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On the pontryagin-steenrod-wu theorem

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Abstract

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: πn(M) →H n(M; ℤ) by the formula degf =f*[S n], where [S n] εH n(M; ℤ) is the fundamental class. The degree map is bijective, if there existsβ εH 2(M, ℤ/2ℤ) such thatβ ·w 2(M) ε 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.

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Authors and Affiliations

  1. Institute for Mathematics, Physics and Mechanics, University of Ljubljana, P. O. Box 2964, 1001, Ljubljana, Slovenia

    Dušan Repovš

  2. Department of Differential Geometry, Faculty of Mechanics and Mathematics, Moscow State University, 119992, Moscow, Russia

    Mikhail Skopenkov

  3. Dipartimento di Matematica, Università di Modena e Reggio Emilia, via Campi 213/B, 41100, Modena, Italy

    Fulvia Spaggiari

Authors
  1. Dušan Repovš
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  2. Mikhail Skopenkov
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  3. Fulvia Spaggiari
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Correspondence to Dušan Repovš.

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Repovš, D., Skopenkov, M. & Spaggiari, F. On the pontryagin-steenrod-wu theorem. Isr. J. Math. 145, 341–347 (2005). https://doi.org/10.1007/BF02786699

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  • DOI: https://doi.org/10.1007/BF02786699

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