The Index of Discontinuous Vector Fields
by
Daniel H. Gottlieb and Geetha Samaranayake
Introduction
At the frontier between the continuous and thediscrete there is a naturally *
*occurring
additive, integral "quantum number" which is preserved under "collisions" of di*
*scontinu-
ities. This quantum number depends only on the basic topological notions of com*
*pactness,
connectedness, dimension, and the concept of pointing inside.
We assume we are in a smooth manifold N. Avector field is an assignment of t*
*angent
vectors to some, not necessarily all, of the points of N. We make no assumptio*
*ns about
continuity. We will call this N the arena for our vector fields. We consider *
*the set of
defects of a vector field V in N, that is the set D which is the closure of the*
* set of all
zeros, discontinuities and undefined points of V. That is we consider a defect *
*to be a point
of N at which V is either not defined,or is discontinuous, or is the zerovector*
*, or which
contains one of those points in every neighborhood.
We are interested in the connected components of thedefects and how they cha*
*nge
in time. Those connected components of D which are compact we will call topolog*
*ical
particles. If we can find an open set about a particle which does not intersect*
* any defect
not in the particle itself,then we say the particle is isolated. If C is an iso*
*lated particle
we can assign an integer which we call the index of C in V. We denote this by I*
*nd(C ).
The key properties of Ind(C) are that it is nontrivial,additive over particl*
*es, easy to
calculate and is conserved under interactions with proper components as V varie*
*s under
time. For example, let V be the electric vector field generated by one electro*
*n in R3.
Then the position of the electron e is the only defect and Ind(e) = 1. Now if V*
* changes
under time in such a way that there are only a finite number of particles at ea*
*ch time,all
contained in some large fixed sphere, then the sum of the indices of the partic*
*les at each
time t is equal to 1. Thus the electron vector field can change to the proton v*
*ector field
only if the set of defects changingunder time is unbounded, since the proton ha*
*s index
+1 which is different from the index of the electron. In this case we will say*
* that the
transformation of the electron to the proton involves " topological radiation".
Vector fields varying under time,and defect components interacting with each*
* other,
can be made precise by introducing the concept of otopy, which is a generalizat*
*ion of the
concept of homotopy. An otopy isa vector field on N I so that each vector is ta*
*ngent
to a slice N t. Thus an otopy is a vector field W on N I so that W(n;t) is tang*
*ent to
N t.We say that V0 is otopic to V1 if V0(n) = W(n;0) and V1(n) = W(n; 1).We say*
* that
a set of components Ciof defects on V0transforms into a set of components of de*
*fects Dj
and T " (N 1) = [Dj. If T is a compact connected component of defects ofW ,which
transforms a set of isolated particles Ciinto isolated particlesDj, then we say*
* there is no
topological radiation and
X X
(1) Ind(Ci) = Ind(Dj):
If T is not compact, we say there is topological radiation.
We define Ind(C) as follows. Since C is an particle, there is an open set Uc*
*ontaining
C so that there are no defects in the closure of U except for C. We can define *
*an index for
any vector field defined on theclosure of an open set so that the set of defect*
*s is compact
and there is no defect on the frontier of the open set. We say such a vector fi*
*eld isproper
with domain the open set. In the case at hand, V restricted to cl(U) is proper *
*with domain
U. Hence we can define Ind(V jU). We set Ind(C) = Ind(VjU ).
Next we define Ind(V ) with domain Uto be equal to the index of V jM where M*
* aeU
is a smooth compact manifold with boundary containing the defects of V in its i*
*nterior.
We can find such an Msince the defects are a compact set in U.
We call a vector field V defined on a compact manifold M proper if there are*
* no defects
on the boundary. Consider the open set of the boundary where V points inside. W*
*e denote
that set by @ M. We define the vector field @ V with domain @ M in the arena *
*@M by
letting @ V be the end product of first restricting V to the boundary and then *
*projecting
each vector so that it is tangent to @M which results in a vector field @V tang*
*ent to @M,
and then finally restricting @V to @ Mto get @ V . Then we define Ind(V ) by *
*the equation
(*) Ind(V ) = \(M) Ind(@ V )
where \(M) denotes the Euler-Poincare number of M. We know that @ V is a proper
vector field with domain @ M since the set of defects is compact unless there i*
*s a defect
at the the frontier of @ M. If there were such a defect,it would be a zero of V*
* tangent to
@M and hence a zero of V on the boundary, so V would not have been proper.
Now @ V isa proper vector field with domain the open set @ M which is one d*
*imension
lower than M. Then Ind(@ V) is defined in turn by finding a compact manifold c*
*ontaining
the defects of @ V and using equation (*). We continue this process until eith*
*er @ M is
a zero dimensional manifold where every point is a defect and so Ind(@ V ) is s*
*imply the
number of points, or where @ M empty in which case Ind(@ V ) = 0.
To summarize, we define the index of a proper vector field V with domain U a*
*ssuming
that the index for vector fields isalready defined for compact manifolds with b*
*oundary.
Then the index of V is defined to be the index of V restricted to a compact s*
*mooth
manifold with boundary of codimension zero containing all the defects of V in U*
*. We will
show this definition is well-defined, that is it does not depend on the chosen *
*manifold with
boundary, by showing that a vector field with no defects defined on a compact m*
*anifold
with boundary has index zero.