Classification of framed links in 3-manifolds

Abstract

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