We present a short and direct proof (based on the Pontryagin-Thom construction) of the following Pontryagin-Steenrod-Wu theorem: (a) LetM be a connected orientable closed smooth (n + 1)-manifold,n≥3. Define the degree map deg: π n (M) →H n (M; ℤ) by the formula degf =f*[S n ], where [S n ] eH n (M; ℤ) is the fundamental class. The degree map is bijective, if there existsβ eH 2(M, ℤ/2ℤ) such thatβ ·w 2(M) e 0. If suchβ does not exist, then deg is a 2-1 map; and (b) LetM be an orientable closed smooth (n+2)-manifold,n≥3. An elementα lies in the image of the degree map if and only ifρ 2 α ·w 2(M)=0, whereρ 2: ℤ → ℤ/2ℤ is reduction modulo 2.