The Index of Discontinuous Vector Fields
by
Daniel H. Gottlieb and Geetha Samaranayake
Introduction
At the frontier between the continuous and the discrete there is a naturall*
*y 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. A vector field is an assignment of*
* tangent
vectors to some, not necessarily all, of the points of N. We make no assumption*
*s 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 th*
*e 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 zero vect*
*or, or which
contains one of those points in every neighborhood.
We are interested in the connected components of the defects and how they c*
*hange
in time. Those connected components of D which are compact we will call topolo*
*gical
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 is*
*olated particle
we can assign an integer which we call the index of C in V . We denote this by *
*Ind(C).
The key properties of Ind(C) are that it is nontrivial, additive over parti*
*cles, 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 electr*
*on 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 *
*vector field
only if the set of defects changing under time is unbounded, since the proton h*
*as 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 ea*
*ch other,
can be made precise by introducing the concept of otopy, which is a generalizat*
*ion of the
concept of homotopy. An otopy is a vector field on N x I so that each vector is*
* tangent
to a slice N x t. Thus an otopy is a vector field W on N x I so that W (n; t) i*
*s tangent to
N xt. 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 Ci of defects on V0 transforms into a set of components of *
*defects Dj
of V1 if there is a connected component T of the defects of W so that T \ (N x *
*0) = [Ci
and T \ (N x 1) = [Dj. If T is a compact connected component of defects of W , *
*which
transforms a set of isolated particles Ci into isolated particles Dj, then we s*
*ay 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 U*
* containing
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 the closure of an open set so that the set of defec*
*ts is compact
and there is no defect on the frontier of the open set. We say such a vector fi*
*eld is proper
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 |U). We set Ind(C) = Ind(V |U).
Next we define Ind(V ) with domain U to be equal to the index of V |M where*
* M U
is a smooth compact manifold with boundary containing the defects of V in its i*
*nterior.
We can find such an M since the defects are a compact set in U.
We call a vector field V defined on a compact manifold M proper if there ar*
*e 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 @- M to get @- V . Then we define Ind(V ) by*
* the equation
(*) Ind(V ) = O(M) - Ind(@- V )
where O(M) denotes the Euler-Poincare number of M. We know that @- V is a pro*
*per
vector field with domain @- M since the set of defects is compact unless there *
*is 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 is a proper vector field with domain the open set @- M which is on*
*e dimension
lower than M. Then Ind(@- V ) is defined in turn by finding a compact manifold *
*containing
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 *
*simply 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 *
*assuming
that the index for vector fields is already defined for compact manifolds with *
*boundary.
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.
2
The well-definedness of this definition will involve the first four section*
*s of this paper.
In section 5 we summarize the useful properties of the index which we have prov*
*ed along
the way, along with a few proved in other papers. The key property is that of *
*a proper
otopy described below.
Suppose that V is a proper vector field with open domain U. A proper otop*
*y is a
proper vector field W defined on N x I with domain an open set where we require*
* W to
be tangent to the slices. Then we say W is a proper otopy of V if V is the rest*
*riction of
W to N x 0 and the domain of W intersects N x 0 in U. The key property of the i*
*ndex of
proper vector fields with open domains is that the index is invariant under pro*
*per otopy.
For connected manifolds the converse is true: Two proper vector fields are prop*
*erly otopic
if and only if they have the same index.
We may generalize the concept of otopy in two ways. Recall an otopy is an o*
*pen set T
on N x I with a vector field W which is tangent to the slices. Now this can be *
*generalized
by considering a fibre bundle E ! B with fibre N and an open set T on E and a v*
*ector
field W whose vectors are tangent to the fibre. It is clear that if W is a prop*
*er vector field,
that is the defects form a compact set and there are no defects on the frontier*
* of T , then
W restricted to any fibre has an index. This index is the same for every fibre.*
* In [B-G],
for the case of continuous W , it is shown that there is an S-map which induces*
* a transfer
on homology with trace equal to this index.
The second way to generalize an otopy is to note that N x I can be thought *
*of as
a manifold S with a natural non-zero vector field. Then W is a vector field wh*
*ich is
orthogonal to this vector field. In fact any vector field can be projected orth*
*ogonal to the
natural vector field. If S is a space-time, there is a field of light cones. If*
* we consider a
space-like vector field W on S, it is like an otopy. W restricts to any space-l*
*ike slice and
projects tangent to it. The index of the defects at any event is thus an invari*
*ant of general
relativity, it is invariant under any change of coordinate system. The defects*
* propagate
through space-time and the index satisfies a conservation law, just like the co*
*nservation
law of electric charges under particle collisions. It is very easy to believe t*
*hat the index of
a vector field, as here exposed, must lead to an explanation of the conservatio*
*n of physical
properties under collision based on the idea of connectivity and continuity and*
* pointing
inside.
1. The definition for one-dimensional manifolds
The inductive definition begins with empty vector fields, that is domains w*
*hich are
empty. This could arise since @- M is empty if V never points inside from the b*
*oundary.
We define the index of an empty vector field to be equal to zero. Zero dimensio*
*nal manifolds
consist of discrete sets of points. The only vectors are zero vectors, so for a*
* vector field to
be proper it must consist of a finite number of zeros. One-dimensional compact *
*manifolds
with boundary consist of a finite disjoint union of compact components which ar*
*e compact
3
intervals. We use the definition (*), that is
Ind(V )= (number of components) - (number of boundary
points where V is pointing inwards):
In the case of components without boundaries, circles in this case, we define t*
*he index to
be O(circle) = 0.
Lemma 1.1. Two vector fields V and V 0are properly otopic if and only if
Ind(@- V ) = Ind(@- V 0) on each component of the boundary:
Proof. Let W be a vector field so that W (m) = V (m)=kV (m)k for m on the b*
*oundary
of M. Assume that W (m) = 0 outside a collar of the boundary, and assume that W
continuously decreases in size from the unit vectors on the boundary to the zer*
*o vectors at
the other end of the collar. Then we define the homotopy tV + (1 - t)W . This i*
*s a proper
homotopy, since at any point m on the boundary V (m) and W (m) both point eithe*
*r inside
or outside so no zero can arise on the boundary. If V should have a defect at s*
*ome m in
the interior, we may alter V by assigning V (m) = 0. Thus the homotopy is defin*
*ed. Now
both V and V 0are properly otopic to W , hence they are otopic to each other.
Lemma 1.2. If M is a finite collection of manifolds with boundary and f is *
*a diffeo-
morphism so that the related vector field is denoted by V *, then
Ind(V ) = Ind(V *):
Proof. Pointing inside is preserved under diffeomorphism.
Lemma 1.3. If V has no defects, then Ind(V ) = 0.
Proof. Each connected component of M is an interval. Since V has no defects*
* on this
interval, V must point outside on one end and inside on the other. Thus Ind(V )*
* = 1-1 = 0
on this interval, and thus on all the intervals. So Ind(V ) = 0 is true for M.
Now suppose that the arena is a connected manifold N with no boundary and n*
*ot
compact. Thus an open interval. Then we define the index of V with open doma*
*ins to
be the index of V restricted to a union of compact intervals which contain the *
*defects of
V . This is well-defined. If M and M0 are two manifolds with boundary contain*
*ing the
defects, there is a compact manifold with boundary M00containing both M and M0.*
* The
vector field V restricted to M00- int(M) is a nowhere zero vector field, and th*
*e previous
lemma and the fact that the index is additive proves that the index is well-def*
*ined.
Next we deal with the case of the arena N being a closed manifold, in this *
*case that is a
finite set of circles. We will consider the case of a single circle, the genera*
*l case will be given
4
by adding the indices for each connected component. The set of defects is close*
*d. If the
defects can be contained in a compact manifold with boundary, in this case diff*
*eomorphic
to a closed interval, we define the index of V to be the index of V restricted *
*to the compact
manifold. On the other hand, if the domain of V is the entire arena, then we de*
*fine
Ind(V ) = O(arena) - Ind(@- V ) = O(circle) - Ind(empty vector field) = *
*0:
These two definitions are consistent. If V has domain the entire circle, *
*then it is
properly homotopic to the zero vector field. Then we homotopic the zero vector *
*field to
V 0which is zero inside a large closed interval and not zero around a point wit*
*h the vectors
thus forced to point in the same sense around the circle. Then V 0restricted to*
* the large
closed interval has index zero which is just what the global definition gives.
We make a few more observations before we finish with the one-dimensional c*
*ase.
Lemma 1.4. Given a connected arena N, two proper vector fields are properly*
* otopic
if and only if they have the same index. For every integer n there is a vector *
*field whose
index equals that integer.
Proof. Suppose we have a proper otopy W with domain T on N x I. Let Vt deno*
*te W
restricted to N x t. We show that there is some interval about t such that Vs h*
*as the same
index for all s in the interval. Since the set of defects of the otopy is compa*
*ct we can find a
compact manifold M so that M x J, for some closed interval J, lies in T and con*
*tains the
defects inside @M x J. Thus the proper homotopy Vt on M x J preserves the index*
* on M,
and hence the proper otopy on N x J preserves the index on N as t runs over J. *
*Thus we
have a finite sequence of vector fields each having the same index as the previ*
*ous vector
field. Hence the first and last vector fields have equal indices. Conversely, f*
*or any integer
n, let Wn be the vector field consisting of |n| vector fields defined on disjoi*
*nt open intervals
in N, each one of index 1 if n > 0 and of index -1 if n < 0. Thus Ind(Wn) = n. *
*Now if
V has index n, we must show that V is properly homotopic to Wn. Now the domain *
*of
V consists of open connected intervals, and only a finite number of them contai*
*n defects.
Each of these intervals has index equal to 1, -1, or 0. Now V is properly otopi*
*c to the
same vector field V whose domain is restricted to only those intervals which ha*
*ve nonzero
indices. Now if two adjacent intervals have different indices, there is a prope*
*r otopy which
leaves the rest of the vector field fixed, and removes the two intervals of opp*
*osite indices.
After a finite number of steps we are left with either an empty vector field, i*
*f n = 0, or a
Wn. The empty vector field is W0. Thus V is properly otopic to Wn.
5
Lemma 1.5. The index of a vector field on an open manifold is invariant und*
*er diffeo-
morphism.
Proof. Immediate from Lemma 1.2 and the definition of index for open manifo*
*lds.
Lemma 1.6. Let V be a vector field over a domain U and suppose that U is th*
*e disjoint
union of U1 and U2. Then if V1 and V2 denote V restricted to U1 and U2 respecti*
*vely, we
have
Ind(V ) = Ind(V1) + Ind(V2):
2. The index defined for compact n-manifolds
The otopy extension property. Let V be a continuous vector field on a clos*
*ed
manifold N. Let U be an open set in N. Any continuous proper otopy of V on the *
*domain
U can be extended to a continuous homotopy of V on all of N.
Proof. The continuous proper otopy implies there is a continuous vector fie*
*ld W on
an open set T in N x I which extends to the closure of T with no zeros on the f*
*rontier
and which is V when restricted to N x 0. This vector field W can be thought o*
*f as a
cross-section to the tangent bundle over N x I defined over a closed subset. I*
*t is well
known that cross-sections can be extended from closed sets to continuous cross-*
*sections
over the whole manifold.
We assume that the index is defined for (n - 1)-manifolds in such a way tha*
*t all the
lemmas of section 1 hold.
First we consider the case of compact manifolds such that every component i*
*s a mani-
fold with boundary. We suppose that V is a proper vector field on such a manifo*
*ld M. We
choose a vector field N on the boundary @M which points outside of M. Every vec*
*tor v
at a point m on @M can be uniquely written as v = t + kN(m) where t is a vector*
* tangent
to @M and k is some real number. We say t is the projection of v tangent to @M.*
* Then
@V is the vector field obtained by projecting V tangent to @M. Now we define @-*
* V by
restricting @V to @- M, the set of points such that V is pointing inward. Then *
*we define
(*) Ind(V ) = O(M) - Ind(@- V ):
Lemma 2.1. Ind(V ) is well-defined.
Proof. We have already defined the index on (n - 1)-dimensional manifolds w*
*ith open
domains for proper vector fields. Note that @- V is proper since V is, since th*
*e frontier of
@- M is a subset of @0M where V is tangent to @M. So a defect of @- V on the fr*
*ontier
must come from a defect of V on @M. Hence Ind(@- V ) is defined. Now the vector*
* field
@- V obviously depends upon the outward pointing N. If we had another outward p*
*ointing
vector field N0 we would project down to a different @- V , call it W . Now the*
* homotopy of
6
vector fields Nt = tN + (t - 1)N0 always points outside of M for every t. Hence*
* it induces
a homotopy from @- V to W and this homotopy is proper. Thus Ind(@- V ) = Ind(W *
*).
We will also allow the case where N is not defined on a closed set of @M wh*
*ich is
disjoint from the frontier of @- M. Then @V has defects, but @- V is still p*
*roper. A
homotopy between N and N0, as in the lemma, still induces a proper otopy betwee*
*n @- V
and W , so the Ind(V ) is still well-defined in this case also. This case aris*
*es when M is
embedded as a co-dimension zero manifold in such a way that it has corners. Th*
*en the
natural outward pointing normal in this situation is not defined on the corners*
*. But we
still have the index defined if none of the corners is on the frontier of @- M.
Now our goal is to prove that non-zero vector fields have index equal to ze*
*ro on compact
manifolds with boundary.
Theorem 2.2. V is properly otopic to W if and only if
Ind(@- V ) = Ind(@- W )
for every connected component of @M. So as a corollary in the case that @M is c*
*onnected,
we have that V is properly otopic to W if and only if Ind(V ) = Ind(W ). If V a*
*nd W are
both continuous, then "otopic" can be replaced by "homotopic" in the above stat*
*ements.
Proof. The theorem is true for manifolds one dimension lower by lemma 1.1. *
*A proper
otopy of V to W induces a proper otopy from @- V to @- W in the arena @M. He*
*nce
Ind(@- V ) = Ind(@- W ). Hence Ind(V ) = Ind(W ) from (*). Conversely, we can*
* find a
smooth collar @M x I of the boundary so that V restricted to this collar has no*
* defects.
Then we otopy V to V 0where V 0is defined by V 0(m; t) = tV (m) for a point in *
*the collar
and V 0= 0 outside the collar. Now since Ind(@- V ) = Ind(@- W ) for each conn*
*ected
component of the boundary, we can find a proper otopy from @- V to @- W . Now*
* this
otopy can be extended to a homotopy of @V to @W by the otopy extension property*
*. This
homotopy in turn can be used to define a proper homotopy from V 0to W 0. Here w*
*e assume
W 0has the same definition relative to W as V 0has to V . Thus W is properly ot*
*opic to V .
Lemma 2.3. Suppose V is a proper vector field on a compact manifold M each*
* of
whose components has a non-empty boundary. Let @M x I be a collar of the bounda*
*ry
so small so that V has no defects on the collar. Then V restricted to M minus t*
*he open
collar @M x (0; 1] has the same index as V .
Proof. Let @Vt denote the projection of V tangent to the submanifold @M xt *
*for every
t in I. Let W be the vector field on the collar defined by W (m; t) = @- Vt if*
* (m; t) is a
point in @- M x t. Then W is a proper otopy, proper since V has no defects on t*
*he collar.
Thus Ind(@- V ) = Ind(@- V0) and hence Ind(V ) = O(M) - Ind(@- V ) equals the i*
*ndex of
V restricted to M0 = M - open collar, because the indices of the @- vector fiel*
*ds are the
same on their respective boundaries and O(M) = O(M0).
7
Lemma 2.4. Let V be a proper continuous vector field on M. Suppose that @- *
*V is
properly otopic to some vector field W on @M. Then there is a proper homotopy o*
*f V to
a proper continuous vector field X so that @- X = W and the zeros of each stage*
* of the
homotopy Vt are not changed.
Proof. Use the otopy extension property to find a homotopy Ht from @V to a *
*vector
field on @M which we shall call @X. Let n(m; t) be a continuous real valued fu*
*nction
on @M x I which is positive on the open set T of the otopy between @- V and W ,*
* zero
on the frontier of T , and negative in the complement of the closure of T , and*
* so that
n(m; 0) = n(m) where V (m) = n(m)N(m) + @V (m) defines n(m). Such a function ex*
*ists
by the Tietze extension theorem. Using n(m; t), we define a vector field X0 on*
* @M x I
by X0(m; t) = n(m; t)N(m) + Ht(m). We adjoin the collar to M as an external co*
*llar
and extend the vector field V by X0 to get the continuous vector field X. Now M*
* with
the external collar is diffeomorphic to M. Under this diffeomorphism X becomes *
*a vector
field which we still denote by X. We may assume this diffeomorphism was so chos*
*en that
X = V outside of a small internal collar. Then the homotopy tX + (1 - t)V is th*
*e required
homotopy which does not change the zeros of V .
Lemma 2.5. If V is a vector field with no defects on an n-ball, then Ind(V *
*) = 0.
Proof. For the standard n-ball of radius 1 and center at the origin, we def*
*ine the homo-
topy Wt(r) = V (tr). This homotopy introduces no zeros and shows that V is homo*
*topic
to the constant vector field. The constant vector field has index equal to zero*
*, as can be
seen by using (*). If we have a ball diffeomorphic to the standard ball, then t*
*he index of
the vector field under the diffeomorphism is preserved, and hence it has the ze*
*ro index. If
the ball is embedded with corners so that the corners are not on the frontier o*
*f the set of
inward pointing vectors of V , then the index is defined and by lemma 2.3 it is*
* equal to the
index of V restricted to a smooth ball slightly inside the original ball. This *
*index is zero.
Theorem 2.6. If V is a vector field with no defects on a compact manifold s*
*uch that
all the components have non-empty boundary, then Ind(V ) = 0.
Proof. Now M can be triangulated and suppose we have proved the theorem for*
* man-
ifolds triangulated by k - 1 n-simplicies. The previous lemma proves the case k*
* = 1. We
divide M by a manifold L of one lower dimension into manifolds M1 and M2 each c*
*overed
by fewer than k n-simplicies so that the theorem holds for them.
We arrange it so that L is orthogonal to @M. We use lemma 2.4 to homotopy V*
* to a
vector field with no defects so that the new V is pointing outside orthogonally*
* to @M at
L \ @M. Then a simple counting argument shows that Ind(V ) = 0 since the restri*
*ctions
of V to M1 and M2 have index zero. This argument works if M has no corners. If *
*M has
corners we find a collar of M which is a smooth embedding of @M x t for all t b*
*ut the last
t = 1. Then by lemma 2.3 above, we find that V , restricted to the manifold bou*
*nded by
@M x t for t close enough to 1, has the same index as V . That is zero.
8
The counting argument goes as follows. By induction, Ind(V |M1) = Ind(V |M*
*2) =
0. Thus Ind(@- V1) = O(M1) and Ind(@- V2) = O(M2). Now Ind(@- V ) = Ind(@- V1*
*) +
Ind(@- V2) - Ind(W ) where W is the projection of V on the common part of the b*
*oundary
of M1 and M2, that is L. This follows from repeated applications of lemma 1.6.*
* Now
Ind(W ) = O(L) since W points outwards at the boundary of L. Hence
Ind(@- V ) = Ind(@- V1) + Ind(@- V2) - Ind(W ) = O(M1) + O(M2) - O(L) = O(M*
*):
Hence Ind(V ) = 0 from (*).
3. The index for open n-manifolds
Let N be an n-manifold and let V be a proper vector field on N with domain *
*U. Then
the set of defects of V in U is compact. Thus we can find a compact manifold M *
*which
contains the defects of V . We define
(**) Ind(V ) = Ind(V |M):
Lemma 3.1. Ind(V ) is well-defined.
Proof. If M and M0 are two manifolds with boundary containing the defects, *
*there is
a compact manifold with boundary M00containing both M and M0. The vector field V
restricted to M00- int(M) is a nowhere zero vector field. Then Theorem 2.6 impl*
*ies that
the index of V restricted to M00- int(M) is zero. Now the index of V restricted*
* to M00
equals the index of V restricted to M by the following lemma.
Lemma 3.2. Suppose M is the union of two manifolds M1 and M2 where the three
manifolds are compact manifolds with boundary so that the intersection of M1 an*
*d M2
consist of part of the boundary of M1 and is disjoint from the boundary of M. S*
*uppose
that V is a proper vector field defined on M which has no defects on the bounda*
*ries of M1
and M2. Then Ind(V ) = Ind(V1) + Ind(V2) where Vi= V |Mi.
Proof.
Ind(V )= O(M) - Ind(@- V )
= O(M) - (Ind(@- V1) + Ind(@- V2) - Ind(@- V1|L) - Ind(@- V2|L))
where L = M1 \ M2. Now
Ind(@- V1|L) + Ind(@- V2|L) = Ind(@- V1|L) + Ind(@+ V1) = O(L):
Thus
Ind(V ) = O(M1) + O(M2) - Ind(@- V1) - Ind(@- V2) = Ind(V1) + Ind(V2);
as was to be proved.
9
Lemma 3.3. Let V be a proper vector field with domain U. Suppose U is the u*
*nion
of two open sets U1 and U2 such that the restriction of V to each of them and t*
*o U1 \ U2
is a proper vector field denoted V1 and V2 and V12 respectively. Then
(***) Ind(V ) = Ind(V1) + Ind(V2) - Ind(V12):
Proof. We choose disjoint compact manifolds M1, M2, and M12 containing the *
*zeros
of V which lie in U1- U12 and U2- U12 and U12 respectively. Then the index of V*
* is equal
to the index of V restricted to the union of M1, M2, and M12. But the index of *
*V1 is the
index of V restricted to M1 and M12, and the index of V2 is the index of V rest*
*ricted to
M2 and M12, and the index of V12 is the index of V restricted to M12. Hence cou*
*nting the
index gives the equation (* * *).
Theorem 3.4. Given a connected arena N, two proper vector fields are propel*
*y otopic
if and only if they have the same index. For every integer n there is a vector *
*field whose
index equals that integer.
Proof. Suppose we have a proper otopy W with domain T on N x I. Let Vt deno*
*te W
restricted to N x t. We show that there is some interval about t such that Vs h*
*as the same
index for all s in the interval. Since the set of defects of the otopy is compa*
*ct we can find
a compact manifold M so that M x J, for some closed interval J, lies in T and c*
*ontains
the defects so that the defects avoid @M x J. Thus the proper homotopy Vt on M*
* x J
preserves the index on M, and hence the proper otopy on N x J preserves the ind*
*ex on
N as t runs over J. Thus we have a finite sequence of vector fields each having*
* the same
index as the previous vector field. Hence the first and last vector fields have*
* equal indices.
Conversely, for any integer k, let Wk be the vector field consisting of |k|*
* vector fields
defined on disjoint open balls in N, each one of index 1 if k > 0 or of index -*
*1 if k < 0.
Thus Ind(Wk) = k. Now if V has index k, we must show that V is properly homotop*
*ic to
Wk. Now the defects of V form a compact set which are contained in a compact ma*
*nifold
with boundary M so that V is defined and has no defects on the boundary. We m*
*ay
proper otopy V first to a continuous vector field, and then to a smooth vector *
*field. Then
we consider V as a cross-section to the tangent bundle of M. Using the transv*
*ersality
theorem, we can smoothly homotopy the cross-section so that it is transversal t*
*o the zero
section of the tangent bundle keeping the cross-section fixed over the boundary*
*. The
dimensions are such that the intersection consists of a finite number of points*
*. Thus we
proper otopy V to a vector field with only a finite number of zeros. Now we put*
* small open
balls around each of these zeros. The index of the vector field on the ball aro*
*und each of
these zeros is either 1 or -1. This follows from transversality, but we do not*
* need that
fact. We may find a diffeomorphic n-ball which contains exactly |k| zeros so th*
*at around
these zeros the vector field restricts to Wk. The two vector fields have the sa*
*me index on
the n-ball and thus are properly homotopic, since from (*) the index on the bou*
*ndary of
10
the inward pointing @- vector fields is the same, and so by induction they are *
*properly
otopic, hence by the otopy extension property the @ vector fields are homotopic*
*. This
homotopy can be extended to a homotopy of the two vector fields originally on t*
*he n-ball.
Then using the sequence of homotopies and otopies, we can piece together a prop*
*er otopy
of V to Wk.
Corollary 3.5. The proper homotopy classes of continuous proper vector fiel*
*ds on
a compact manifold with connected boundary is in one-to-one correspondence with*
* the
integers via the index.
Lemma 3.8. The index of a vector field on an open manifold is invariant und*
*er diffeo-
morphism.
Lemma 3.9. The index of a vector field V on a closed manifold M whose domai*
*n is
the whole of M is equal to O(M).
Proof. First otopy V to the zero vector field. Then homotopy the zero vecto*
*r field to
a vector field V 0so that it is a non-zero vector field on a small n-ball B abo*
*ut a point.
Now let V1 be V 0on the n-ball and let V2 be V 0on the complement. Then Ind(V1)*
* = 0,
so Ind(@- V1) = 1. Now Ind(@- V2) = (-1)n-1. So
Ind(V2) = O(M - B) - (-1)n-1 = O(M) - (-1)n - (-1)n-1 = O(M):
Hence Ind(V ) = Ind(V1) + Ind(V2) = 0 + O(M).
4. The Index of particles
Let V be a vector field on an arena N. Let D be the set of defects of V .*
* Then D
breaks up into a set of connected components Di. We define an index for each co*
*mponent
Di which is compact and is an open set in the subspace topology of D. That is,*
* in the
terminology of the Introduction, we define the index of an isolated particle. F*
*or isolated
particles we can find a compact manifold M containing Di and no other defects. *
*Then we
define
(****) Ind(Di) = Ind(V |M):
P Now if we have a finite number of particles Di in the domain of V , then In*
*d(V ) =
iInd(Di). However it is possible that V is a proper vector field and there ar*
*e an infinite
number of Di. Then at least one of the Di is not isolated in D. But the index o*
*f V is still
defined. This event is very rare in practical situations. A one dimensional exa*
*mple occurs
when M is the interval [-1; 1] and the vector field V is defined by V (x) = x s*
*in(1=x) for
x 6= 0 and V (0) = 0. Then 0 is a connected component of the defects which is n*
*ot open in
the set of zeros of V .
11
If we have an otopy Vt, we imagine the components of the defects Dt as chan*
*ging under
time. We can say that Dtiat time t transforms without radiation into Dsj at ti*
*me s if
there is a compact connected component T of the defects of the otopy from time *
*t to time
s so that T intersects N x t in exactly Dtiand T intersects N x s exactly at Ds*
*i. The
index of Dtiis the same as the index of Dsj if T is compact. In other words if*
* a finite
number of particles Di at time t are transformed into a finite number of partic*
*les Cj at
time s by a compact T , the sum of the indices are conserved. That is
X X
(1) Ind(Ci) = Ind(Dj):
Thus the idea of otopy allows us to make precise the concept of defects mov*
*ing with
time and changing with time and undergoing collisions. The index is conserved u*
*nder these
collisions as long as the "world line" T of the component is compact. That is, *
*as long as
there are is no radiation.
5. Properties of the Index
(2) Ind(V ) + Ind@- V = O(M)
This is in fact the equation (*) which defines the index.
(3) Let N be a connected arena. V is a properly otopic to W if and only if Ind*
*V = IndW .
For any integer n there is a vector field W so that n = Ind W .
(4) Suppose M is a compact manifold so that @M is connected, and suppose V and*
* W
are continuous proper vector fields on M. Then V is properly homotopic to W if *
*and only
if Ind V = Ind W . For any integer n there is a continuous proper vector field *
*W so that
n = Ind W .
(5) If M is a closed compact manifold and V is a vector field whose domain is *
*all of M,
then Ind V = O(M).
Proof. Property (3) and (4) are Theorem 3.4 and Corollary 3.5 respectively *
*for the
homotopy part. For the fact that n = Ind W for some vector field W , we apply (*
*2) and
induction starting with Lemma 1.4. The proof of (5) is Lemma 3.9.
(6) Let A and B be open sets and let V be a proper vector field on A [ B so th*
*at V |A
and V |B are also proper. Then Ind(V |A [ B) = Ind(V |A) + Ind(V |B) - Ind(V |A*
* \ B).
Proof of (6). Lemma 3.3
(7) Suppose V us a vector field with no defects. Then Ind V = 0.
Proof. Theorem 2.6 for compact manifolds with boundary.
12
(8) Suppose V is a proper vector field andPthe set of defects consists of a fi*
*nite number
of connected components Di. Then Ind V = Ind(Di).
i
Proof. This follows from the definition of Ind(Di) and (3).
(9) Let V and W be proper vector fields on A and B respectively. Let V x W be *
*a vector
field on AxB defined by V xW (s; t) = (V (s); W (t)). Then Ind(V xW ) = (Ind V *
*).(Ind W ).
Proof. We can assume that A and B are open sets in their arenas. Then V is *
*otopic to
Vn where Vn is restricted to a finite set of open sets in A homeomorphic to the*
* interior of
Ik when k = dim A and so that Vn(t1; : :;:tk) = (t1; t2; : :;:tk) where the +t*
*1 is taken
if Ind V is positive and -t1 is taken if Ind V is negative. The index ofPthe Vn*
*|Ik is 1
respectively (by induction on (9)). So Ind (V x W ) = (Ind Vn x Wn) = Ind(Vn*
*|Iki) x
i;j
(Wn|I`j). Now it is easy to see that Ind(Vn|Iki) x (Wn|I`j)) = Ind(Vn|Iki) . In*
*d(Wn|Ikj)).
(10) (-1)nInd(V ) = Ind(-V ) where n = dim M.
Proof. The theorem is true for n = 1. Assume it is true for (n - 1)-manifol*
*ds. Now
using (2) we have
Ind(-V )= O(M) - Ind(@- (-V )) by (2)
= O(M) - Ind(-@+ V ) by definition of @- V and @+ V
= O(M) - (-1)n-1Ind(@+ (V )) by induction
= O(M) + (-1)n(O(@M) - Ind(@- V ))
since
O(@M) = Ind(@- V ) + Ind(@+ V ):
If n is even then
Ind(-V ) = O(M) + (0 - Ind(@- V )) = Ind V by (2):
If n is odd then
Ind (-V )= O(M) - (2O(M) - Ind(@- V ))
= -(O(M) - Ind(@- V )) = -Ind V by (2)
(11) Suppose M is a compact sub-manifold of Rn of 0-codimension. Let f : M ! R*
*n be
a map so that f(@M) does not contain the origin. Define a proper vector field V*
* fon M
by V f(m) = f(m). Then Ind V f= deg f0 where f0 : @M ! Sn-1 by f0(m) = _f(m)_kf*
*(m)k.
Proof. We homotopy f if necessary so that "0is a regular value. Then f-1 ("*
*0) is a finite
set of points. There is a neighborhood of f-1 (0) of small balls so that f : @(*
*ball) ! Rn-0 ~=
13
Sn-1 . Now, in each of these small balls, f has either degree 1 or -1. If degre*
*e equals 1,
then f|@(ball) is homotopic to the identity. If degree = -1, then f|@(ball) is *
*homotopic
to reflection about the equator. In these cases Ind(V f|ball) = 1 = deg f|@(bal*
*l). Now
X
Ind(V f)= IndV f|(ball) by proper otopy
X
= deg f|@(balls) = deg f0:
(12) Suppose f : M ! Rn where M Rn is a codimension zero compact manifold.
Define Vf(m) = m - f(m). Then Ind Vf = fixed point index of f (assuming no fix*
*ed
points on @M)
Proof. The fixed point index is defined to be the degree of the map m ! _m-*
*f(m)__km-f(m)k
from @M ! Sn-1 . Hence by (11) we have the result
(13) Let f : M ! N where M and N are Riemannian manifolds and f is a smooth ma*
*p.
Let V be a vector field on M. Define the pullback vector field f*(V ) by
= :
Then if f : Mm ! Rn so that f*|@M has maximal rank and f(@M) contains no zeros*
* of
V , then X
Ind f*V = viwi+ (O(M) - deg N^)
where vi= Ind(xi) where xi is the ith zero of V , wi is the winding number of f*
*|@M about
xi, and N^: @M ! Sn-1 is the normal (or Gauss) map.
Proof. In paper [G5].
14
REFERENCES
[G1] Daniel H. Gottlieb, A certain subgroup of the fundamental group, Amer. J.*
* Math., 87 (1966), pp.
1233-1237.
[G2] _______________, A de Moivre formula for fixed point theory, ATAS do 5O E*
*ncontro Brasiliero
de Topologia, Universidade de S"ao Paulo, S"ao Carlos, S.P. Brasil, 31 (1*
*988), pp. 59-67.
[G3] _______________, A de Moivre like formula for fixed point theory, Proceed*
*ings of the Fixed
Point Theory Seminar at the 1986 International Congress of Mathematicians*
*, R. F. Brown (editor),
Contemporary Mathematics, AMS Providence, Rhode Island, 72, pp. 99-106.
[G4] _______________, On the index of pullback vector fields, Proc. of the 2nd*
* Siegen Topology Sym-
posium, August 1987, Ulrich Koschorke (editor), Lecture Notes of Mathemat*
*ics, Springer Verlag,
New York.
[G5] _______________, Zeroes of pullback vector fields and fixed point theory *
*for bodies, Algebraic
topology, Proc. of Intl. Conference March 21-24, 1988, Contemporary Mathe*
*matics, 96, pp. 168-180.
[G6] _______________, Vector fields and classical theorems of topology, Rencon*
*ti del Seminario Matem-
atico e Fisico, Milano.
[M] Marston Morse, Singular points of vector fields under general boundary co*
*nditions, Amer. J. Math,
51 (1929), pp. 165-178.
[P] Charles C. Pugh, A generalized Poincare index formula, Topology, 7 (1968)*
*, pp. 217-226.
Purdue University
15