The classical (absolute) Hurewicz theorem states that, for \(X\) a simply connected space, the first non-vanishing homotopy group and the first non-vanishing homology group are isomorphic. The isomorphism is realized by the Hurewicz homomorphism \(h\colon \pi_i(X) \to H_i(X)\) in the appropriate degree. Now \(h\) induces the rationalized Hurewicz homomorphism \(H\colon \pi_i(X)\otimes\mathbb{Q} \to H_i(X; \mathbb{Q})\) and in this rational setting the statement may be strengthened as follows: If \(X\) is simply connected and \(\pi_i(X)\otimes\mathbb{Q} = 0\) for \(1< i < r\), then the rationalized Hurewicz homomorphism is an isomorphism for \(1 \leq i < 2r-1\) and is onto for \(i = 2r-1\). This paper gives a proof of the latter result that is elementary in the sense that it avoids relying on spectral sequence arguments.
The basic ingredients used in this proof are: the classical Hurewicz theorem; the Künneth theorem; the long exact homology sequence of a pair; a sporadic topological fact, namely that \(K(\mathbb{Z}/p, 1)\) may be realized as a suitable infinite-dimensional Lens space. From these ingredients, the rational Thom isomorphism and rational Gysin and Wang sequences are developed in an accessible and essentially self-contained way. Then easy induction arguments using familiar fibrations, such as a path-loop fibration over an Eilenberg-Mac Lane space, obtain the results. As a bonus, the basic facts concerning rational cohomology of Eilenberg-Mac Lane spaces and rational homotopy groups of spheres are also established.