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 $\deg:L_1(M)\to H_1(M;\Z)$bethemaptakingaframedlinktoitshomologyclass\mathrm{.}Thenforeach$\alpha\in H_1(M;\Z)$thereisa1-1correspondencebetweentheset$\deg\nolimits^{-1}\alpha$andthegroup$\Bbb Z_{2d(\alpha)}$,where$d(\alpha)$isthedivisibilityoftheprojectionof$\alpha$ to the free part of $H_1(M;\mathbb{Z})$.