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 gives a concise proof of a classical theorem of Pontryagin concerning the classification of framed links in a closed oriented smooth three‑manifold M. A framed link is a collection of disjoint circles embedded in M together with a trivialization of the normal bundle (a framing). Two framed links are considered equivalent if there exists a framed cobordism between them, i.e., a smoothly embedded surface in M ×