The concept of the index of a vector field is one of the oldest inAlgebraicTopology. First stated by Poincare and then perfected by Heinz Hopf andS.Lefschetz and Marston Morse, it is developed as the sum of local indicesof thezeros of the vector field, using the idea of degree of a map andinitiallyisolated zeros. The vector field must be defined everywhere and becontinuous. Akey property of the index is that it is invariant under properhomotopies.In this paper we extend this classical index to vector fields which arenotrequired to be continuous and are not necessarily defined everywhere. Inthismore general situation, proper homotopy corresponds to a new conceptwhich wecall proper otopy. Not only is the index invariant under proper otopy,but theindex classifies the proper otopy classes. Thus two vector fields areproperlyotopic if and only if they have the same index. This allows us to goback to the continuous case and classify globally defined continuous vector fields up toproper homotopy classes. The concept of otopy and the classificationtheorems allow us to define the index for space-like vector fields onLorentzian space-time where it becomes an invariant of generalrelativity.
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