The Index of Discontinuous Vector Fields

摘要: The concept of the index a vector eld is one oldest in Algebraic Topology. First stated by Poincare and then perfected Heinz Hopf S. Lefschetz Marston Morse, it developed as sum local indices zeros eld, using idea degree map initially isolated zeros. must be dened everywhere continuous. A key property that invariant under proper homotopies. In this paper we extend classical to elds which are not required continuous necessarily everywhere. more general situation, homotopy corresponds new call otopy. Not only otopy, but classies otopy classes. Thus two properly otopic if they have same index. This allows us go back case classify globally up classication theorems allow dene for space-like on Lorentzian space-time where becomes an relativity.