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

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