 

Daniel H. Gottlieb and Geetha Samaranayake
The Index of Discontinuous Vector Fields


Published: 
August 3, 1995 
Keywords: 
manifolds, homotopy, degree, fixed point, critical point, spacetime 
Subject: 
55M20, 55M25, 57R25 


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


Author information
Daniel H. Gottlieb:
Department of Mathematics, Purdue
University}, West Lafayette
Indiana, 47907
gottlieb@math.purdue.edu
http://www.math.purdue.edu/~gottlieb/
Geetha Samaranayake:
Department of Mathematics, Ohio State Universith, Newark, Ohio

