^{n-1} \cup J^{n-1}\text{,}\) and the two pieces meet in \(I^{n-1} \cap J^{n-1} = \partial I^{n-1}\text{.}\)

Definition 6.2.

Let \((X, A)\) be a pair of spaces and \(x_0 \in A\text{.}\) A map of triples \(f \colon (I^n, \partial I^n, J^{n-1}) \to (X, A, x_0)\) is a continuous map \(f \colon I^n \to X\) with
\begin{equation*} f(\partial I^n) \subseteq A \qquad\text{and}\qquad f(J^{n-1}) = \{x_0\}. \end{equation*}
A homotopy of triples between two such maps is a homotopy \(H \colon I^n \times I \to X\) such that each \(H(-,t)\) is a map of triples, i.e.
\begin{equation*} H(\partial I^n \times I) \subseteq A \qquad\text{and}\qquad H(J^{n-1} \times I) = \{x_0\}. \end{equation*}
For \(n \geq 1\text{,}\) the \(n\)th relative homotopy group of the pair \((X, A)\) at \(x_0\) is the set
\begin{equation*} \pi_n(X, A, x_0) \;=\; \bigl\{\, f \colon (I^n, \partial I^n, J^{n-1}) \to (X, A, x_0) \,\bigr\} \,\big/\, \text{homotopies of triples}, \end{equation*}
with the class of the constant map \(c_{x_0}\) as a distinguished element.
The key feature of a map of triples \(f\) is its restriction to the distinguished face: since \(\partial I^{n-1} \subseteq J^{n-1}\text{,}\) the restriction
\begin{equation*} f|_{I^{n-1}} \colon (I^{n-1}, \partial I^{n-1}) \to (A, x_0) \end{equation*}
is a based map, i.e. it represents an element of \(\pi_{n-1}(A, x_0)\text{.}\)

Remark 6.3.

Collapsing \(J^{n-1}\) to a point gives a homeomorphism of triples
\begin{equation*} \bigl(I^n / J^{n-1},\; \partial I^n / J^{n-1},\; J^{n-1}/J^{n-1}\bigr) \;\cong\; (D^n, S^{n-1}, s_0), \end{equation*}
since \(I^n / J^{n-1}\) is an \(n\)-disc whose boundary is \(I^{n-1}/\partial I^{n-1} \cong S^{n-1}\text{.}\) A map of triples \((I^n, \partial I^n, J^{n-1}) \to (X, A, x_0)\) is constant on \(J^{n-1}\text{,}\) so it factors uniquely through this quotient. Hence elements of \(\pi_n(X, A, x_0)\) may equivalently be represented by maps of triples
\begin{equation*} f \colon (D^n, S^{n-1}, s_0) \to (X, A, x_0), \end{equation*}
modulo homotopies of triples \(H \colon D^n \times I \to X\) with \(H(S^{n-1} \times I) \subseteq A\) and \(H(\{s_0\} \times I) = \{x_0\}\text{.}\) We will switch freely between the cube model and the disc model, using whichever is more convenient.

Example 6.4.

Two extreme cases are worth recording.
  1. If \(A = \{x_0\}\text{,}\) then a map of triples is just a map \(f \colon I^n \to X\) with \(f(\partial I^n) = \{x_0\}\text{,}\) and a homotopy of triples is a based homotopy. Hence
    \begin{equation*} \pi_n(X, \{x_0\}, x_0) = \pi_n(X, x_0). \end{equation*}
    In this sense the relative homotopy groups generalise the absolute ones.
  2. If \(A = X\text{,}\) then \(\pi_n(X, X, x_0) = 0\) for all \(n \geq 1\text{,}\) i.e. it consists only of the class of \(c_{x_0}\text{.}\) Indeed, given a map of triples \(f\text{,}\) define
    \begin{equation*} H(s_1, \ldots, s_n, t) = f\bigl(s_1, \ldots, s_{n-1},\, (1-t)s_n + t\bigr). \end{equation*}
    This “pushes \(f\) towards the top face”. The condition \(H(\partial I^n \times I) \subseteq X\) is vacuous. If \((s_1, \ldots, s_n) \in J^{n-1}\text{,}\) then either \(s_n = 1\text{,}\) or \(s_i \in \{0,1\}\) for some \(i < n\text{;}\) in both cases \((s_1, \ldots, s_{n-1}, (1-t)s_n + t)\) again lies in \(J^{n-1}\text{,}\) so \(H(J^{n-1} \times I) = \{x_0\}\text{.}\) Finally \(H(-,0) = f\) and \(H(-,1) = f(-, 1) = c_{x_0}\text{,}\) because the top face lies in \(J^{n-1}\text{.}\)

The group structure.

Remark 6.5.

For \(n \geq 2\) we define a sum on \(\pi_n(X, A, x_0)\) by concatenation in the \(s_1\)-direction, exactly as for absolute homotopy groups in Chapter 5:
\begin{equation*} (f + g)(s_1, s_2, \ldots, s_n) = \begin{cases} f(2 s_1, s_2, \ldots, s_n), & 0 \leq s_1 \leq \tfrac{1}{2}, \\[2pt] g(2 s_1 - 1, s_2, \ldots, s_n), & \tfrac{1}{2} \leq s_1 \leq 1. \end{cases} \end{equation*}

Remark 6.6.

It is important that we concatenate in a direction \(s_i\) with \(i < n\text{:}\) the coordinate \(s_n\) is reserved for the distinguished face and cannot be used for this purpose. In particular the sum is only defined for \(n \geq 2\text{.}\)

Proof.
This is the Eckmann–Hilton argument again.

Remark 6.8.

In contrast with the absolute case, \(\pi_2(X, A, x_0)\) is in general not abelian. We will see an explicit example in Example 6.19 below.

Remark 6.9.

The set \(\pi_1(X, A, x_0)\) is not a group in general. Its elements are homotopy classes of maps
\begin{equation*} f \colon (I,\, \{0,1\},\, \{1\}) \to (X,\, A,\, x_0), \end{equation*}
i.e. paths in \(X\) that start somewhere in \(A\) and end at \(x_0\text{,}\) where during a homotopy the starting point may move around inside \(A\) while the endpoint stays fixed at \(x_0\text{.}\) There is no way to concatenate two such paths in general, since the starting point of one need not agree with the endpoint of the other. We regard \(\pi_1(X, A, x_0)\) as a pointed set, with basepoint the class of the constant path.
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