Classification of framed links in 3-manifolds

TL;DR

This paper provides a concise proof of Pontryagin's theorem relating framed links in 3-manifolds to their homology classes, establishing a clear correspondence with cyclic groups based on divisibility.

Contribution

It offers a simplified, detailed proof of a classical theorem connecting framed links and homology in 3-manifolds, previously known through complex and unpublished methods.

Findings

Establishes a bijection between framed links and cyclic groups based on homology divisibility.

Clarifies the structure of the set of framed links up to cobordism in terms of algebraic invariants.

Provides a more accessible proof of a fundamental theorem in 3-manifold topology.

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 $\deg:L_1(M)\to H_1(M;\Z)$ 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 $H_1(M;\Bbb Z)$.