New questions related to the topological degree

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23 Scopus citations


Degree theory for continuous maps has a long history and has been extensively studied, both from the point of view of analysis and topology. If f ∈ C 0(S n, S n), deg f is a well-defined element of ℤ, which is stable under continuous deformation. Starting in the early 1980s, the need to define a degree for some classes of discontinuous maps emerged from the study of some nonlinear PDEs (related to problems in liquid crystals and superconductors). These examples involved Sobolev maps in the limiting case of the Sobolev embedding; see Sections 2 and 3 below. (Topological questions for Sobolev maps strictly below the limiting exponent have been investigated in [15] and [14].) In these cases, the Sobolev embedding asserts only that such maps belong to the space VMO (see below) and need not be continuous.

Original languageEnglish (US)
Title of host publicationProgress in Mathematics
PublisherSpringer Basel
Number of pages18
StatePublished - 2006

Publication series

NameProgress in Mathematics
ISSN (Print)0743-1643
ISSN (Electronic)2296-505X

All Science Journal Classification (ASJC) codes

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology


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