\magnification=\magstep1
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%\def\br{{\Bbb R}}
\def\deg{\operatorname{deg}}
\def\Ind{\operatorname{Ind}}
\def\tr{\operatorname{tr}}
\def\Pf{\operatorname{Pf}}
\def\Proj{\operatorname{Proj}}
\def\Diff{\operatorname{Diff}} 
\def\Dim{\operatorname{Dim}}
\def\grad{\operatorname{grad}}
\def\leaderfill{\leaders\hbox to 1em{\hss.\hss}\hfill}
\def\Chi{\chi}
\def\TI{\operatorname{Ind}}
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\centerline{\bf TOPOLOGY AND THE NON-EXISTENCE OF MAGNETIC MONOPOLES} 
%\centerline{\bf v. 1.11}
\bigskip
\centerline{by}
\bigskip
\centerline{Daniel Henry Gottlieb}
\bigskip
\centerline{Department of Mathematics}
\centerline{Purdue University}
\centerline{West Lafayette, Indiana 47906}
\vskip .25truein
\noindent
{\bf Abstract}

Most of the work being done to unify General Relativity and Quantum
Mechanics tries to represent General Relativity in the Quantum Mechanics
language. We propose here an approach to represent
Quantum Mechanics in the language of Relativity. In order to introduce
discretness into the language of Relativity we consider the classical
invarients of homotopy theory, in particular the index of a vector
field. We insist that these invariants be treated as physical
quantities, independent of choices of observers and conventions.
Following this prescription we found 
an argument that pseudo-vectorfields should have zero index. Hence 
magnetic monopoles should not exist. 
We give extended philosophical
arguments that the index should play an important role in Mathematics, and
hence Physics, based
on a novel definition of Mathematics and the meaning of the underlying
unity of Mathematics.

\vfill\eject
\baselineskip=18pt
%\input mssymb
\parskip=6pt
\noindent
{\bf 1.\ \ Introduction}

There are two successful theories
in Physics: Quantum Mechanics and General Relativity.
There is considerable work being done to unify them. Mostly, this work tries
to represent Relativity as a form of Quantum Mechanics. To a mathematician,
however, General Relativity seems to make Physics well-defined and clear,
{\cite O},\cite {S--W},\cite {P},
whereas Quantum Mechanics is full of tricks and arguments with coordinate
systems and formal manipulations,\cite {Su}.
So why not try to make Quantum Mechanics
look like Relativity ?


To do this, we want some way of introducing discreteness and quantum numbers
into smooth and continuous space-time. In Quantum Mechanics this is
done by
eigenvalues and eigenvectors and symmetries. We believe that in Relativity,
one can do it by using the homotopy invariants of Algebraic Topology.
A good way
to proceed is to insist that any homotopy invariant which arises in a
physical way should be treated as a physical object. The easiest invariant
and most flexible to use is the index of a vector field. Hence the Index
Principle.

\proclaim{Principle of Invariance of Index}
The index of any ``physical'' vector
field is invariant under changes of coordinates and orientation of
space--time. If the index is
undefined, it signals either radiation or unrealistic physical
hypotheses.
\endproclaim


\proclaim{Consequence} Every ``physical'' pseudo--vector
field has index zero or the index is undefined.
 \endproclaim

 Now the magnetic vector field $\overrightarrow{B}$ is a pseudo--vector field.
 That means if we change the orientation of space,
 $\overrightarrow{B}$ changes to
 $-\overrightarrow{B}$.
 Now Ind$(-V)=(-1)^n$ Ind$(V)$ where $n$ is the dimension of the
 manifold on which $V$ is defined.
 Thus Ind$(-\overrightarrow{B})=(-1)^3$ Ind$(\overrightarrow{B})$.
 So either Ind$(\overrightarrow{B})$ is not defined
 or Ind$(\overrightarrow{B})=0 $.
 Now a magnetic monopole will give
 rise to a $\overrightarrow{B}$ with index $\pm 1$.
 As this is inconsistent with the Invariance of Index Principle, we predict
 that magnetic monopoles do not exist.

We remark that since the magnetic vector field $\overrightarrow{B}$ changes
for each observer, we provide a more sophisticated version of the 
argument. This involves the ``fibre bundle of space-time'' and will be 
discussed in section 6. But the essence of the argument is the one above.



Magnetic monopoles were predicted by Dirac 
based on an alteration of
Maxwell's equations which made them more symmetric, {\cite F}.
Despite Dirac's ideas, magnetic monopoles have not been found in nature.
More recently magnetic monopoles were
predicted using the topological nontriviality of certain principal
bundles,\ {\cite B}. 
Our non-existence argument is also based on Topology, but our
argument is more direct. Note that it does not depend on Maxwell's 
equations. It is a new kind of argument for Physics and so
we devote considerable philosophical discussion in sections 2 and 
3 as to why it is reasonable.


In section 2, we discuss why the index of vector fields should play an
important role in Mathematics. This involves the question of the
underlying unity of Mathematics. In section 3, methods of applying the
index in Mathematics are discussed. In section 4, the definition and key
properties of the index are listed. In section 5, natural extension
of the ideas of index and vector fields to fibre bundles are made. In
section 6, we give a more sophisticated argument that magnetic monopoles do
not exist using the electromagnetic field tensor $F$. 
We 
introduce a fibre bundle of space--time which
clarifies what we mean by the index undefined. Finally in section 7, we
discuss how the index of different zeros of a vector field act like 
electric charge of particles, and following this suggestion we speculate
that the transfer theorem for vertical vector bundles combined with the
existence of antiparticles gives  inductive evidence that particles may
be discribed by appropriate vertical vector fields on certain fibre bundles.

It is a pleasure to thank Solomon Gartenhaus for numerous discussions and
suggestions.

\bigskip\noindent
{\bf 2.\ \ The Unity of Mathematics}

We take the following definition of Mathematics:

\proclaim{Definition} Mathematics is the study of well--defined concepts.
\endproclaim

Now well--defined concepts are creations of the human mind.
And most of those creations can be quite arbitrary.
There is no limit to the well--defined imagination.
So if one accepts the definition that Mathematics is the study of the 
well--defined, then how can Mathematics have an underlying unity?
Yet it is a fact that many savants see a underlying unity in Mathematics,
so the
key question to consider is:

\proclaim{Question}
Why does Mathematics appear to have an underlying unity?
\endproclaim

If mathematical unity really exists then it is reasonable to hope that there
are a few basic principles which explain the occurrence of those phenomena
which persuade us to believe that Mathematics is indeed unified;
just as the various phenomena of Physics
seem to be explained by a few fundamental laws.
If we can discover these principles it would give us great insight into the
development of Mathematics and perhaps even insight into Physics.

Now what things produce the appearance of an underlying 
unity in Mathematics? 
Mathematics appears to be unified when
a concept, such as the Euler characteristic, appears over and over in
interesting results; or an idea, such as that of a group, is involved in
many different fields and is used in Science to predict or make
phenomena precise; or an equation, like De Moivre's formula
$$
e^{i\theta} = \cos\theta + i\sin\theta
$$
yields numerous interesting relations among important concepts in
several
fields in a mechanical way.

Thus the appearance of 
underlying unity comes from the ubiquity of certain concepts and objects,
such as the numbers $\pi$ and $e$ and concepts such as groups and rings,
and invariants such as the Euler characteristic and eigenvalues, which
continually appear in striking relationships and in diverse fields of
Mathematics and Physics.
We use the word {\it broad} to describe these concepts.

Compare broad concepts with {\it deep} concepts.
The depth of an idea seems to be a function of time.
As our understanding of a field
increases, deep concepts become elementary concepts, deep
theorems are transformed into definitions and so on.
But something broad, like the Euler characteristic, remains broad, or becomes
broader as time goes on.
The relationships a broad concept has with other concepts are forever.


\proclaim{The Function Principle}
Any concept which arises from a simple construction
of functions will appear over and over again throughout Mathematics.
\endproclaim


We assert the principle that function is one of 
the broadest of all mathematical
concepts, and any concept or theorem derived in a natural way from that of
functions must itself be broad.
We will use this principle to assert that the underlying unity of Mathematics
at least partly stems from the breadth of the concept of function.
We will show how the breadth of category and functor and equivalence and $e$
and $\pi$ and de Moivre's formula and groups and rings and Euler Characteristic
all follow from this principle.
We will subject this principle to the rigorous test of a scientific theory:\ \
It must predict new broad concepts.
We make such predictions and report on evidence that the predictions are
correct.

The concept of a function as a mapping $f\colon X\rightarrow Y$ from a source
set $X$ to a target set $Y$ did not develop until the Twentieth Century.
The modern concept of a function did not even begin to emerge until the middle
ages.
The beginnings of Physics should have given a great impetus to the notion of
function, since the measurements of the initial conditions of an experiment
and the final results gives implicitly a function from the initial states of an
experiment to the final outcomes; but historians say that the early physicists 
and mathematicians never thought this way.
Soon thereafter calculus was invented.
For many years afterwards functions were thought to be always given by some
algebraic expression.
Slowly the concept of a mapping grew.
Cantor's set theory gave the notion a good impulse but the modern notion was
adopted only in the Twentieth Century. See \cite {ML} for a good account
of these ideas.

The careful definition of function is necessary so that the definition of the
composition of two functions can be defined.
Thus $f \circ g$ is only defined when the target of $g$ is the source of $f$.
This composition is associative:\ \ $(f \circ g) \circ h=
f \circ (g \circ h)$ and $f$ composed with
the identity of either the source or the target is $f$ again.
We call a set of functions a {\it category} if it is closed under compositions
and contains the identity functions of all the sources and targets.

Category was first defined by S.\ Eilenberg and S.\ MacLane and was employed
by Eilenberg and N.\ Steenrod in the 1940's to give homology theory its 
functorial character.
Category theory became a subject in its own right, it's practitioners joyfully
noting that almost every branch of Mathematics could be organized as a category.
The usual 
definition of category is merely an abstraction of functions closed
under composition.
The functions are abstracted into things called morphisms and composition
becomes an operation on sets of morphisms satisfying exactly the same properties
that functions and composition satisfy.
Most mathematicians think of categories as very abstract things and are 
surprised to find they come from such a homely source as functions closed
under composition.

A {\it functor} is a function whose source and domain are categories and
which preserves composition.
That is, if $ F$ is the functor, then $ F(f \circ g)= F(f) \circ F(g)$.
This definition also is abstracted and one says category and functor in the
same breath.

Now consider the question:\ \ What statements can be made about a function $f$
which would make sense in every possible category?
There are basically 
only four statements since the only functions known to exist in every
category are the identity functions.
We can say that $f$ is an identity, or that $f$ is a {\it retraction} by which
we mean that there is a function $g$ so that $f \circ g$ is an identity, or that
$f$ is a {\it cross--section} by which we mean that there is a function $h$
so that $h \circ f$ is an identity, or finally that $f$ is an {\it equivalence}
by which we mean that $f$ is both a retraction and a cross--section.
In the case of equivalence the function $h$ must equal the function $g$ and it
is called the {\it inverse} of $f$ and it is unique.

Retraction and cross--section induce a partial ordering of the sources and
targets of a category, hereafter called the {\it objects} of the category.
Equivalences induce an equivalence relation on the objects and give us the
means of making precise the notion that two mathematical structures are the
same.

Now consider the self equivalences of some object $X$ in a category of
functions.
Since $X$ is both the source and the target, composition is always defined for
any pair of functions, as are inverses.
Thus we have a {\it group}.
The definition of a group in general is just an abstraction, where the functions
become undefined elements and composition is the undefined operation which
satisfies the group laws of associativity and existence of identity and
inverse, these laws being the relations that equivalences satisfy.
The notion of functor restricted to a group becomes that of
{\it homomorphism}.
The equivalences in the category of groups and homomorphisms are called
{\it isomorphisms}.

The concept of groups arose in the solution of polynomial equations, with the
first ideas due to Lagrange in the late eighteenth century, continuing through
Abel to Galois.
Felix Klein proposed that Geometry should be viewed as arising from groups of
symmetries in 1875.
Poincare proposed that the equations of Physics should be invariant under the
correct symmetry groups around 1900.
Since then groups have played an increasingly important role in Mathematics
and in Physics.
The increasing appearance of this broad concept must have fed the feeling of
the underlying unity of Mathematics.
Now we see how naturally it follows from the Function Principle.

If we consider a set of functions $S$ from a fixed object $X$ into a group $G$,
we can induce a group structure on $S$ by defining the multiplication of two
functions $f$ and $g$ to be $f*g$ where $f*g(a)=f(a)\cdot g(a)$. Here $a$ runs
through all the elements in $X$ and ``$\cdot$''
is the group multiplication in $G$.
This multiplication can be easily shown to satisfy the laws of group 
multiplication.
The same idea applied to maps into the Real Numbers or the Complex Numbers
gives rise to addition and multiplication on functions.
These satisfy properties which are abstracted into the concepts of abelian rings.
If we consider the set of self homomorphisms 
of abelian groups and use composition and
addition of functions, we get an important 
example of a non--commutative ring.
The natural functors for rings should be ring homomorphisms.
In the case of a ring of functions into the Real or Complex numbers we note
that a ring homomorphism $h$ fixes the constant maps.
If we consider all functions which fix the constants and preserve the
addition, we get a category of functions from rings to rings; that is, these
functions are closed under composition.
We call these functions {\it linear transformations}.
They contain the ring homomorphisms as a subset.
Study the equivalences of this category.
We obtain the concepts of {\it vector spaces} and linear transformations
after the usual abstraction.

Now we consider a category of homomorphisms of abelian groups.
We ask the same question which gave us equivalence and groups, namely:\ \
What statements can be made about a homomorphism $f$ which would make sense
in every possible category of abelian groups?
Now between every possible abelian group there is the trivial homomorphism
$0\colon A\rightarrow B$ which carries all of $A$ onto the identity of $B$.
Also we have for every integer $N$ the homomorphism from $A$ to itself which
adds every element to itself $N$ times, that is multiplication by $N$.

Thus for any homomorphism $h\colon A\rightarrow B$ there are three statements
we can make which would always make sense.
First $N \circ h$ is the trivial homomorphism 0, second that there is a
homomorphism $\tau\colon B\rightarrow A$ so that $h \circ \tau$ 
is multiplication
by $N$, and third that $\tau \circ h$ is multiplication by $N$.
So we can give to any homomorphism three non--negative integers:\ \ The
{\it exponent}, the {\it cross--section degree}, and the {\it retraction
degree}.
The {\it exponent} is the smallest positive integer such that $N \circ h$ is the
trivial homomorphism $0$.
If there is no such $N$ then the exponent is zero.
Similarly the {\it cross--section degree} is the smallest positive $N$ such
that there is a $\tau$, called a {\it cross--section transfer}, so that
$h \circ \tau$ is multiplication by $N$.
Finally the {\it retraction degree} is the smallest positive $N$ such that
there is a $\tau$, called a {\it retraction transfer}, so that $\tau \circ h$ 
is
multiplication by $N$.

In accordance with the Function Principle,
we predict that these three numbers will be
seen to be broad concepts.
Their breadth should be less than the breadth of equivalence, retraction and
cross--section because the concepts are valid only for categories of abelian
groups and homomorphisms.
But exponent, cross--section degree and retraction degree can be pulled back
to other categories via any functor from that category to the category of
abelian groups.
So these integers potentially can play a role in many interesting categories.
In fact for the category of topological spaces and continuous maps we can say
that any continuous map $f\colon X\rightarrow Y$ has exponent $N$ or
cross--section degree $N$ or retraction degree $N$ if the induced
homomorphism $f_*\colon H_*(X)\rightarrow H_*(Y)$ on integral homology has
exponent $N$ or cross--section degree $N$ or retraction degree $N$
respectively.

As evidence of the breadth of these concepts we point out that for integral
homology, cross--section transfers already play an important role for fibre
bundles.
There are natural transfers associated with many of the important classical
invariants such as the Euler characteristic and the index of fixed points and
the index of vector fields,\cite{B--G},\cite {G1}
and the Lefschetz number and coincidence number and
most recently the intersection number, \cite{G--O}.
And a predicted surprise relationship occurs in the case of cross--section 
degree for a map between two spaces.
In the case that the two spaces are closed oriented manifolds of the same
dimension, the cross--section degree is precisely the absolute value of the
classical Brouwer degree.
The retraction degree also is the Brouwer degree for closed manifolds if we
use cohomology as our functor instead of homology, \cite{G1}.

The most common activity in Mathematics is solving equations.
There is a natural way to frame an equation in terms of functions.
In an equation we have an expression on the left set equal to an expression
on the right and we want to find the value of the variables for which the two
expressions equal.
We can think of the expressions as being two function $f$ and $g$ from $X$ to
$Y$ and we want to find the elements $x$ of $X$ such that $f(x)=g(x)$.
The solutions are called {\it coincidences}.
Coincidence makes sense in any category and so we would expect the elements
of any existence or uniqueness theorem about coincidences to be very broad 
indeed.
But we do not predict the existence of such a theorem.
Nevertheless in Topology there is such a theorem.
It is restricted essentially to maps between
closed oriented manifolds of the same dimension.
It asserts that locally defined coincidence indices add up to a globally defined
coincidence number which is given by the action of $f$ and $g$ on the homology
of $X$.
In fact this coincidence number is the 
alternating sum of traces of the composition of the umkehr
map $f_!$, which is defined using Poincare Duality, and $g_*$, the homomorphism
induced by $g$.
We predict, at least in Topology and Geometry, more frequent appearances of
both the coincidence number and also the local coincidence index and they 
should relate with other concepts.

If we consider self maps of objects, a special coincidence is the fixed point
$f(x)=x$.
From the point of view of equations in some algebraic setting, the coincidence
problem can be converted into a fixed point problem, so we do not lose any
generality in those settings by considering fixed points.
In any event the fixed point problem makes sense for any category.
Now the relevant theorem in Topology is the Lefschetz fixed point theorem.
In contrast to the coincidence theorem, the Lefschetz theorem holds essentially
for the wider class of compact spaces.
Similar to the coincidence theorem, the Lefschetz theorem has locally defined
fixed point indices which add up to a globally defined Lefschetz number.
This Lefschetz number is the alternating sum of traces of $f_*$, the
homomorphism induced by $f$ on homology.
This magnificent theorem is easier to apply than the coincidence theorem and so
the Lefschetz number and fixed point index are met more frequently in various
situations than the coincidence number and coincidence indices.

In other fields fixed points lead to very broad concepts and theorems.
A linear operator gives rise to a map on the one dimensional subspaces.
The fixed subspaces are generated by {\it eigenvectors}.
Eigenvectors and their associated eigenvalues play an important role
in Mathematics and Physics and are to be found in the most surprising places.

Consider the category of $C^\infty$ functions on the Real Line.
The derivative is a function from this category to itself taking any function
$f$ into $f^\prime$.
The derivative practically defines the subject of calculus.
The fixed points of the derivative are multiples of $e^x$.
Thus we would predict that the number $e$ appears very frequently in calculus
and any field where calculus can be employed.
Likewise consider the set of analytic functions of the Complex Numbers.
Again we have the derivative and its fixed point are the multiples of $e^z$.
Now it is possible to relate the function $e^z$ defined on a 
complex plane with real valued functions by
$$
e^{(a+ib)}=e^a (\cos(b)+i\sin (b)).
$$
We call this equation de Moivre's formula.
This formula contains an unbelievable amount of information.
Just as our concept of space--time separation is supposed to break down near
a black hole in Physics, so does our definition--theorem view of Mathematics
break down when considering this formula.
Is it a theorem or a definition?
Is it defined by sin and cos or does it define those two functions?

Up to now the function 
principle predicted only that some concepts and objects will
appear frequently in undisclosed relationships with important concepts
throughout Mathematics.
However the de Moivre equation gives us methods for discovering the precise
forms of some of the relationships it predicts.
For example, the natural question ``When does $e^z$ restrict to real valued
functions?'' leads to the ``discovery'' of $\pi$.
From this we might predict that $\pi$ will appear throughout calculus type
Mathematics, but not with the frequency of $e$.
Using the formula in a mechanical way we can take complex roots, prove
trigonometric identities, etc.

There is yet another fixed point question to consider:\ \ What are the fixed
points of the identity map?
This question not only makes sense in every category; it is solved in every
category!
The invariants arising from this question should be even broader than those from
the fixed point question.
But they are very uninteresting.
However, if we consider
the fixed point question for functions which are equivalent
to the identity under some suitable equivalence relation in a suitable category
we may find very broad interesting things.
A suitable situation involves the fixed points of maps homotopic to the identity
in the topological category.
For essentially compact spaces the {\it Euler characteristic} (also called the
Euler--Poincare number) is an invariant of a space whose nonvanishing results
in the existence of a fixed point.
This Euler characteristic is the most remarkable of all mathematical invariants.
It can be defined in terms simple enough to be understood by a school boy, and
yet it appears in many of the star theorems of Topology and Geometry.
A restriction of the concept of the Lefschetz number, its occurrence far exceeds
that of its ``parent'' concept.
First mentioned by Descartes, then used by Euler to 
study regular polyhedra, the Euler characteristic slowly proved its importance.
Bonnet showed in the 1840's that the total curvature of a closed surface
equaled a constant times the Euler characteristic of the two dimensional sphere.
Poincare gave it its topological invariance by showing it was the alternating
sum of Betti numbers.
In the 1920's Lefschetz showed that it determined the existence of fixed points
of maps homotopic to the identity, thus explaining, according to the
Function Principle,
its remarkable history up to then and predicting
the astounding frequency of its subsequent appearances in Mathematics.

The Euler characteristic is equal to the sum of the local fixed point indices
of the map homotopic to the identity.
We would predict frequent appearances of the local index.
Now on a smooth manifold we consider vector fields and regard them as
representing infinitesimally close maps to the identity.
Then the local fixed point index is the local index of the vector field.

These considerations lead us to the prediction that a certain equation due
to Marston Morse, {\cite M},
will play a very active role in Mathematics, and by
extension Physics.
This equation, which we call the Law of Vector Fields was discovered in 1929
and has not played a role at all commensurate with our prediction up until now.

We describe the Law of Vector fields.  Let
$M$ be a compact manifold with boundary.  Let $V$ be a vector field on
$M$
with no zeros on the boundary. Then consider the open set of the
boundary
of $M$ where $V$ is pointing inward.  Let $\partial_{-}V$ denote the
vector
field defined on this open set on the boundary which is given by
projecting
$V$ tangent to
the boundary.  The Euler characteristic of $M$ is denoted by
$\chi(M)$, and $\Ind(V)$ denotes the index of the vector field.  Then the
Law of Vector Fields is
$$
\Ind(V) + \Ind(\partial_{-}V) = \chi(M)
$$

We propose two methods of applying the law of vector fields to get new results
and we report on their successes.
These successes and the close bond between Physics and Mathematics encourage
us to predict that the Law of Vector Fields and its attendant concepts must
play a vital role in Physics.


\bigskip\noindent
{\bf 3.\ \ The Law of Vector Fields}

Just as de Moivre's formula 
gives us  mechanical 
methods which yields precise relationships among broad concepts,
we predict that the Law of Vector Fields will give mechanical 
methods which will
yield precise relationships among broad concepts.

One method is:

\item{1.}Choose an interesting vector field $V$ and manifold $M$.

\item{2.}Adjust the vector field if need be to eliminate zeros on the boundary.

\item{3.}Identify the global and local Ind $V$.

\item{4.}Identify the global and local index Ind$(\partial_- V)$.

\item{5.}Substitute 3 and 4 into the Law of Vector Fields.

We predict that this method will succeed because the Law of Vector
Fields is morally the definition of index, so all features of the index
must be derivable from that single equation.  We measure success in the
following descending order: 1. An important famous theorem generalized;
2.
A new proof of an important famous theorem; 3. A new, interesting
result.
We put new proofs before new results because  it may not be apparent at
this time that the new result will famous or important.

In category 1 we already have the extrinsic Gauss-Bonnet theorem of
differential Geometry \cite{G3}, the Brouwer fixed point 
theorem of Topology \cite{G3}, and
Hadwiger's formulas of Integral Geometry,\ \cite{G3},
\ \cite{Had},\ \cite{Sa}.  In category 2 we have the
Jordan separation theorem, The Borsuk-Ulam theorem, the Poincare-Hopf
index
theorem of Topology; Rouche's theorem and the Gauss-Lucas theorem in complex
variables;
the fundamental
theorem of algebra and the intermediate value theorem of elementary
Mathematics; and the not so famous Gottlieb's theorem of group homology,
\cite{G2}.
Of course we have more results in category 3, but it is not so easy to
describe them with a few words.  One snappy new result is the following:
Consider any straight line and smooth surface of genus greater than 1 in
three dimensional Euclidean space.  Then the line must be contained in a
plane which is tangent to the surface, ({\cite{G3}}, theorem 15).

We will discuss the Gauss-Bonnet theorem since that yields results in
all
three categories as well as having the longest history of all the
results
mentioned.  One of the most well-known theorems from ancient times is
the
theorem that the sum of the angles of a triangle equals 180 degrees.
Gauss showed for a triangle whose sides are geodesics on a surface $M$
in
three-space that the sum of the angles equals $\pi + \int_{M}KdM$, where
$K$ is the Gaussian curvature of the surface.  Bonnet and Van Dyck pieced these
triangles
together to prove that for a closed surface $M$ the total curvature
$\int_{M}KdM$ equals $2\pi\chi(M)$.  Hopf proved that $\int_{M}KdM$,
where
$M$ is a closed hypersurface in odd dimensional Euclidean space and $K$
is
the product of the principal curvatures must equal the degree of the
Gauss
map $\hat N: M^{2n}\to S^{2n}$ times the volume of the unit sphere.
Then
he proved $2\deg(\hat N) = \chi(M^{2n})$.  (Morris Hirsch in \cite{Hi}
gives credit to Kronicker and Van Dyck for Hopf's result.)
For a history
of
the Gauss-Bonnet theorem see \cite{Gr}, pp. 89-72 or \cite{Sp}, p. 385.
Or see \cite {G4} for a history close to the point of view given here.


Let $f: M\to R^{n}$ be a smooth map from a compact Riemannian manifold
of
dimension $n$ to $n$-dimensional Euclidean space so that $f$ near the
boundary $\partial M$ is an immersion. 
The index of the
gradient of
$x \circ f: M\to R$, where $x$ is the projection of $R^{n}$ onto the $x$-axis,
is equal to the difference between the Euler Characteristic and the
degree
of the Gauss map.  Thus
$$
\Ind(\grad(x  \circ f))= \chi(M)-\deg \hat N.
$$
This equation leads to an immediate proof of the Gauss-Bonnet Theorem,
since for odd dimensional $M$ and any vector field $W$, the index
satisfies
$\Ind(-W)= -\Ind(W)$.  Thus the left side of the equation reverses sign
while the right side of the equation remains the same.  Thus $\chi(M)$
equals the degree of the Gauss-map, which is the total curvature over
the
volume of the standard $n-1$ sphere.  Now $2\chi(M) = \chi(\partial M)$,
so
we get Hopf's version of the Gauss-Bonnet theorem.

Note as a by-product we also get $\Ind(\grad(x \circ f)) = 0$ which  is a new
result thus falling into category 3.  Another consequence of the
generalized
Gauss-Bonnet theorem follows when we assume the map $f$ is an immersion.
In
this case the gradient of $x \circ f$ has no zeros, so its index is zero so
the
right hand side in zero and so again $\chi(M)= \deg \hat N$.  This is
Haefliger's theorem \cite{Hae}, a category 2 result.  Please note in
addition that the Law of Vector Fields applied to odd dimensional closed
manifolds, combined with the category 2 result $\Ind(-W)= -\Ind
(W)$, implies that the Euler characteristic of such manifolds is zero,
(category 2).  So the Gauss-Bonnet theorem and this result have the same
proof in some strong sense. Also our prediction of the 
non-existence of magnetic monopoles follows from the same result! 

Just as the Gauss-Bonnet theorem followed 
from considering pullback vector fields, the
Brouwer fixed point theorem is generalized by considering the following
vector field.  Suppose $M$ is an $n$-dimensional body in $R^{n}$ and
suppose that $f: M\to R^{n}$ is a continuous map.  Then let the vector
field $V_{f}$ on $M$ be defined by drawing a vector from $m$ to the
point
$f(m)$ in $R^{n}$.  
If the map $f$
satisfies the transversal property, that is the line between $m$ on the
boundary of $M$ and $f(m)$ is never tangent to $\partial M$, than $f$
has
a fixed point if $\chi(M)$ is odd (category 1).  This last sentence is
an
enormous generalization of the Brouwer fixed point theorem, yet it
remains
a small example of what can be proved from applying the Law of Vector
Fields
to $V_{f}$.  In fact the Law of Vector Fields applied to $V_{f}$ is the
proper generalization of the Brouwer fixed point theorem.

A second method of producing Mathematics from the Law of Vector Fields
involves making precise the statement that the Law defines the index of
vector fields, \cite{G--S}.
In this method we learn from the Law.  The Law teaches us
that there is a generalization of homotopy which is very useful.  This
generalization, which we call {\it otopy}, not only allows the vector
field to
change under time, but also its domain of definition changes under time.
An otopy is what $\partial_{-}V$  undergoes when $V$ is undergoing a
homotopy.  A {\it proper otopy} is an otopy which has a compact set of zeros.
The proper otopy classes of vector fields on a connected manifold is in
one
to one correspondence with the integers via the map which takes a vector
field to its index.  This leads to the fact that homotopy classes of
vector
fields on a manifold with a connected boundary where no zeros appear on
the
boundary are in one to one correspondence with the integers.  This is
not
true if the boundary is disconnected.

We find that we do not need to assume that vector fields are continuous.
We can define the index for vector fields which have discontinuities and
which are not defined everywhere.  We need only assume that the set of
``defects" is compact and never appears on the boundary or frontier of
the
sets for which the vector fields are defined.  We
then can define an index for any compact connected component of defects
(subject only to the mild condition that the component is open in the
subspace of defects).  Thus under an otopy, it is as if the defects
change
shape with time and collide with other defects, and all the while each
defect has an integer associated with it.  This integer is preserved
under
collisions.  That is, the sum of the indices going into a collision
equals
the sum of the indices coming out of a collision, provided no component
``radiates out to infinity", i.e. loses its compactness.

This picture is very suggestive of the way charged particles are
supposed
to interact.  Using the Law of Vector Fields as a guide we have defined
an
index which satisfies a conservation law under collisions.  The main
ideas
behind the construction involve dimension, continuity, and the concept
of
pointing inside.  We suggest that those ideas might lie behind all the
conservation laws of collisions in Physics. 

\bigskip


\noindent {\bf 4. Properties of Index}



The Law of Vector Fields is the following:  Let $M$ be a compact smooth
manifold and let $V$ be a vector field on $M$ so that $V(m)\not=\vec 0$ for
all $m$ on the boundary $\partial M$ of $M$.  Then $\partial M$ contains
an open set $\partial_-M$ which consists of all $m\in\partial M$ so that $V(m)$
points
inside.  We define a vector field, denoted $\partial_-V$
on $\partial_-M$, so that
for every $m\in\partial_-M$ we have $\partial_-V(m)=$
Projection of $V(m)$ tangent
to $\partial_-M$.  Under these conditions we have
$$
\TI\ V+ \TI\ \partial_-V=\chi(M)\tag1
$$
where $\TI\ V$ is the index of the vector field and $\chi(M)$ is the Euler
characteristic of $M$.  (\cite{M}, \cite{G--O}, \cite{P}).

The Law of Vector Fields can be used to define the index of vector fields,
so the whole of index theory follows from (1).  The definition of index is
not difficult, but proving it is well-defined is a little involved, \cite{G--S}.
The definition proceeds as follows:

\item{a)}
The index of an empty vector field is
zero.

\item{b)}
If $M$ is a finite set of points and $V$ is defined on all of the
$M$ (the vectors are necessarily zero), then $\TI(V)=\text{ number of points
in } M$.

\item{c)}
If $V$ is a {\it proper} vector field on a compact $M$, by which we mean $V$
has no defects on $\partial M$, then we set
$$
\TI\ V=\chi(M)-\TI(\partial_-V).
$$

\item{d)}
If $V$ is defined on the closure of an open subset $U$ of a smooth manifold
$M$ so that the set of defects $D$ is compact and $D\subset U$, then we say
$V$ is a {\it proper vector} field.  The index $\TI\ V$ is defined to be
$\TI(V|M)$ where $M$ is any compact manifold such that $D\subset M\subset U$.

\item{e)}
If $C$ is a connected component of $D$ and $C$ is compact and open in $D$ define
$\TI_C(V)$ to be the index of $V$ restricted to an open set containing $C$
and no other defects of $V$.

A key idea in proving this definition is well-defined is a generalization
of the concept of homotopy which we call otopy.  An {\it otopy} is what
$\partial_-V$ undergoes when $V$ undergoes a homotopy.  The formal definition
is as follows:  An {\it otopy} is a vector field $V$ defined on the closure
of an open set $T\subset M\times I$ so that $V(m,t)$ is tangent to the slice
$M\times t$.  The otopy is {\it proper} if the set of defects $D$ of $V$ is
compact and contained in $T$.  The restriction of $V$ to $M\times 0$ and
$M\times 1$ are said to be properly otopic vector fields.  Proper otopy is
an equivalence relation.

The following properties hold for the index:

\item{(2)}
Let $M$ be a connected manifold.  The proper otopy classes of proper vector
fields on $M$ are in one to one correspondence via the index to the
integers.  If $M$ is a compact manifold with a connected boundary, then a
vector field $V$ is properly homotopic to $W$ if and only if
$\TI\ V=\TI\ W$.
\medskip

\item{(3)}
$\TI(V|A\cup B)=\TI(V|A)+\TI(V|B)-\TI(V|A\cap B)$
\medskip

\item{(4)}
$\TI(V\times W)=\TI(V)\cdot\TI(W)$
\medskip

\item{(5)}
$\TI(-V)=(-1)^{\text{dim } M}\TI(V)$
\medskip

\item{(6)}
If $V$ has no zeros, then $\TI(V)=0$
\medskip

\item{(7)}
$\TI(V)=\sum_C \TI_C(V)$ for all compact connected components $C$, assuming
$D$ is the union of a finite number of compact connected components.
\bigskip
For certain vector fields the index is equal to classical invariants. Suppose
$f:R^n\to R^n$.  Let $M$ be a compact $n$ submanifold.  Define
$V^f$ by $V^f(m)=f(m)$.  If $f:\partial M\to R^n-\vec 0$, then
\medskip

\item{(8)}
$\TI\ V^f=\text{deg }f$.
\medskip

Suppose $f:U\to R^n$ where $U$ is an open set of $ R^n$.  Let
$V_f(m)=\vec m-\overset\longrightarrow\to{f(m)}$.  Then
\medskip
\item{(9)}
$\TI\ V_f=$ fixed point index of $f$ on $U$.
\bigskip

Suppose $f:M\to N$ is a smooth map between two Riemannian manifolds.  Let
$V$ be a vector field on $N$.  Let $f^*V$ be the pullback of $V$ on $M$.  We
define the pullback by
$$
\langle f^*V(m),\vec v_m\rangle=\langle V(f(m)), f_*(\vec v_m)\rangle.
$$
Note that for $f:M\to R$ and $V=\frac d{dt}$, we have $f^*V=
\text{ gradient } f$.

Now suppose that $f:M^n\to  R^n$ where $M^n$ is compact and $V$ is a
vector field on $ R^n$ so that $f$ has no singular points near
$\partial M$ and $V$ has no zeros on $f(\partial M)$.  Then if $n>1$
\medskip
\item{(10)}
$\TI\ f^*V=\sum w_iv_i+(\chi(M)-\text{deg } \hat N)$
\medskip
\noindent
where $\hat N:\partial M\to S^{n-1}$ is the Gauss map defined by the
immersion of $\partial M$ if $ R^n$ under $f$, and
$v_i={\text Ind}_{c_i}(V)$ for the i${}^{\text{th}}$ zero of $V$ and $w_i$
is the winding number of the i${}^{\text{th}}$ zero
with respect to $f:\partial M\to R^n$.
The winding number is calculated by sending
a ray out from the i${}^{\text{th}}$ zero and noting where it hits the
immersed $n-1$
manifold $\partial M$.  At each point of intersection the ray is either
passing inward or outward relative to the outward pointing normal $N$.  Add up
these point assigning $+1$ if the ray is going from inside to outside, and
$-1$ if the ray goes from outside to inside. This is the generalized Gauss-
Bonnet theorem.



Using the property that the Euler--Poincare number is an invariant of
homotopy type and the above properties of the index, the following
useful properties of the Euler--Poincare number can easily be proved.

\item{1.}\ \ $\chi (M) = 1$ where $M$ is contractible.

\item{2.}\ \ $\Chi (M_1 \cup M_2) = \Chi (M_1) + \Chi (M_2) - \Chi
(M_1 \cap M_2)$
where $M_1$ and $M_2$ and $M_1 \cap  M_2 $ are submanifolds of
$M_1 \cup M_2$.

\item{3.}\ \ $\Chi (M_1 \times M_2) = \Chi (M_1) \times \Chi (M_2) $

\item{4.} \ \  $\Chi(M) = 0 $ when $M$ is a closed odd dimensional
manifold.

\item{5.} \ \ $\Chi (\partial M) = 2 \Chi (M)$ if $M$ is an even
dimensional manifold.

\bigskip
\noindent

\noindent {\bf 5. Vertical Vector Fields}

Let $F\to E\overset p\to\longrightarrow B$ be a fibre bundle whose fibre
$F$ is a smooth manifold and whose structure group is the group of
diffeomorphisms of $F$.  Then we have a vector bundle $\alpha$ over $B$ of
vectors tangent to the fibres.  That is $\alpha |F=$
tangent bundle of $F$.  A
vertical vector field $V$ on $E$ is an assignment to each point $e$ of $E$ a
vector in $\alpha$ at $e$. $V$ might be empty or it might be defined on
part of $E$.  A more precise way to express this is that $V:
S\to\alpha$ is a cross-section from some subset $S$ of $B$ into $\alpha$.

If $B= I$, the unit interval, then $V$ is called an otopy.  If $V$ is an
otopy which is continuous and defined over all of $E$, then $V$ is called a
homotopy.

A vertical vector field $V$ is {\it proper} if $D\cap p^{-1}(C)$ is
compact for all compact subsets $C$ of $B$ where $D$ is the set of defects
of $V$.

A vertical vector field $V$ is {\it proper with respect to an open
set} $U\subset E$ if $(D\cap U)\cap p^{-1}(C)$ is compact for all compact
$C$ in $B$ and if $V$ can be extended continuously over the frontier
$\overline U -U$ so that there are no zeros on $\overline U-U$.

If $F\to E\overset p\to\longrightarrow B$ is a fibre bundle so that $F$
is compact and
has boundary ${\overset \cdot\to F}$, we say a vertical vector field $V$ is
{\it proper with respect to the boundary} if $D\cap {\overset \cdot\to
E}
=\emptyset$ where ${\overset \cdot\to E}\subset E$ is the set of points in $E$
on the boundary of some fibre.

Note that $V$ proper with respect to the boundary implies that $V$ is
proper with respect to the open set $E- {\overset \cdot\to E}$.

The above definitions restrict to the concepts of proper homotopy and
proper otopy when $B = I$.



If $W$ is a vertical vector field, and if $V$ is the restriction of $W$
to a
fixed fibre, then we say the defects of $V$ {\it interact} via $W$ if they
are
contained in a connected set of defects of $W$.
An important class of questions is the following.
If $F @>>> E @>p>> B$ is a fibre bundle and $V$ is a vector field on a
fibre
$F$, is it possible to extend $V$ to a vertical vector field $W$ so that
certain defects of $V$ do not interact, or so that the defects of $W$
satisfy
some condition such as they are compact?


The extension of vector fields with nonzero indices puts strong
conditions on the homology of the fibre bundle as the following results
from [B--G] show:

Let $F\overset{i}\to\longrightarrow E\overset{p}\to\longrightarrow B$ be a
smooth fibre bundle with $F$ a compact manifold with boundary
$\partial F$ and $B$ a finite complex.  Let $V$ be a proper vertical vector
field defined
on an open set of $E$.  We assume that $V$ has no zeros on
$\dot {E}$.  We will call such vector fields
{\it vertical} vector fields for short in the next theorems.

In [B--G], we defined an $S$-map $\tau^{}_{V}: B^{+}\rightarrow E^{+}$
associated with
$V$.  This transfer $\tau^{}_{V}$ has the usual properties:

\noindent {\it a) If $V$ is homotopic to a vertical vector field $V'$ by a
homotopy of
vertical vector fields, so in particular no zeros appear on ${\overset
\cdot\to E}$,
then $\tau^{}_{V'}$ is homotopic to $\tau^{}_{V}$.

\noindent b) $\tau^{*}_{V}(p^{*}\alpha\cup \beta) = \alpha\cup\tau^{*}_{V}(\beta
)$
for cohomology theories $h^{*}$ with cup products.

\noindent c) For ordinary homology or cohomology, $p_{*}\circ\tau^{}_{V_{*}}$
and
$\tau^{*}_{V}\circ p^{*}$ is multiplication by the index of $V$ restricted
to a fibre $F$, denoted $\Ind (V|F)$.}
$$
p_{*}\circ\tau^{}_{V_{*}}={\text Ind}(V|F) \tag11
$$



Also in [B--G] the following theorem is shown. Given fibre bundle
$F\overset{i}\to\longrightarrow E\overset{p}\to\longrightarrow B$ where
$V$ is a vertical vector field, that
$$
0 = \Ind  (V|F)\omega_{*}: \{X, \Omega B\}\rightarrow \{X,F\}
$$
is trivial.  Here we assume that $X$ is a finite complex, $\Omega
B\overset{\omega}\to\longrightarrow F$ is the transgression map induced by
the fibre bundle, $\{X,Y\}$ denotes the group of stable homotopy classes
from $X$ to $Y$.

It follows that
$$
0 = \Ind  (V|F)\omega_{*}: H_*(\Omega B)\rightarrow H_*(F) \tag12
$$

\bigskip
\noindent
{\bf 6.\ \ The Nonexistence of Magnetic Monopoles}



We have the following picture immerging out of the previous sections. A
vector field has a set of connected components of defects.
Now under a homotopy these components move around and collide with one another.
There is a conservation law which says that the sum of the indices of the
components going into a collision is equal to the sum of the indices of the
components at the collision is equal to the sum of the indices after the
collision,
if during the homotopy all the components remain compact and there are
only a finite number of them.
Thus the index remains conserved unless some component ``radiates out to 
infinity.''
This suggests particles bearing charges could be modeled as defects
of vector fields.

The fact that charge--like conservation follows from a simple topological
construct, which depends only on continuity and dimension and pointing inside,
suggests that the topological concept of index has physical content.

There is another compelling reason to consider the index as a physical
quantity. It is an invariant of General Relativity. This is made precise
in the following theorem.

\proclaim{theorem} Suppose $V$ is a space--like vector field in a
space--time
$S$. Suppose  $M$ and $N$ are two time--like slices of $S$ which can be
smoothly
deformed into each other. Suppose $D$, the the set of 
defects of $V$, is compact in the
region of $S$ where the deformation takes place. Then the index of $V$
projected
onto $M$ is equal to the index of $V$ projected onto $N$.
\endproclaim

This theorem is true since we can set up a proper otopy between the two
projected vector fields given the hypotheses of the theorem. This means for any
space--like
vector field, the index is invariant under any choice of space--like
slices.
Thus it is mathematically true that the index is an invariant of general
relativity, just like proper time, unless there is topological radiation
or
there is a singularity or strange topology between the two slices so
that there is no deformation possible.

Now we will give an argument that Magnetic Monopoles do not exist using
Classical Field Theory. See \cite {M--T--W}, {\cite P}, {\cite T},
{\cite F} for the relevant formalism. 

Let $F$ be an electromagnetic 2-form on space-time $S$.
Let $\hat {F}$ denote the associated linear transformation on the tangent
space of $S$. Let $u$ denote a time-like unit vector field.
Then the electric vector field associated to $u$ is a space-like vector field
given by  $\overrightarrow{E} = \hat{F}u$.
Now consider the 2-form $*F$. Here the $*$ denotes the Hodge dual which
depends on the choice of orientation made on $S$. Now the magnetic field
relative to $u$ is given by
$- \overrightarrow{B} = (\hat{F}^*)u$. Note that $\overrightarrow{B}$
reverses direction if the orientation is changed. The Index Principle
asserts that either the index of $- \overrightarrow{B} = (\hat{F}^*)u$
is zero, or the vector field is not defined. Now a magnetic monopole will
have either index $+1$ or $-1$. Hence the monopole cannot exist.

We make three remarks on the above argument. The first deals with the
question: Why doesn't $\overrightarrow{E}$ reverse sign if we change our
conventions of positive charge to negative charge? We say that the
vector field $u$ represents a swarm of test charges. If the sign of the
test charge changes we still expect that the test charge will
accelerate in the same direction because the signs of every charge is
reversed. If we insisted on keeping the sign of the test charge the same
and changed the signs on all the real charges it would be as if we
changed the experiment, so we should not expect that the index of 
$\overrightarrow{E}$ would remain the same.

The second remark deals with the orientation of space-time.
We are assuming that changing the orientation of space-time does not
intail a change in some experiment used to define the electric and
magnetic vectors. If it did, if for example some particle interacts with
$F$ by means of an ``intrinsic choice of orientation", then the Index
Principle would no longer apply. So it is worthwhile to think a bit
about changes in orientation.

The violation of parity by the weak force is illustrated by describing an
experiment which cannot take place if reflected in a mirror. Sudbury
does it in describing the beta decay of Cobalt 60 
on page 273 of \cite {Su}.
and Feynman does it on page 17--11
in volume III of \cite{F--L--S} where he describes the
$\Lambda^0$ decay. In both cases the experiment would be possible if one
changed the handedness conventions in the mirror. That is, assume the
reflection carries the change from the right hand rule to the left hand
rule. In that case, the axial vectors spin and angular momentum would be
reflected the correct way. Feynman describes the mirror reflection very
carefully in volume I , chapter 35, section 5 of 
\cite{F--L--S}. There is no overriding reason why he chooses not to
change the handedness in the reflected world.

In Quantum Mechanics the transformation   P  is given by reflecting the
coordinate system. Since Quantum Mechanics concerns itself with formal
manipulation of coordinate systems, there is no question about
reversing the handedness. It does not seem that changing orientation
fits into Quantum Mechanics. But
Classical Electrodynamics is unequivocal that the handedness be changed.
The change of coordinates of Quantum Mechanics is replaced by the more
intrinsic reflection diffeomorphism $R$. Now $R$ is an isometry of the
Lorentzian metric which reverses orientation. Since every measurement
must be the same, if we reverse the orientation we can imagine $R$ as an
orientation preserving isometry onto the target space--time and every
statement involving electromagnetism on the source space--time is in
one to one correspondence with those of the target space--time. Then if
we change the orientation back so that $R$ is not orientation
preserving, the possibility exists that there is some statement no
longer true. Thus the 
arguments of Sudbury and of Feynman are not so convincing since
they arbitrarily choose the orientation in the mirror's world.

The third remark deals with the fact that there may be local zeros of
$\overrightarrow{B}$ with nonzero indices. This seems to violate the Index
Principle. Also, $\overrightarrow{B}$ changes for changing observers.
So we need to modify the index principle by making precise the concept of 
undefined index. We shall do that below.

The correct generalization of proper otopy, as we mentioned in section 5,
is that of a {\it proper
vertical vector field along the fibres of a fibre bundle}
$$ M \rightarrow E \rightarrow B $$
where $M$ is a smooth manifold and $V$ is a vector field along the fibres
and proper means that the defects of $V$ are compact over any compact
subset of $B$. Then $V$ restricted to any fibre has the same index.

Now there is a very natural fibre bundle whose fibres are diffeomorphic
to a standard space--like slice $M$ of a space-time $S$. Let $B$ denote 
the space of smooth space--like imbeddings of $M$ in $S$. Let $G$ denote the 
group of self diffeomorphisms $M$. Then $G$ acts as a transformation
group on $B$ by composition on the right. Also $G$ acts on $B \times M$
diagonally. The projection from $B \times M$ to $B$ commutes with these
actions and hence induces a map on the quotient spaces which gives rise
to the fibre bundle
$$ M \rightarrow (B \times M)/G \rightarrow B/G$$
We will call this fibre bundle the {\it space-time fibre bundle}.

Now there is a natural way to put vertical vector fields on the space-time fibre
bundle given an
antisymmetric 2-tensor $F^*$ on $S$.
Now for each $M$ in
$S$ there is a normal unit future pointing time-like vector field $u$.
Then $\hat F^*u$ is tangent to that slice since $\hat F^*u$ is 
orthogonal to $u$. See {\cite P}, {\cite T}. We now think of
$\hat F^*u$ as being tangent to
the fibre $M$ in the fibre bundle. Thus we have our vertical vector field.


Now $ - \overrightarrow{B} = \hat F^*u$ where $*F$ is the Hodge dual of the
electromagnetic tensor $F$. Here the $*$ denotes the Hodge dual which
depends on the choice of orientation made on $S$. Now the vector field
along the fibre on the space-like fibre bundle arising from $*F$ is the
magnetic vector field $\overrightarrow{B}$ on each fibre, 
and it reverses
direction if the orientation is changed. The Index Principle 
asserts that either the index on each fibre is zero, or the vector field
is not proper.

\proclaim{Principle of Invariance of Index}
The index of any ``physical'' proper vertical vector
on the space--time fibre bundle 
is invariant under changes of orientation of
space--time. Otherwise the vector field is not proper and 
it signals either radiation or unrealistic physical
hypotheses.
\endproclaim

A careful discription of the electromagnetic forms and their associated
operators can be found in \cite {G5} and \cite {G6}. A main result in 
\cite {G5} is stated
in terms of the degree of map, in the generality mentioned previously. Thus
it is a new result predicted by the function principle.

\bigskip
\noindent
{\bf 7. \  \ The index as a quantum number}

As we mentioned in section 5, a vector field changing under an otopy
has zeros and defects moving under time and colliding and interacting
with each other. Each defect has an index, and the sum of the indices
coming into a collision is equal to the sum of the indices of the 
outgoing indices. If we could identify what particles correspond to
which defects of the appropriate vector fields, we could derive as a 
consequence the conservation of the particle's indices. 

Now vector fields along the fibre of a fibre bundle are a generalization
of the
concept of otopy. We can think of these vertical vector fields as
representing a collection of possible
otopies under
certain circumstances. We say that a set of defects {\it
potentially interact}
on a fibre if they are contained in a connected component of defects in
the total space. 



Suppose we have a fibre bundle and a proper vector field
defined on
the total space so that every vector is tangent along the fibre. What
kind of
vector fields on a given fibre could be the restriction of the global
vector
field along the fibre? Equation (11) of section 5
 states that there is a transfer on
homology whose trace is the index of the vector field restricted to the
fibre.
Thus the homology of the fibre bundle restricts the possible indices of
vector
fields on the fibre. For example, consider the principal $SU(2)$ --
bundle whose
total space is the 7 dimensional sphere and whose base space is the 4
dimensional sphere. This is the Hopf fibration. The homology only
permits
transfers of trace 0. Hence the index of the restriction of any vector
field
along the fibre restricted to a fibre must be zero.

So if we could find a physical meaning for the Hopf bundle and if there is
a vertical vector field on it which represents all the possible processes,
then any particle with index not zero must be annihilated 
under some process. As an example, consider a classical electron in three
space made compact, hence the three sphere. The defects of the 
electric field are the electron and and the point at infinity. The electron
has index $-1$ and the zero at infinity has index $+1$. If there is a 
universal vertical vector field for the Hopf bundle which restricts to 
this coulomb field in one  fibre, then the electron must interact with
the zero at infinity. Since all particles are said to have antiparticles
which annihilate them, we have inductive evidence that just maybe, principle
bundles over the four dimensional sphere with vertical vectorfields 
may describe particles.

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\vfill\eject
\enddocument

