### 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 three-manifolds to the two-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.

Original language | English (US) |
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Pages (from-to) | 121-181 |

Number of pages | 61 |

Journal | Communications in Contemporary Mathematics |

Volume | 12 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1 2010 |

### Keywords

- Faddeev model
- Hopf invariant
- Integrality
- Pontrjagin invariant
- Skyrme model
- Sobolev maps
- Variational problems

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Communications in Contemporary Mathematics*,

*12*(1), 121-181. https://doi.org/10.1142/S0219199710003695