On the Pontryagin-Steenrod-Wu theorem

We present a short and direct proof (based on the Pontryagin-Thom construction) of the following Pontryagin-Steenrod-Wu theorem: (a) Let M be a connected orientable closed smooth (n+1)-manifold, n>=3. Define the degree map deg: \pi^n(M) \to H^n(M;Z) by the formula deg f=f*[S^n], where [S^n] \in H^n(M;Z) is the fundamental class. The degree map is bijective if there exists \beta \in H_2(M,Z/2Z) such that \beta \cdot w_2(M)\ne 0. If such \beta does not exist, then deg is a 2-1 map; and (b) Let M be an orientable closed smooth (n+2)-manifold, n>=3. An element \alpha lies in the image of the degree map if and only if \rho_2 \alpha \cdot w_2(M)=0, where \rho_2 :Z \to Z/2Z is reduction modulo 2.