Classification of framed links in 3-manifolds

We present a short proof of the following Pontryagin theorem, whose original proof was complicated and has never been published in details: {\bf Theorem.} Let $M$ be a connected oriented closed smooth 3-manifold. Let $L_1(M)$ be the set of framed links in $M$ up to a framed cobordism. Let $\deg:L_1(M)\to H_1(M;\Z)$ be the map taking a framed link to its homology class. Then for each $\alpha\in H_1(M;\Z)$ there is a 1-1 correspondence between the set $\deg\nolimits^{-1}\alpha$ and the group $\Bbb Z_{2d(\alpha)}$, where $d(\alpha)$ is the divisibility of the projection of $\alpha$ to the free part of $H_1(M;\Bbb Z)$.

💡 Research Summary

The paper addresses the classification problem for framed links in a closed, oriented, connected smooth 3‑manifold M. Let L₁(M) denote the set of framed 1‑dimensional submanifolds of M modulo framed cobordism, and let
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