Classification of framed links in 3-manifolds |
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| Authors: | Matija Cencelj Dušan Repovš Mikhail B Skopenkov |
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| Institution: | (1) Institute for Mathematics, Physics and Mechanics and Faculty of Education, University of Ljubljana, P.O. Box 2964, 1001 Ljubljana, Slovenia;(2) Department of Differential Geometry, Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119992, Russia |
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| Abstract: | We present a short and complete proof of the following Pontryagin theorem, whose original proof was complicated and has never
been published in detail. Let M be a connected oriented closed smooth 3-manifold, L
1(M) be the set of framed links in M up to a framed cobordism, and deg: L
1(M) → H
1(M; ℤ) be the map taking a framed link to its homology class. Then for each α ∈ H
1(M; ℤ) there is a one-to-one correspondence between the set deg−1
α and the group ℤ2d(α), where d(α) is the divisibility of the projection of α to the free part of
H
1(M; ℤ). |
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| Keywords: | Framed link framed cobordism framing normal bundle normal Euler class homotopy classification of maps cohomotopy set degree of a map Pontryagin-Thom construction |
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