Introduction

For a generalized homology theory, one of the Eilenberg–Steenrod axioms is excision. It says that for an excisive triad \((X; A, B)\text{,}\) the inclusion \((A, A \cap B) \hookrightarrow (X, B)\) induces an isomorphism on homology. One of the reasons that homotopy groups are so much harder to compute than homology groups is that the analogous map on homotopy groups is not in general an isomorphism. In Chapter 10 we worked around this failure using fibrations, for which \(\pi_n(E, F) \cong \pi_n(B)\text{.}\) In this chapter we explore both the failure and a second, partial replacement, the Homotopy Excision Theorem. It says that excision does hold for homotopy groups in a range of degrees controlled by connectivity. We derive Freudenthal’s Suspension Theorem as a consequence, and use it to introduce stable homotopy groups and spectra. The moral is that the failure of excision is an unstable phenomenon, which disappears after enough suspensions (Remark 11.16).