• Mikhail B Skopenkov

      Articles written in Proceedings – Mathematical Sciences

    • Classification of Framed Links in 3-Manifolds

      Matija Cencelj Dušan Repovš Mikhail B Skopenkov

      More Details Abstract Fulltext PDF

      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 $\deg: L_1(M)\to H_1(M;\mathbb{Z})$ be the map taking a framed link to its homology class. Then for each $\alpha\in H_1(M;\mathbb{Z})$ there is a one-to-one correspondence between the set $\deg^{-1}\alpha$ and the group $\mathbb{Z}_{2d(\alpha)}$, where $d(\alpha)$ is the divisibility of the projection of 𝛼 to the free part of $H_1(M;\mathbb{Z})$.

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