The Pontrjagin-Hopf invariants for Sobolev maps

TL;DR

This paper develops a complete set of homotopy invariants for Sobolev maps from 3-manifolds to the 2-sphere, extending classical invariants to less regular maps and ensuring they match smooth case classifications.

Contribution

It introduces a new framework for homotopy invariants applicable to Sobolev maps, aligning with classical invariants for smooth maps and extending their applicability to finite energy maps.

Findings

Invariants extend to Sobolev maps with finite Faddeev energy.

Invariants for finite energy maps match those of smooth maps.

Provides applications to the Faddeev model.

Abstract

Subtle issues arise when extending homotopy invariants to spaces of functions having little regularity, e.g., Sobolev spaces containing discontinuous functions. Sometimes it is not possible to extend the invariant at all, and sometimes, even when the formulas defining the invariants make sense, they may not have expected properties (e.g., there are maps having non-integral degree). In this paper we define a complete set of homotopy invariants for maps from 3-manifolds to the 2-sphere and show that these invariants extend to finite Faddeev energy maps and maps in suitable Sobolev spaces. For smooth maps, our description is proved to be equivalent to Pontrjagin's original homotopy classification from the 1930's. We further show that for the finite energy maps the invariants take on exactly the same values as for smooth maps. We include applications to the Faddeev model. The techniques that we use would also apply to many more problems and/or other functionals. We have tried to make the paper accessible to analysts, geometers and mathematical physicists.