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\topmatter
\title  The Index of Discontinuous Vector Fields
\endtitle
\rightheadtext{Vector Fields} 

         % Put all authors' names within the next environment.
\author Daniel H. Gottlieb and Geetha Samaranayake
\endauthor
\leftheadtext{Gottlieb and Samaranayke} 

         % Put all acknowledgments for support in the next
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  % Give separate address and email, entries for each author.
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\address Department of Mathematics, Purdue University, West Lafayette
Indiana, 47907       
\endaddress
\email gottlieb@math.purdue.edu
\address Department of Mathematics, Ohio State University, Newark, Ohio
\endadress
\email geetha@math.ohio-state.edu      
\endemail

       % Keywords are important, as they may be used by browsers to
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\keywords manifolds, homotopy, degree, fixed point, critical point, space-time
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\subjclass 55M20, 55M25, 57R25
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       % Abstracts are required, and will be distributed by a listserv list.
\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 space-like vector fields on
Lorentzian space-time where it becomes an invariant of general
relativity.


\endabstract 
% Place any personal macros used in the document here.
\define\TI{\text{Ind}}
\define\TIU{\text{Ind}_U}
\define\TIM{\text{Ind}_M}
\define\TIN{\text{Ind}_N}
\define\CL{\text{cl}}
\define\Ti{\text{ind}}

\endtopmatter
\document %%%% **** The text of the paper starts here **** %%%%





 

%\baselineskip=18pt    
\noindent
\head  Introduction
\endhead 

The authors would like to thank James C. Becker for many excellent suggestions
which improved the definitions and eliminated subtle errors of
exposition.

\subhead
A. The Results
\endsubhead

We generalize the notion of homotopy of vector fields to that of otopy
of vector fields.
Using otopy we can:
\subsubhead $1$
\endsubsubhead
Classify the proper homotopy classes of vector fields.
The index is a proper homotopy class invariant but two vector fields with
the same index may not be proper homotopic. (See (4) in Section 5)

\subsubhead $2$
\endsubsubhead
Show that two vector fields are properly otopic if and only if they
have the same index. (See (3))

\subsubhead $3$
\endsubsubhead
Extend the definition of index to any vector field, without
hypotheses. (See Subsection C in the Introduction)
\subsubhead $4$
\endsubsubhead
Define a local 
index for any connected set of defects for any vector field
instead of merely for isolated zeros. (See the penultimate paragraph of
C
or Section 4)
Under an otopy these defects move and interact.
The following conservation law holds.
The sum of the indices of the incoming defects is equal to the sum of the
indices of the outgoing defects. (See (1))
\subsubhead $5$
\endsubsubhead
Demonstrate that the concept of the Index of a vector field depends
only on elementary differential topology, the concept of pointing inside,
and the Euler-Poincare number. This is done in Sections 1, 2, 3, 4.
The classical approach depends on the degree of a map.
In this paper we show how the degree of a map might be defined via the index
of a vector field by using (11).
\subsubhead $6$
\endsubsubhead
We can study vector fields along the fibre on fibre bundles. An otopy
generalizes to a vector field along the fibre $V$ restricted to an open set. For
a proper $V$ only certain values of the index of $V$ restricted to a fibre are
possible, (See (14)). For example the Hopf fibrations of spheres admit only the
index zero.
\subsubhead $7$
\endsubsubhead
We can study space-like vector fields on a Lorentzian space-time,
 $M$. The concept of otopy
 generalizes to a space-like vector field restricted to an open set
of $M$. For a
proper space-like vector field $V$ the index of $V$ restricted to a space-like
slice is independent of the slice. 
Thus the index is an invariant of General Relativity. Hence
it should be used to describe physical phenomena. 
For example, the Coulomb electric vector
field $E$ of an electron or a proton has index $-1$ or $1$ respectively.
This remains true no matter what what coordinate system is used to 
describe the field. 

\subsubhead 
\endsubsubhead

\head  B. Organization of the paper.
\endhead
%%%%%%%%%%%%%%%%%%


There are a few features to
be explicitly noted. First we are actually defining two types of indices.
These are usually denoted  $\TIU (V)$, and $\Ti (P)$. The first 
takes values in the integers and $\infty$ and the last takes on the value
$ - \infty$ as well. Second, there are two different definitions of these
indices. The advanced definition 
is based on the definition of index already defined for
continuous vector fields and is found in Subsection C of the Introduction. The
elementary 
definition is given inductively in Sections 1, 2, 3, 4. This definition
is equivalent to the first and the proof that it is well-defined is completely
self contained, using only pointset topological methods. The only
algebraic topological notion is that of the Euler Poincare number.

Subsection C of the Introduction establishes notation and the formal
concept of proper otopy as well as the key example of otopy which forces
the concept on us.
If the reader draws a few pictures and understands
what is to be formalized, the formal definitions will be obvious
except for a few small details. The formal
definition of otopy is in Subsection C along with the advanced 
definition of index.

Subsection D of the Introduction contains guides for three different ways
for reading the paper. Especially contained in D is a simplified description of
the index which if combined with the list of key properties of Section 5 should
give the reader the essence of the subject without the technicalities of the
proof.


The main burden of the paper is the definition of the elementary
definition of the index. It is here that the two different types of
index, $\TIU (V)$, and $\Ti (P)$, are carefully defined. The
definition is made inductively on the dimension of the manifolds and are
shown to be well-defined. This takes up Sections 1, 2, 3, and 4. 

In Section 5 we write a list of 14 properties of the index. There are
short proofs of them. It is hoped that this list will be easy to use for
the mathematician or physicist who needs to apply the idea of index in
their work.
%%%%%%%%%%%%%%%%%%

\head  C.  Definition of otopy and index
\endhead



The concept of otopy arises from homotopy in a natural way.
Consider a smooth compact manifold $M$ with boundary $\partial M$.
Let $V$ be a continuous vector field defined on $M$.
There is an associated vector field $\partial V$ on $\partial M$ given by
projecting the vectors of $V$ on $\partial M$ to vectors which are tangent to
$\partial M$.
(See the paragraph above Lemma 2.1 for the definition of projection.)
Denote by
$\partial_- M$
the open set of $\partial M$ where the vectors of $V$ point inside. 
Let $\partial_- V$ denote $\partial V$ restricted to $\partial_- M$.
For outward pointing vectors we define $\partial_+ M$ and $\partial_+ V$.
Let $\partial_0 M$ denote the closed set of $\partial M$ where the vectors of $V$ are tangent to $\partial M$.

Now consider a homotopy $V_t$.
It induces a homotopy $\partial V_t$ on $\partial M$.
Now $\partial_- V_t$ is varying with $t$, but it is not a homotopy.
We say it is an {\it otopy}. It is the key example of an otopy.

\remark{The key observation about otopies} Consider a zero of 
$\partial V$ which
passes from $\partial_- M$ to $\partial_+ M$ in $\partial M$. 
As it passes over $\partial_0 M$ it coincides with a zero of $V$ which is 
passing through $\partial M$ .

Thus there is a connection between the zeros of $V_t$ which pass inside and
outside of $M$ through $\partial M$ and the zeros of $\partial V_t$ which
pass inside and outside of $(\partial_- M)_t$.
The concepts of {\it proper} homotopies and {\it proper} otopies and
{\it proper} vector fields are introduced so that no zeros appear on $\partial M$ or $\partial_0 M$.
\endremark

\definition {Definition of continuous otopy}
Let $N$ be a manifold and let $V$ be a continuous vector field defined on
$N\times I$ so that $V$ is tangent to the slices $N\times t$.
Then we say that $V$ is a 
continuous {\it homotopy} and that $V_0=V(m,0)$ and $V_1=V(m,1)$
are homotopic vector fields.
Suppose that $T$ is an open set on $N\times I$ and $V$ is a continuous vector
field defined on $T$ so that $V$ is tangent to the slices $N\times t$.
Then we say that $V$ is a continuous {\it otopy} and that $V_0$ and $V_1$ are 
{\it otopic}.
Note that $V_0$ or $V_1$ are vector fields 
defined only on the open sets in $M$ given by the intersection of 
$T$ with $M \times i$ for $i = 0$ or $1$.
Thus $V_0$ or $V_1$ can be ``empty'' vector fields. Also note that ``otopy"
gives an equivalence relation on the set of vector fields defined on open
sets in $N$. This follows just as in the homotopy case. But it is a
trivial equivalence relation, since every vector field is homotopic to the
zero vector field and every vector field field defined on an open set is
otopic to the empty vector field.
\enddefinition

\definition{Definition of proper continuous otopy}
If $U$ is an open set in a manifold with
boundary we will adopt the convention that the 
{\it Frontier} of  $U$ includes
that part of the boundary $\partial M$ of $M$ which lies in $U$, as well
as
the usual frontier. The capital F will distinguish these two different
notions of Frontier and frontier.
We say that $V$ defined on an open set $U$ in $N$ is a {\it proper} vector
field on the domain $U$ if the zeros of $V$ form a compact set in $U$ and if
$V$ extends continuously to a vector field on $\overline{U}$ with no zeros
on the Frontier of $U$. 
Thus if $V$ is defined on a compact manifold $M$ with boundary $\partial M$, we
say $V$ is {\it proper} if there are no zeros on $\partial M$.
A {\it proper otopy} with domain $T$ is an otopy $V$ 
defined on the open set $T$ with a compact set of zeros 
whose restriction to any slice is a
proper vector field.
A {\it proper homotopy} is an proper otopy $V$ defined on all of
 $N \times I$. 
\enddefinition

Now the index for proper continuous 
vector fields on a compact manifold with or without
boundary, as well as the index for proper continuous
vector fields defined on an open
set $U$, were defined in \cite{B-G}.  
If $V$ is a continuous proper vector field defined on a compact manifold $M$
with boundary $\partial M$ and Euler-Poincare number $\chi(M)$, then
$\partial_- V$ is also proper and the index satisfies
$$
\text{Ind} V=\chi(M)-\text{Ind}(\partial_-V)
$$

Now we consider any arbitrary, possibly discontinuous, vector field $V$ on $N$.
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 assumptions about continuity. 
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 {\it defect} to be a point of $N$ at which $V$ is either not
defined, or is discontinuous, or is the zero vector, or which contains
one
of those points in every neighborhood.

We extend the notion of {\it proper} to arbitrary tangent vector fields by
replacing the word zero by defect.

\definition {Definition of discontinuous proper otopy}
We say that $V$ is a {\it proper} vector field on an open set $U$ if 
the
defects of $V$ in $U$ form a compact set and if $V$ can be continuously
extended to $\overline{U}$ so that there are no defects on the Frontier
 of $U$. Thus for $N$ a compact manifold with boundary we say that
 $V$ is a 
{\it proper vector field} if there are no defects on the boundary.
A {\it proper otopy} $W$ with domain $T$ is an otopy in
$N \times I$ whose defects form a compact set and whose restriction to
every slice is a proper vector field for that slice.
A globally defined
otopy is still called a {\it homotopy}. We will modify the word 
homotopy
to  discontinuous
homotopy
if needed.
\enddefinition

\remark{remarks}
1. 
As before, the concept of proper discontinuous otopy is an equivalence
relation on the locally defined vector fields of $N$. It is a simple
exercise of pointset topology to show that every discontinuous vector
field is otopic to a continuous vector field. Also, if two continuous
locally defined vector fields are otopic, they are continuously otopic.
So the extension of index theory from continuous to discontinuous
vector fields is not
mathematically challenging. But discontinuous vector fields arise very
naturally in mathematics and physics and now the results of index
theory can be applied to them without any mental anguish.
2. In order to avoid confusion between points at which the
vector field $V$ is undefined inside the open set $U$ and outside the open
set $U$ we can restrict our attention without loss of generality to vector
fields which are defined everywhere on $U$, but are not necessarily
continuous. Any vector field on $U$ which is not defined at some points in
$U$ can be replaced by the same vector field returning the zero vector at
those undefined places. 
In fact, for those readers who are uncomfortable with the notion of
discontinuous vector fields, Remark 1 offers a way to proceed by thinking
of only continuous vector fields.
3. 
Note that the defects of an otopy need not
be a defect of the vector field defined on the slice. For example,
consider the unit vector field pointing to the right on the real line and
otopy it to the unit vector field pointing to the left by letting the
field reverse direction when $t=1$. The vector field at $t=1$ has no
defects thought of as a vector field on the real line, but the otopy
defects are located at all the points of the $t=1$ slice. Thus this otopy
is not proper since the set of defects is not compact. Replacing the line
by a closed interval, the above example has a compact set of defects, but
it still is not a proper otopy because defects are on the Frontier of
the $t=1$ slice. If we replace the real line by a circle in our example,
we again get defects on the top circle, but they form a compact set and
there is no Frontier, so we consider this as a proper otopy, indeed a
proper discontinuous homotopy.
\endremark

\remark{ The advanced definition of index}
If $V$ is a proper discontinuous vector field defined on a compact $M$ with
boundary, then $\partial_- V$ is continuously defined on $\partial_- M$.
So Ind$(\partial_- V)$ is defined.
Then Ind$(V)$ is defined by 
$ \text{Ind} V=\chi(M)-\text{Ind}(\partial_-V) $.
If $V$ is a proper discontinuous vector field ``defined'' on the open domain
$U$, we can find a compact manifold $M$ which contains the
compact set of defects of $V$ and has none on $\partial M$.
Then $\text{Ind}_U V:=\text{Ind}_M V$ where $\text{Ind}_M(V)$ means the
Index of $V$ restricted to $M$.

If $V$ is not a proper vector field on $U$ , then we define
$\text{Ind}_U (V):=\infty$ .
We introduce $\infty$ to avoid saying that the Index is undefined, since there
is information when $V$ is not proper.

Now let $P$ be a connected component of the set of defects $D$ of a vector
field $V$ on $N$.
We will define the local index of $P$, which we denote by ind$(P)$, as follows:
Let $U$ be an open set containing $P$ and no other defects, so that $V$ is
proper on $U$.
Then ind$(P):=\text{Ind}_U (V)$. If there is no such $U$, but $P$ is contained
in an open set on which $V$ is proper, then ind$(P):=-\infty$.
If there is no open set $U$ containing $P$ on which $V$ is proper, then
ind$(P):=\infty$.
\endremark

The relationship between $\text{Ind}_U (V)$ and ind$(P)$ is very striking.
If all the indices involved are finite
$$
\text{Ind}_U(V)=\sum \text{ ind}(P)\text{ where the sum is over all connected
components } P \text{ of } D.
$$

%\vfill\eject


%%%%%%%%%%%%%%%%%%%%%%
\head  D. Guides
\endhead

\subsubhead 1. The intuitive picture
\endsubsubhead

%1. {\it A simplified view:}
Consider the defects of a vector field as ``topological particles''
$P_i$ endowed with a ``charge'' denoted $\text {ind} (P_i)$.
These $P_i$ move and interact as the vector field evolves in time (that is under
otopy and homotopy).
These ``charges'' are preserved under collisions just as electric charge is.
See (1), the conservation law.
Then for a region of space $U$ or $M$, we have $\text{Ind}_U (V)=\sum
\text{ind} (P_i)$ for $P_i$ contained in $U$, ((8), the summation equation).
The list of properties 1--14 in Section 5 can then be used to calculate the
index.
Particularly useful is the Law of Vector Fields, (2). The classification
of otopy by index means that any set of defects $P_i$ can be transformed to any
other set of $P_j$'s if and only if the sum of the indices of the $P_i$
is equal to sum of the indices of the $P_j$.
Note that $\text{ind}(P_i)$ in dimension one can only take on the values, 
$-1, 0,1, \infty, -\infty$.
In higher dimensions $\text{ind}(P_i)$ can be any integer and $\infty$ and
$-\infty$.
The value $-\infty$ will most probably not appear in a physical application.

\subsubhead 2. The elementary definition
\endsubsubhead

%2. {\it The elementary definition:}    
This is the {\it only}
modern complete account of indices that the authors are aware of.
In $\S$1 are listed 6 lemmas.
The $\text{Ind}_M (V)$ and $\text{Ind}_U (V)$ are assumed to be defined in
dimension $n-1$ and satisfy the 6 lemmas.
Then for $M$ a compact manifold we define $\text{Ind}_M (V):=\chi (M)-\text{Ind}_{\partial_- M} (\partial_-V)$.
In $\S$2 we show that this is well-defined.
Then for $U$ an open set, we define in 
$\S$3 $\text{Ind}_U (V):=\text{Ind}_M (V)$
where $M\subset U$ contains the defects of $V$ in $U$.
In $\S$2 and $\S$3 we prove the lemmas in $\S$1 for dimension $n$.
The most subtle property to prove is the ``existence of defects'' (7).
The lemmas for dimension 1 are proved in $\S$1.
After $\text{Ind}_M (V)$ and $\text{Ind}_U (V)$ are established, we define in
$\S$4 the 
local index for a ``topological particle,'' that is a connected component
$P$ of the set of defects of $V$, then $\text{ind}(P):=\text{Ind}_U (V)$ where
$U$ is an open set containing $P$ and no other defects.
The two cases where such a $U$ cannot exist are given by $\text{ind}(P)=\pm
\infty$.
In the rest of the paper $\text{Ind}_U$ and $\text{Ind}_M$ will frequently
be shortened to Ind.

The prerequisites for this development of index are elementary topology and
differential topology.
The only ``sophisticated'' results used are: The Tietze Extension Theorem;
the existence of triangulations for smooth manifolds; transversality;
smooth approximation to continuous cross-sections; the additivity of the
Euler-Poincare number.
Most of these can be found in \cite{G-P}.

In $\S$5 all the key properties of the index are listed.
Properties (1) to (8) are basically proved in the earlier sections.
Properties (9) and (10), the product and sign rules, are proved as simple
consequences of properties (1) to (8).
Property (11) requires knowledge of the degree of a map.
It is this result which shows that the index defined this way agrees with
the other definitions as in \cite{B-G} or \cite{M}.
It should be mentioned that property (11) could stand as a definition of the
degree, and presumably most of the properties of degree could be proved from
properties (1) to (11).
The main point is this: The Index is independent of degree, and also
intersection number, fixed point index, and coincidence number.
Properties (12), (13), (14) are proved elsewhere.
The proofs employ the previous properties and sophisticated algebraic topology.
Each one is a generalization of a famous theorem.

\subsubhead 3. The advanced definition
\endsubsubhead

% 3. {\it The advanced definition:}
For the Expert who knows homotopy theory and differential
topology well  \cite{B-G} will be accessible.
The concept of otopy was introduced in that paper and the invariance of index
under otopy was established.
(Although otopy was first published there, its actual discovery came from
the underlying motivation of this paper: To define the Index by means of the
Law of Vector fields).
One should read the definition in $\S$C in the Introduction to extend
the definition of
the index for discontinuous vector fields.
Then to prove the classification theorems, (3) and (4), use the properties
of \cite{B-G} where needed
and the otopy extension property, which is proved in $\S$2.

\subsubhead  4. An advantage for the elementary definition
\endsubsubhead
% 4. {\it An advantage for the elementary definition:}

The elementary definition is based directly on the vector field, unlike
the other definitions.
In \cite{B-G} 
a map is constructed and the degree of the map is the index.
In Hopf's definition the vector field must be deformed until there are only
a finite set of zeros.
G. Samaranayake makes use of this advantage in her thesis \cite{S}.
She has a computer program  which estimates the index of a zero using the Law of
Vector Fields, (2).
It works well because she does not need to prepare the vector field in any
substantial way.
Using this program she can  search for 
zeros of a static coulomb electric field
generated with a finite number of electrons 
and protons whose index is not $-1,\ 0$, or 1.
Placing protons at the vertices of the Platonic solids; Tetrahedron,
Octahedron, Cube, Icosahedron, and Dodecahedron, she estimates the index
of the central zero to be --3, --5, 5, --11, and 11 respectively.

\head 1. The definition for one-dimensional manifolds
\endhead
%%%%%%%%%%%%%%%%%%%%%%%%%

First we describe the organization of the definition of Index and the way
we will show it is well-defined. In most situations $M$ will denote a
compact smooth manifold with boundary $\partial M$ which is possibly
empty.
 $N$ will usually denote an arbitrary smooth
manifold with or without boundary, and $U$ will denote an open set in $N$
or $M$. We usually consider vector fields $V$ as 
globally defined over $M$ or
locally defined over $N$. If $V$ is locally defined it is
associated to an open set $U$ in $N$ on which it is globally
defined. We say $U$ is
the {\it domain} of $V$.

\definition {The inductive definition of index}
Let $\phi$ denote the empty domain. Define $\text{Ind}_{\phi}(V) := 0 $.
If $M$ is a 
compact connected manifold and $V$ is globally defined on $M$ with no
zeros on $\partial M$, then define $\text{Ind}_{M}(V)$ by
$$
\TIM(V):=\chi(M)-\TIU(\partial_-V) {\text { where }} 
U = \partial_-M.\tag*
$$
Let $M$ be a smooth manifold with a globally
defined $V$. Then $\text{Ind}_{M}(V) := \text{sum of indices on each path
component}$.
Let $V$ be a proper vector field on the open set $U$
in $N$. Define $\TIU(V) := \TIM(V)$ where $M$ is a
compact manifold in $U$ containing the defects of $V$ .

\enddefinition

\remark{remark}
It will be clear that by Lemma 1.6, the equation (*) will hold for
non-connected manifolds also. We shall refer below to (*) without the
connectedness hypothesis .
\endremark
 
We begin the induction at dimension $-1$, the empty manifold. Here the
index is zero. For dimension 0, the connected manifold is a point and
the vector field $V$ consists of the zero vector. Applying (*) we see
that $\text{Ind}_{M}(V) $ equals 1. Thus 
$\text{Ind}_{U}(V)$ equals the number of points in $U$.

In dimension 1 there are two compact connected manifolds: The circle
and the closed interval. Let $V$ be a vector field globally defined on a
 circle $M$.
Then $\TIM (V) = 0$ follows from (*). Note that if $V$ were
the zero vector field, it is proper when $N$ is a circle. This
contrasts to the fact that a zero vector field can never be proper on an $M$
or a $U$ with non empty Frontier.

Let $M$ be an closed interval and let $V$ be a proper vector field. Then
$\text{Ind}_{M}(V) = 1 - (\text{number of points on the boundary where
 $V$
points inside)}$. Thus $\TIM (V)$ can take on the values 1, 0, $-1$.

Let $M$ be a general compact 1-dimensional manifold with a globally
defined $V$. Then 
$M$ is a finite union of closed intervals and circles
and
$\text{Ind}_{M}(V) := \text{sum of indices on each path component}$.

So we have a definition for $\text{Ind}_{M}(V)$ which is obviously
well-defined in one dimension. It will be necessary, however, to prove
that $\text{Ind}_{M}(V)$ is well-defined beginning with dimension 2 for
each step of the induction. We must show in dimension 1 already that 
$\TIU$ is well-defined. 
 We prove three lemmas about the $M$ case and
then after Lemma 1.3 we can show that $\text{Ind}_{U}(V)$ is 
well-defined.
We state the lemmas in this Section for general manifolds. 
The proofs will be for dimension one. 
Frequently $\TI$ will stand for either $\TIM$ or $\TIU$.


\proclaim{Lemma 1.1}
Two vector fields $V$ and $V'$ globally defined on $M$ 
are properly homotopic if and only if
$$
\TI(\partial_-V)=\TI(\partial_-V')\text{ on each component of the boundary}.
$$
\endproclaim


\demo{Proof}
We may assume that $M$ is connected. If $M$ is a circle, every globally 
defined $V$ is properly otopic to any other globally defined $V'$. On the
other hand, there is no path component of the circle's empty boundary. So
the result is trivially true for the circle.
Next assume that $M$ is a closed interval.
Let $W$ be a vector field so that $W(m)=V(m)/\Vert V(m)\Vert$ for $m$ on
the boundary 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 zero vectors at the other end of the collar.
Then we define the homotopy $tV+(1-t)W$.  This is a proper homotopy, since
at any point $m$ on the boundary  $V(m)$ and $W(m)$ both point either inside
or outside so no zero can arise on the boundary. 
Now both $V$ and $V'$ are properly homotopic to $W$, hence
they are properly homotopic to each other.
\qed
\enddemo

\remark{remark} Note that if $V$ and $V'$ are continuous vector fields,
there is a continuous proper homotopy between them. If they are smooth,
then there is a smooth proper homotopy between them. Also note that for a
vector field $V$ globally defined on an interval, there are only four
proper homotopy classes. In higher dimensions there are infinitely many
proper homotopy classes. The corresponding result in higher dimensions is
Theorem 2.2.
\endremark

\proclaim{Lemma 1.2}
If $M$ is a compact manifold diffeomorphic to $M'$
and the
vector field related to $V$ by the diffeomorphism $f$ 
is denoted by $V^*$, then
$$
\text{Ind}_{M}(V) = \text{Ind}_{M'}(V^*)
$$
\endproclaim

\demo{Proof}
Pointing inside is preserved under diffeomorphism.
\qed
\enddemo

\proclaim{Lemma 1.3}
If $V$ has no defects, then $\TI(V)=0$.
\endproclaim

\demo{Proof}
We may assume that $M$ is connected.
Let $M$ be an interval.  Since $V$ has no defects
on this interval, $V$ must point outside on one end and inside on the
other.  Thus $\TI(V)=1-1=0$ on this interval.
For $M$ a circle the
globally defined $V$ must always have index zero.
\qed
\enddemo

Now we can show that $\TIU(V)$ is well-defined. If $M$
and $M'$ are two compact manifolds containing the defects,
and contained in $U$, there is a
compact manifold $M''$ also contained in $U$ and 
containing both $M$ and $M'$.  The
vector field $V$ restricted to $M''-\text{int}(M)$ is a nowhere zero vector
field, and the previous lemma and the fact that the index is additive proves
that $\TIU(V)$ is well-defined for those vector fields for which the
defects sit inside a compact manifold with boundary.



\proclaim{Lemma 1.4}
Given a connected $N$, two proper locally defined (continuous) 
vector fields are properly otopic (by a continuous otopy)
if and only if they have the same index.  For every integer $n$ there is a
vector field whose index equals that integer (provided $N$ has positive
dimension).
\endproclaim

\demo{Proof}
Suppose we have a proper otopy $W$ with domain $T$ on $N\times I$.  Let $V_t$
denote $W$ restricted to $N\times t$.  We show that there is some interval
about $t$ such that $V_s$ has the same index for all $s$ in the interval.
Since the set of defects of the otopy is compact we can find a compact manifold
$M$ so that $M\times J$, for some closed interval $J$, lies in $T$ and contains
the defects inside $\partial M\times J$.  Thus the proper homotopy $V_t$ on
$M\times J$ preserves the index on $M$, and hence the proper otopy on
$N\times J$ 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 previous
vector field.  Hence the first and last vector fields have equal indices.
Conversely, for any integer $n$, let $W_n$ be the vector field consisting
of $|n|$ vector fields defined on disjoint open intervals in $N$, each one of
index $1$ if $n>0$ and of index $-1$ if $n<0$.  Thus $\TI(W_n)=n$.  Now if
$V$ has index $n$, we must show that $V$ is properly homotopic to $W_n$.  Now
the domain of $V$ consists of open connected intervals, and only a finite
number of them contain defects.  Each of these intervals has index equal to
$1$, $-1$, or $0$.  Now $V$ is properly otopic to the same vector field $V$
whose domain is restricted to only those intervals which have nonzero indices.
Now if two adjacent intervals have different indices, there is a proper otopy
which leaves the rest of the vector field fixed, and removes the two
intervals of opposite indices.  After a finite number of steps we are left
with either an empty vector field, if $n=0$, or a $W_n$.  The empty vector
field is $W_0$.  Thus $V$ is properly otopic to $W_n$.
\qed
\enddemo

\proclaim{Lemma 1.5}
$\TIU(V)$ on $N$ is invariant under
diffeomorphism.
\endproclaim

\demo{Proof}
Immediate from Lemma 1.2 and the definition of index for 
locally defined vector fields.
\qed
\enddemo

\proclaim{Lemma 1.6}
Let $V$ be a vector field over a domain $U$ and suppose that $U$ is the
disjoint union of $U_1$ and $U_2$.  Then if $V_1$ and $V_2$ denote $V$
restricted to $U_1$ and $U_2$ respectively, we have
$$
\TI(V)=\TI(V_1)+\TI(V_2).
$$
\endproclaim

%%%%%%%%%%%%%%%%%%%%
\head 2.  The index defined for compact $n$-manifolds
\endhead

\proclaim {The otopy extension property}
Let $V$ be a continuous vector field on a closed 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 $N$. In fact, if $V$ and $W$ are continuous
vector fields with a proper continuous otopy between restrictions of them
to open sets, then the otopy can be extended to a continuous homotopy of
$V$ to $W$.
\endproclaim
\demo{Proof}
The continuous proper otopy implies there is a continuous vector field $W$
on an open set $T$ in $N\times I$ which extends to the closure of $T$ with
no zeros on the Frontier and which is $V$ when restricted to $N\times 0$.
This vector field $W$ can be thought of as a cross-section to the tangent
bundle over $N\times I$ defined over a closed subset. It is well known that
cross-sections can be extended from closed sets to continuous cross-sections
over the whole manifold.
\qed
\enddemo

We assume that the index is defined for $(n-1)$-manifolds and that
all the lemmas of Section 1 hold.

First we consider the case of connected compact manifolds 
$M$ 
We suppose that $V$ is a 
globally defined proper vector field on
such a manifold $M$.  We choose a vector field $N$ on the boundary
$\partial M$ which points outside of $M$.  Every vector $v$ at a point $m$
on $\partial M$ can be uniquely written as $v=t+kN(m)$ where $t$ is a vector
tangent to $\partial M$ and $k$ is some real number.  We say $t$ is the
{\it projection} of $v$ tangent to $\partial M$.  Then $\partial V$ is the
vector field obtained by projecting $V$ tangent to $\partial M$.  Now we
define $\partial_-V$ by restricting $\partial V$ to $\partial_-M$, the set
of points such that $V$ is pointing inward.  Then we define
$$
\TIM(V)=\chi(M)-\TIU(\partial_-V) {\text { where }} U = \partial_-M.\tag*
$$

\proclaim{Lemma 2.1}
$\TIM(V)$ is well-defined.
\endproclaim

\demo{Proof}
We assume already defined the index on $(n-1)$-dimensional manifolds with
open domains for proper vector fields.  Note that $\partial_-V$ is proper
on $\partial M$ if $V$ is proper on $M$, 
because the Frontier of $\partial_-M$ is a subset of
$\partial_0M$, the subset
where $V$ is tangent to $\partial M$.  So a defect of
$\partial_-V$ on the Frontier must come from a defect of $V$ on $\partial M$.
Hence $\TIU(\partial_-V)$ is defined.  Now the vector field $\partial_-V$
obviously depends upon the outward pointing $N$.  If we had another outward
pointing vector field $N'$ we would project down to a different
$\partial_-V$, call it $W$.  Now the homotopy of vector fields
$N_t=tN+(t-1)N'$ always points outside of $M$ for every $t$.  Hence it
induces a homotopy from $\partial_-V$ to $W$ and this homotopy is proper.
Thus $\TI(\partial_-V)=\TI(W)$ by 1.1. Hence $\TIM (V)$ is well-defined
for connected manifolds with boundary. If $M$ has empty boundary, then 
$\TIM (V) = \chi (M)$ by (*). Hence $\TIM (V)$ is well-defined for all
connected manifolds, and hence is well-defined for all $N$-manifolds.
\qed
\enddemo

\remark {remark}
The above lemma is also true in
the case where the normal vector field 
$N$ is not defined on a closed set of
$\partial M$ which is disjoint from the Frontier of
$\partial_-M$.  Then $\partial V$  is not everywhere defined,
but $\partial_-V$ is still proper.  A homotopy between $N$ and
$N'$, as in the lemma, still induces a proper otopy between $\partial_-V$
and $W$, so the $\TI(V)$ is still well-defined in this case also.  This
case arises when $M$ is embedded as a co-dimension zero manifold in such a
way that it has corners.  Then 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 $\partial_-M$. This 
point arises in
Theorem 2.6.
\endremark

Now our goal is to prove that non-zero vector fields have index equal to
zero on compact manifolds with boundary.

\proclaim{Theorem 2.2}
On $M$ the globally defined vector field 
$V$ is properly homotopic to $W$ if and only if
$$
\TI(\partial_-V)=\TI(\partial_-W)
$$
for every connected component of $\partial M$.  So as a corollary in the
case that $\partial M$ is connected, we have that $V$ is properly homotopic
to $W$ if and only if $\TI(V)=\TI(W)$.  If $V$ and $W$ are both continuous,
then ``homotopic'' can be replaced by ``continuously homotopic'' 
in the above theorem.
\endproclaim

\demo{Proof}
We may assume that $M$ is connected. If $M$ has empty boundary, the
theorem is true since every globally defined vector field is properly
otopic to any other globally defined vector field. So assume that $M$ has
non-empty boundary.
The theorem is true for manifolds one dimension lower by lemma 1.1.  A proper
homotopy of $V$ to $W$ induces a proper otopy from $\partial_-V$ to
$\partial_-W$ in the manifold $\partial M$.  Hence $\TI(\partial_-V)=
\TI(\partial_-W)$.  Hence $\TI(V)=\TI(W)$ from $(*)$.  Conversely, we can
find a smooth collar $\partial M\times I$ of the boundary so that $V$
restricted to this collar has no defects.  Then we homotopy $V$ to $V'$ where
$V'$ is defined by $V'(m,t)=tV(m)$ for a point in the collar and $V'=0$ outside
the collar.  Now since $\TI(\partial_-V)=\TI(\partial_-W)$ for each
connected component of the boundary, we can find a proper otopy from
$\partial_-V$ to $\partial_-W$.  Now this otopy can be extended to a homotopy
of $\partial V$ to $\partial W$ by the otopy extension property.  This 
homotopy in turn can be used to define a proper homotopy from $V'$ to $W'$.
Here we assume $W'$ has the same definition relative to $W$ as $V'$ has to $V$.
Thus $W$ is properly homotopic to $V$.
\qed
\enddemo

\proclaim{Lemma 2.3}
Suppose $V$ is a proper vector field on a compact manifold $M$ .  
Let $\partial M\times I$ be a collar
of the boundary so small so that $V$ has no defects on the collar.  Then $V$
restricted to $M$ minus the open collar $\partial M\times (0,1]$ has the same
index as $V$.
\endproclaim

\demo{Proof}
Let $\partial V_t$ denote the projection of $V$ tangent to the submanifold
$\partial M\times t$ for every $t$ in $I$.  Let $W$ be the vector field on
the collar defined by $W(m,t)=\partial_-V_t$ if $(m,t)$ is a point in
$\partial_-M\times t$.  Then $W$ can be regarded as a proper otopy, proper since $V$ has no
defects on the collar.  Thus $\TI(\partial_-V)=\TI(\partial_-V_0)$ and
hence $\TI(V)=\chi(M)-\TI(\partial_-V)$ equals the index of $V$ restricted
to $M'=M- open\ collar$, because the indices of the $\partial_-$
vector fields are the same on their respective boundaries and $\chi(M)=
\chi(M')$.
\qed
\enddemo

\proclaim{Lemma 2.4}
Let $V$ be a proper continuous vector field on $M$.  Suppose that $\partial_-V$
is properly otopic to some locally defined 
vector field $W$ on $\partial M$.  Then there
is a proper homotopy of $V$ to a proper continuous vector field $X$ so that
$\partial_-X=W$ and the zeros of each stage of the homotopy $V_t$ are 
equal.
\endproclaim

\demo{Proof}
Use the otopy extension property to find a homotopy $H_t$ from $\partial V$
to a vector field on $\partial M$, which we shall call $\partial X$. Let
$n(m,t)$ be a continuous real valued function on $\partial M\times I$ which
is positive on the open set $T$ of the otopy between $\partial_-V$ and $W$,
zero on the Frontier of $T$, and negative in the complement of the closure
of $T$, and so that $n(m,1)=n(m)$ where $V(m)=n(m)N(m)+\partial V(m)$
defines $n(m)$.  Such a function exists by the Tietze extension theorem.
Using $n(m,t)$, we define a vector field $X'$ on $\partial M\times I$ by
$X'(m,t)=n(m,t)N(m)+H_t(m)$.  We adjoin the collar to $M$ as an external
collar and extend the vector field $V$ by $X'$ 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 chosen that $X=V$ outside of a small
internal collar.  Then the homotopy $tX+(1-t)V$ is the required homotopy which
does not change the zeros of $V$.
\qed
\enddemo

\proclaim{Lemma 2.5}
If $V$ is a vector field with no defects on an $n$-ball, then $\TI(V)=0$.
\endproclaim

\demo{Proof}
For the standard $n$-ball of radius $1$ and center at the origin, we define
the homotopy $W_t(r)=V(tr)$. This homotopy introduces no zeros and shows
that $V$ is homotopic 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 the index of the vector field
under the diffeomorphism is preserved by Lemma 
1.2, and hence it has the zero index.  If
the ball is embedded with corners so that the corners are not on the
Frontier of 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.
\qed
\enddemo

\proclaim{Theorem 2.6}
If $V$ is a vector field with no defects on a compact manifold $M$
then $\TIM(V)=0$.
\endproclaim

\demo{Proof}
Now $M$ can be triangulated and suppose we have proved the theorem for
manifolds 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 $M_1$ and $M_2$ each covered by fewer than $k$ $n$-simplicies
so that the theorem holds for them.

We arrange it so that $L$ is orthogonal to $\partial M$.  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 $\partial M$ at $L\cap\partial M$. Then a simple
counting argument shows that $\TIM(V)=0$ since the restrictions of $V$ to $M_1$
and $M_2$ have index zero.  This argument works if $M$ has no corners.  If
$M$ has corners we find a collar of $M$ which gives a smooth embedding of
$\partial M\times t$ for all $t$ but the last $t=1$. Then by lemma 2.3 above, we
find that $V$, restricted to the manifold bounded by $\partial M\times t$ for
$t$ close enough to $1$, has the same index as $V$.  That is zero.

The counting argument follows.  By induction, $\TI(V|M_1)=\TI(V|M_2)=0$.
Thus $\TI(\partial_-V_1)=\chi(M_1)$ and $\TI(\partial_-V_2)=\chi(M_2)$.
Now we have the following equation
$\TI(\partial_-V)=\TI(\partial_-V_1)+\TI(\partial_-V_2)-\TI(W)$ where
$W$ is the projection of $V$ on the common part of the boundary of $M_1$ and
$M_2$, that is $L$.  This follows from repeated applications of lemma 1.6.
Now $\TI(W)=\chi(L)$ since $W$ points outwards at the boundary of $L$.  Hence
$$
\TI(\partial_-V)=\TI(\partial_-V_1)+\TI(\partial_-V_2)-\TI(W)=
\chi(M_1)+\chi(M_2)-\chi(L)=\chi(M).
$$  
Hence $\TIM(V)=0$ from $(*)$.
\qed
\enddemo

%%%%%%%%%%%%%%%%
\head 3.  The index for locally defined vector fields
\endhead
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
$$
\text{Ind}_U(V):=\text{Ind}_M(V).\tag**
$$

\proclaim{Lemma 3.1}
$ \text{Ind}_U(V)$ is well-defined.
\endproclaim

\demo{Proof}
If $M$ and $M'$ are two 
compact manifolds containing the defects, there
is a compact manifold $M''$ containing both $M$ and $M'$.  The
vector field $V$ restricted to $M''-\text{int}(M)$ is a nowhere zero vector
field.  Then Theorem 2.6 implies that the index of $V$ restricted to
$M''-\text{int}(M)$ is zero.  Now the index of $V$ restricted to $M''$ equals
the index of $V$ restricted to $M$ by the following lemma.
\qed
\enddemo

\proclaim{Lemma 3.2}
Suppose $M$ is the union of two manifolds $M_1$ and $M_2$ where the three
manifolds are compact manifolds so that the intersection of
$M_1$ and $M_2$ consist of part of the boundary of $M_1$ and is disjoint
{}from the boundary of $M$.  Suppose that $V$ is a proper vector field defined
on $M$ which has no defects on the boundaries of $M_1$ and $M_2$.  Then
$\text{Ind}_M(V) = \text{Ind}_{M_{1}}(V_1) + \text{Ind}_{M{_2}}(V_2) $
where $V_i=V|M_i$.
\endproclaim

\demo{Proof}
$$
\aligned
\TI(V) &= \chi(M)-\TI(\partial_-V)\\
&= \chi(M)-(\TI(\partial_-V_1)+\TI(\partial_-V_2)-\TI(\partial_-V_1|L)-\TI(
\partial_-V_2|L))
\endaligned
$$
by 1.6 where $L=M_1\cap M_2$.  Now
$$
\TI(\partial_-V_1|L)+\TI(\partial_-V_2|L)=\TI(\partial_-V_1|L)+\TI(\partial_+V_1)=\chi(L).
$$
Thus 
$$
\TI(V)=\chi(M_1)+\chi(M_2)-\TI(\partial_-V_1)-\TI(\partial_-V_2)=
\TI(V_1)+\TI(V_2),
$$
as was to be proved.
\qed
\enddemo

\proclaim{Lemma 3.3}
Let $V$ be a proper vector field with domain $U$.  Suppose $U$ is the union
of two open sets $U_1$ and $U_2$ such that the restriction of $V$ to each of
them and to $U_1\cap U_2$ is a proper vector field denoted $V_1$ and
$V_2$ and $V_{12}$ respectively. Then
$$
\TIU(V)=\text{Ind}_{U_1}(V_1)+\text{Ind}_{U_2}(V_2)
-\text{Ind}_{U_{12}}(V_{12}) .\tag***
$$
\endproclaim

\demo{Proof}
We choose disjoint compact manifolds $M_1$, $M_2$, and $M_{12}$ containing
the zeros of $V$ which lie in $U_1-U_{12}$ and $U_2-U_{12}$ and $U_{12}$
respectively.  Then the index of $V$ is equal to the index of $V$ restricted
to the union of $M_1$, $M_2$, and $M_{12}$.  But the index of $V_1$ is the
index of $V$ restricted to $M_1$ and $M_{12}$, and the index of $V_2$ is the
index of $V$ restricted to $M_2$ and $M_{12}$, and the index of $V_{12}$ is
the index of $V$ restricted to $M_{12}$.  Hence counting the index gives the
equation $(***)$.
\qed
\enddemo

\proclaim{Corollary 3.4}
The index of a vector field $V$ on a closed manifold $M$ whose domain 
is the
whole of $M$ is equal to $\chi(M)$.
\endproclaim

\demo{Proof}
This is true by (*)
{\it a priori}. 
We note that Lemma 3.1 implies that any other way to calculate the
Index of $V$ will give the same answer. We illustrate:
Let
 $V$ be a vector field to a non-zero vector field on a
small $n$-ball $B$ about a point.  Now let $V_1$ be $V$ on the
 $n$-ball and
let $V_2$ be $V$ on the complement.  Then $\TI(V_1)=0$, so
$\TI(\partial_-V_1)=1$.  Now $\TI(\partial_-V_2)=(-1)^{n-1}$.  So
$$
\TI(V_2)=\chi(M-B)-(-1)^{n-1}=\chi(M)-(-1)^n-(-1)^{n-1}=\chi(M).
$$
Hence $\TI(V)=\TI(V_1)+\TI(V_2)=0+\chi(M)$.
\qed
\enddemo

\proclaim{Theorem 3.5}
Given a connected manifold $N$, two locally defined (continuous)
proper vector fields are properly otopic (by a continuous otopy) if
and only if they have the same index.  For every integer $n$ there is a vector
field whose index equals that integer (provided $N$ has positive
dimension).

\endproclaim

\demo{Proof}
Suppose we have a proper otopy $W$ with domain $T$ on $N\times I$.  Let
$V_t$ denote $W$ restricted to $N\times t$.  We show that there is some
interval about $t$ such that $V_s$ has the same index for all $s$ in the
interval.  Since the set of defects of the otopy is compact we can find a
compact manifold $M$ so that $M\times J$, for some closed interval $J$, lies
in $T$ and contains the defects so that the defects avoid
$\partial M\times J$.  Thus by Theorem 2.2, the
proper homotopy $V_t$ on $M\times J$ preserves the index on $M$, and hence
the proper otopy on $N\times J$ 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 previous vector field.  Hence the first and last vector fields
have equal indices.


Conversely, for any integer $k$, let $W_k$ be the locally defined 
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 $\TI(W_k)=k$.  Now if $V$ has
index $k$, we must show that $V$ is properly otopic to $W_k$.  Now the
defects of $V$ form a compact set which are contained in a compact manifold
with boundary $M$ so that $V$ is proper and has no defects on the boundary.
We may 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 transversality theorem, we can smoothly homotopy
the cross-section so that it is transversal to 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 around each of these zeros is either
$1$ or $-1$.  Classically
this follows from transversality, but we do not need that
fact.  We may find a diffeomorphic $n$-ball which contains exactly $|k|$
zeros so that around these zeros the vector field restricts to $W_k$.  The
two vector fields have the same index on the $n$-ball and thus are properly
homotopic, since from $(*)$ the index on the boundary of the inward pointing
$\partial_-$ vector fields is the same, and so by induction they are properly
otopic, hence by the otopy extension property the $\partial$ vector fields
are homotopic.  This homotopy can be extended to a homotopy of the two vector
fields originally on the $n$-ball.  Then using the sequence of homotopies
and
otopies, we can piece together a proper otopy of $V$ to $W_k$.
\qed
\enddemo


\remark{remark} Note that this proof is more complicated than it need be
because it does not use the concept of degree of a map or of intersection
number.
\endremark

\proclaim{Corollary 3.6}
The proper homotopy classes of continuous proper vector fields on a compact
manifold with connected boundary is in one-to-one correspondence with
the integers via the index.
\endproclaim

\proclaim{Lemma 3.7}
The index of a locally defined 
vector field on a manifold $N$ is invariant under
diffeomorphism.
\endproclaim




%%%%%%%%%%%%%%%%%%
\head  4.  The index of a defect
\endhead

Let $V$ be a vector field on an manifold $N$.  Let $D$ be the set of
defects
of $V$. Then $D$ breaks up into a set of connected components $D_i$.
If a component $D_i$ is compact and is an open
set in the subspace topology of $D$, we can define an index denoted
$\text{ind}(D_i)$. Note the lower case `i' here as opposed to the
uppercase
`I' in the definition of the global and local indices. We call 
$\text{ind}(D_i)$ the {\it index of the defect (or zero)} $D_i$.

\definition {Definition}
If the defect set $D$  is connected, compact and isolated, then we can
find
a open set  $U$ of $N$ containing $D$ and no other defects of $V$. Then
we define the index of D by
$$
\text{ind}(D):=\text{Ind}_U(V).\tag****
$$
If $D$ is not isolated, then every open set containing $D$ must contain
another defect of $V$. In this case we say $\text{ind}(D):= -\infty$.
If $D$ is not compact, we say $\text{ind}(D):= \infty$.  
\enddefinition

Now if the set of defects of $V$ on $N$ consists of a  
finite number of compact $D_i$, 
then
$\TIN(V)=\sum_i \text{ind}(D_i)$.  However it is possible that $V$ is a
proper
vector field and there are an infinite number of $D_i$.  Then at least one
of the $D_i$ is not isolated in $D$.  But the index of $V$ is still
defined.
A one dimensional example
occurs when $M$ is the interval $[-1,1]$ and the vector field $V$ is
defined
by $V(x)=x\sin(1/x)$ for $x\not=0$ and $V(0)=0$.  Then $0$ is a connected
component of the defects which is not open in the set of defects of $V$.
Thus $\text{ind}(0):=-\infty$, whereas $\text{Ind}_M(V)=1$.




If we have an otopy $V_t$, we imagine the components of the defects $D_t$ as
changing under time.  We can say that $D_{ti}$ at time $t$ transforms 
{\it without topological radiation} into
$D_{sj}$ at time $s$ if there is a compact connected component $T$ of the
defects of the otopy from time $t$ to time $s$ so that $T$ intersects 
$N\times t$ in exactly $D_{ti}$ and $T$ intersects
$N\times s$ exactly at $D_{si}$.
The index of $D_{ti}$ is the same as the index of $D_{sj}$ if $T$ is compact.
In other words if a finite number of ``particles'' $D_{i}$ at time $t$ are
transformed into a finite
number of particles $C_{j}$ at time $s$ by a compact $T$,
the sum of the indices are
conserved. 

\remark{(1) Conservation Law}

%\noindent (1) {\it Conservation Law:}
$$
\sum \text{ind}(C_i)=\sum \text{ind}(D_j).
$$
\endremark


Thus the idea of otopy allows us to make precise the concept of defects
moving with time and changing with time and undergoing collisions.  The
index is conserved under these collisions as long as the ``world line'' $T$
of the component is compact. That is, as long as there are is no
``topological radiation'', that is as long as the relevant component in
the otopy is compact. 

As we mentioned in Section A of the Introduction, the concept of otopy can
be thought of as a space-like vector field in a space time. So if a
physicist wants to model something by defects of a vector field, there is
a conservation law preserving an invariant of General Relativity which
automatically comes along with the model.

\head  5.  Properties of the Index
\endhead

\remark{(2)  Law of Vector Fields}

%\noindent
\quad$\TI(V)+\TI\ \partial_-V=\chi(M)$

This is in fact the equation (*) which defines the index. We remark here
that any theory of index in which the Law of Vector Fields holds must
agree with our definition. 
\endremark
%\medskip
\remark {(3) Classification of proper otopy by the Index}

Let $N$ be a connected manifold.  $V$ is properly otopic to $W$ if and
only if $\TI\ V=\TI\ W$.  If $V$ and $W$ are continuous vector fields, 
then the otopy can be continuous. For any integer $n$ there is a 
continuous vector field $W$ so
that $n=\TI\ W$.
%\medskip
\remark{ (4) Classification of proper homotopy}


Suppose $M$ is a compact manifold so that $\partial M$ is connected,
and
suppose $V$ and $W$ are continuous globally defined
proper vector fields on $M$.  Then
$V$ is properly homotopic to $W$ if and only if $\TI\ V=\TI\ W$.  For any
integer $n$ there is a continuous proper vector field $W$ so that
$n=\TI\ W$.  In general, $V$ is proper homotopic to $W$ if and only if
$\TI\ (\partial_-V) = \TI\ (\partial_-W)$ on every connected component of $\partial M$.
\endremark

\remark{ (5) Poincare-Hopf Theorem}


If $M$ is a closed compact manifold and $V$ is a vector field whose
domain is all of $M$, then $\TI\ V=\chi(M)$.
\endremark

\demo{Proofs}
Property (3) and (4) are Theorem 3.5, Theorem 2.2, and Corollary 3.6 respectively for the
homotopy part.  For the fact that $n=\TI\ W$ for some vector field $W$, we
apply (2) and induction starting with Lemma 1.4.  The proof of (5) is
Corollary 3.4.
\qed
\enddemo

\remark {(6)  Additivity}


Let $A$ and $B$ be open sets and let $V$ be a proper vector field on
$A\cup B$ so that $V|A$ and $V|B$ are also proper.  Then $\TI(V|A\cup B)=
\TI(V|A)+\TI(V|B)-\TI(V|A\cap B)$.
\endremark

\demo{Proof}
Lemma 3.3
\qed
\enddemo
%\medskip

\remark {(7)  Existence of defects}

If $\TI\ V \neq 0$ then $V$ has a defect.
\endremark

\demo{Proof}
Theorem 2.6 for compact manifolds with boundary.
\qed
\enddemo

\remark {(8)  Summation equation}


Suppose $V$ is a proper vector field and the set of defects consists
of a finite number of connected components $D_i$.  Then
$\TI\ V=\underset i\to\sum\ \text{ind}(D_i)$.
\endremark

\demo{Proof}
This follows from the definition of $\TI(D_i)$ and (3).
\qed
\enddemo

\remark{ (9) Product rule}


Let $V$ and $W$ be proper vector fields on $A$ and $B$ respectively.
Let $V\times W$ be a vector field on $A\times B$ defined by $V\times W(s,t)=
(V(s),W(t))$.  Then $\TI(V\times W)=(\TI\ V)\cdot(\TI\ W)$.
%%%%%%%%%%%%%%%%%%%%%%
\endremark

\demo{Proof}
We can assume that $A$ and $B$ are open sets in their respective
manifolds.  Then $V$ is
otopic to $V_n$ where $V_n$ is restricted to a finite set of open sets
in $A$ homeomorphic to the interior of $J^k$ when $k=\text{ dim }A$ and
$J = [-1, 1]$, so
that $V_n(t_1,\dots,t_k)=(\pm t_1,t_2,\dots,t_k)$ where the $+t_1$ is
taken
if $\TI\ V$ is positive and $-t_1$ is taken if $\TI\ V$ is negative.
The
index of the $V_n|J^k$ is $\pm 1$ respectively by (2).  So
$\TI\ (V\times W)=\TI (V_n\times W_n)=\underset {i,j}\to\sum\
\TI(V_n|J_i^k)
\times(W_n|J_j^\ell)$.  Now it is easy to see that $\TI((V_n|J^k_i)\times
(W_n|J_j^\ell))=\TI(V_n|J_i^k)\cdot\TI(W_n|J_j^k)$.
\qed
\enddemo

\remark{(10)  Sign rule}


$(-1)^n\TI(V)=\TI(-V)$ where $n=\text{ dim } M$.

\endremark

\demo{Proof}
The theorem is true for $n=1$.  Assume it is true for $(n-1)$-manifolds.
Now using (2) we have
$$
\aligned
\TI(-V) &=\chi(M)-\TI(\partial_-(-V))\quad \text{ by (2) }\\
&=\chi(M) -\TI(-\partial_+V)\quad \text{by definition of $\partial_-V$ and
$\partial_+V$}\\
&=\chi(M)-(-1)^{n-1}\TI(\partial_+(V))\quad \text{by induction}\\
&=\chi(M)+(-1)^n(\chi(\partial M)-\TI(\partial_-V))
\endaligned
$$
since
$$
\chi(\partial M)=\TI(\partial_-V)+\TI(\partial_+V).
$$
If $n$ is even then
$$
\TI(-V)=\chi(M)+(0-\TI(\partial_-V))=\TI\ V\quad \text{ by (2)}.
$$
If $n$ is odd then
$$
\aligned
\TI(-V) &=\chi(M)-(2\chi(M)-\TI(\partial_-V))\\
&=-(\chi(M)-\TI(\partial_-V))=-\TI\ V\quad \text{ by (2)}
\endaligned
$$
\qed
\enddemo

\remark{ (11) Index defines degree}


Suppose $M$ is a compact sub-manifold of ${\Bbb R}^n$ of codimension
$0$.
Let $f:M\to {\Bbb R}^n$ be a map so that $f(\partial M)$ does not
contain the origin.  Define a proper vector field $V^f$ on $M$ by $V^f(m)=f(m)$.
Then $\TI\ V^f=deg\ f'$ where $f':\partial M\to S^{n-1}$ by
$f'(m)=\frac{f(m)}{\Vert f(m)\Vert}$.

\endremark

\demo{Proof}
We homotopy $f$ if necessary so that $\vec 0$ is a regular value.  Then
$f^{-1}(\vec 0)$ is a finite set of points. There is a neighborhood of
$f^{-1}(0)$ of small balls so that $f:\partial(\text{ball})\to{\Bbb R}^n-0
\cong S^{n-1}$.  Now, in each of these small balls, $f$ has either degree
$1$ or $-1$.  If degree equals $1$, then $f|\partial(\text{ball})$
is homotopic to the identity.  If degree $=-1$, then $f|\partial(\text{ball})$
is homotopic to
reflection about the equator.  In these cases $\TI(V^f|\text{ball})=\pm 1=
deg\ f|\partial(\text{ball})$.  Now
$$
\aligned
\TI(V^f) &= \sum\ \TI\ V^f|(\text{balls})\qquad \text{ by proper otopy}\\
&= \sum deg\ f|\partial(\text{balls})= deg\ f'.
\endaligned
$$
\qed
\enddemo

\remark{ (12) Brouwer Fixed Point Theorem}

Suppose $f:M\to {\Bbb R}^n$ where $M\subset{\Bbb R}^n$ is a
codimension zero compact manifold.  Define $V_f(m)=m-f(m)$.  Then
$\TI\ V_f=$ fixed point index of $f$ (assuming no fixed points on $\partial M$).
\endremark

\demo{Proof}
The fixed point index is defined to be the degree of the map $m\to\frac{m-f(m)}
{\Vert m-f(m)\Vert}$ from $\partial M\to S^{n-1}$.  Hence by (11) we have
the result
\qed
\enddemo

\remark{ (13) Gauss-Bonnet Theorem}

Let $f:M\to N$ where $M$ and $N$ are Riemannian manifolds and
$f$ is a smooth map.  Let $V$ be a vector field on $M$.  Define the pullback
vector field $f^*(V)$ by
$$
\langle f^*V(m),\vec v_m\rangle =\langle V(f(m)),f_*(\vec v_m)\rangle.
$$
Then if $f:M^n\to {\Bbb R}^n$ so that $f_*|\partial M$ has maximal rank
and $f(\partial M)$ contains no zeros of $V$, then
$$
\TI\ f^*V=\sum v_iw_i+(\chi(M)-deg\ \hat N)
$$
where $v_i=\TI(x_i)$ where $x_i$ is the i${}^{\text{th}}$ zero of $V$, $w_i$
is the winding number of $f|\partial M$ about $x_i$, and $\hat N:\partial M\to
S^{n-1}$ is the normal (or Gauss) map.

\endremark

\demo{Proof}
In paper \cite{G${}_5$}.
\qed
\enddemo

\remark{ (14) Transfer Theorem}

Let $F @> i >> E @> p >> B$ be a smooth fibre bundle with $F$ a compact manifold and $B$ a closed manifold. Let $V$ be a proper vector field on $E$ with vectors tangent to the fibres. Then there is an $S$-map 
$\tau : B^+ \to E^+$ so that in ordinary homology 
$p_* \circ {\tau}_* $ (cohomology ${\tau}^* \circ  p^*$) is 
multiplication by the Index of $V$ restricted to a fibre,  $\TI (V|F)$.
\endremark

\demo{Proof}
In paper \cite{B-G}.
\qed
\enddemo


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%\vskip.75truein
%Purdue University

%\enddocument
\end

