The Homotopy Excision Theorem

Section 11.2 The Homotopy Excision Theorem

Although excision fails in general, it does hold in a range of dimensions controlled by the connectivity of the pairs.

Theorem 11.6. Homotopy Excision.

Let \((X; A, B)\) be a CW triad (Remark 11.3) such that \(C := A \cap B\) is nonempty and connected. If \((A, C)\) is \(m\)-connected and \((B, C)\) is \(n\)-connected (with \(m, n \geq 1\)), then the inclusion-induced map

\begin{equation*} \pi_i(A, C) \to \pi_i(X, B) \end{equation*}

is an isomorphism for \(i < m + n\) and a surjection for \(i = m + n\text{.}\)

We will not prove Theorem 11.6 here; the standard reference is Hatcher §4.2. Its consequences, however, are the main content of this chapter.

In homology the corresponding map is an isomorphism in all degrees (Remark 11.2). So the theorem says that in low degrees, relative to the connectivity of the pieces, homotopy groups behave like homology groups.

Example 11.7.

Return to Example 11.4. Give each sphere the CW structure with one \(0\)-cell (the wedge point), one \(1\)-cell (the rest of the equator) and two \(2\)-cells (the open hemispheres). Then the unthickened \(A\) and \(B\) are subcomplexes, \((X; A, B)\) is a CW triad, and \(C = A \cap B = S^1 \vee S^1\) is connected. Since \(A \simeq *\text{,}\) the long exact sequence gives \(\pi_i(A, C) \cong \pi_{i-1}(S^1 \vee S^1)\text{,}\) so \((A, C)\) is \(1\)-connected but not \(2\)-connected, and likewise for \((B, C)\text{.}\) With \(m = n = 1\text{,}\) Theorem 11.6 predicts an isomorphism for \(i < 2\) and a surjection for \(i = 2\text{.}\)

For \(i = 2\) the map is a homomorphism

\begin{equation*} F_2 = \pi_1(S^1 \vee S^1) \cong \pi_2(A, C) \longrightarrow \pi_2(X, B) \cong \pi_2(S^2 \vee S^2) \cong \mathbb{Z}^2. \end{equation*}

One can show that this map is the abelianization \(F_2 \to \mathbb{Z}^2\text{.}\) It is surjective, as predicted, but not injective, so the range in Theorem 11.6 cannot be improved.