• Classification of Framed Links in 3-Manifolds

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/117/03/0301-0306

• # Keywords

Framed link; framed cobordism; framing; normal bundle; normal Euler class; homotopy classification of maps; cohomotopy set; degree of a map; Pontryagin–Thom construction.

• # Abstract

We present a short and complete proof of the following Pontryagin theorem, whose original proof was complicated and has never been published in detail. Let 𝑀 be a connected oriented closed smooth 3-manifold, $L_1(M)$ be the set of framed links in 𝑀 up to a framed cobordism, and $\deg: L_1(M)\to H_1(M;\mathbb{Z})$ be the map taking a framed link to its homology class. Then for each $\alpha\in H_1(M;\mathbb{Z})$ there is a one-to-one correspondence between the set $\deg^{-1}\alpha$ and the group $\mathbb{Z}_{2d(\alpha)}$, where $d(\alpha)$ is the divisibility of the projection of 𝛼 to the free part of $H_1(M;\mathbb{Z})$.

• # Author Affiliations

1. Institute for Mathematics, Physics and Mechanics and Faculty of Education, University of Ljubljana, P.O. Box 2964, 1001 Ljubljana, Slovenia
2. Department of Differential Geometry, Faculty of Mechanics and Mathematics, Moscow State University, Moscow 119992, Russia

• # Proceedings – Mathematical Sciences

Volume 131, 2021
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019