Elementary methods of calculation
Hatcher §4.2 (pages 360–392). Mapped, not yet decomposed.
Homotopy groups have no excision, but they have a substitute: the homotopy
excision theorem of Blakers and Massey holds in a range of dimensions, and the
Freudenthal suspension theorem follows, giving πₙ(Sⁿ) ≅ ℤ and the existence
of stable homotopy groups. The Hurewicz theorem then pins the first nontrivial
homotopy group to the corresponding homology group, which is the main bridge
between Chapters 2 and 4. Fiber bundles and their long exact sequence supply
the remaining computations, including π₃(S²) ≅ ℤ from the Hopf bundle.
Nothing here is upstream. The section depends on Chapter 2's homology and on homotopy groups.