The Whitehead Theorem and the Hurewicz Theorem

Abstract

Chapter 4 introduces the homotopy groups of a space with a base point and establishes several basic results about these groups. The Hurewicz homomorphism from these groups to the homology groups is defined. Whitehead’s theorem that a map between CW complexes inducing an isomorphism on homotopy groups is a homotopy equivalence is stated and proved. Brouwer’s theorem computing the $\mathrm{n}$ th homotopy group of the n-sphere is stated and proved. The Hurewicz theorem, which states that for a simply connected space the first nonzero homotopy group and reduced homology group are isomorphic, is proved. Lastly, the homotopy exact sequence of a fibration is stated and proved.