ar X iv : 0 80 8 . 12 09 v 1 [ m at h . G T ] 8 A ug 2 00 8 ON THE PONTRYAGIN - STEENROD - WU THEOREM

This paper is on the homotopy classification of maps of (n+1)-dimensional manifolds into the n-dimensional sphere. For a continuous map f : M → S define the degree deg f ∈ H1(M;Z) to be the class dual to f[S], where [S] ∈ H(S;Z) is the fundamental class. We present a short and direct proof of the following specific case of the Pontryagin-Steenrod-Wu theorem: Theorem. Let M be a connected orientable closed smooth (n + 1)-manifold, n ≥ 3. Then the map deg : π(M) → H1(M ;Z) is 1-to-1 (i. e., bijective), if w2(M) · ρ2H2(M ;Z) 6= 0; 2-to-1 (i. e., each element α ∈ H1(M ;Z) has exactly 2 preimages) — otherwise. The proof is based on the Pontryagin-Thom construction and a geometric definition of the Stiefel– Whitney classes wi(M).