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.

On the Pontryagin-Steenrod-Wu theorem / D., Repovs; M., Skopenkov; Spaggiari, Fulvia. - In: ISRAEL JOURNAL OF MATHEMATICS. - ISSN 0021-2172. - STAMPA. - 145(2005), pp. 341-347.

On the Pontryagin-Steenrod-Wu theorem

SPAGGIARI, Fulvia
2005

Abstract

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.
145
341
347
On the Pontryagin-Steenrod-Wu theorem / D., Repovs; M., Skopenkov; Spaggiari, Fulvia. - In: ISRAEL JOURNAL OF MATHEMATICS. - ISSN 0021-2172. - STAMPA. - 145(2005), pp. 341-347.
D., Repovs; M., Skopenkov; Spaggiari, Fulvia
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11380/304136
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