On the Pontryagin-Steenrod-Wu theorem

TL;DR

This paper provides a concise proof of a specific case of the Pontryagin-Steenrod-Wu theorem, classifying homotopy classes of maps from (n+1)-manifolds to spheres based on Stiefel-Whitney classes and homology.

Contribution

It offers a direct proof of the bijective or double-cover nature of the degree map for certain manifolds, linking topology and homotopy classification.

Findings

The degree map is bijective when w_2(M) x r_2 H_2(M;Z) is nonzero.

The degree map is 2-to-1 otherwise.

The proof utilizes the Pontryagin-Thom construction and geometric definitions of Stiefel-Whitney classes.

Abstract

This paper is on homotopy classification of maps of (n+1)-dimensional manifolds into the n-dimensional sphere. For a continuous map f of an (n+1)-manifold into the n-sphere define the degree deg f to be the class dual to f^*[S^n], where [S^n] 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>2. Then the map deg:\pi^n(M)\to H_1(M;Z) is 1-to-1 (i.e., bijective), if the product w_2(M) x r_2 H_2(M;Z) is nonzero, where r_2 is the mod2 reduction; 2-to-1 (i.e., each element of H_1(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 w_2(M).