On the pontryagin-steenrod-wu theorem |
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| Authors: | Email author" target="_blank">Du?an?Repov?Email author Mikhail?Skopenkov Fulvia?Spaggiari |
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| Institution: | (1) Institute for Mathematics, Physics and Mechanics, University of Ljubljana, P. O. Box 2964, 1001 Ljubljana, Slovenia;(2) Department of Differential Geometry, Faculty of Mechanics and Mathematics, Moscow State University, 119992 Moscow, Russia;(3) Dipartimento di Matematica, Università di Modena e Reggio Emilia, via Campi 213/B, 41100 Modena, Italy |
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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: π
n
(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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| Keywords: | |
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