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\title Functions and the Unity of Mathematics
\endtitle
\rightheadtext{Unity of Math.}
\author  Daniel Henry Gottlieb       
\endauthor
\leftheadtext{D. H. Gottlieb}
         
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\address Department of Mathematics, Purdue University, West Lafayette,  Indiana 47907
\endaddress
\email gottlieb\@math.purdue.edu
\endemail

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\keywords Index, vector fields, magnetic monopoles,
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\abstract 
We give a definition of Mathematics. In the context of this definition we
investigate the question: Why does Mathematics appear to have an underlying
unity? We suggest in large part it is because of the modern notion of function.
We give a brief history of the concept of functions. Then we examine the
principle that any general concept easily expressed in terms of functions must
appear in many branches of mathematics. We explain the importance of groups and
other central concepts by the means of this function principle, and we predict
that other new concepts will also come to enjoy the same frequency of
application. In particular we discuss the de Moivre--Euler formula from the
standpoint of the function principle. We predict another formula will play the
the same kind of role. We call it the Law of Vector Fields. We examine the
successes of Law of Vector Fields. We show how it gives a new perspective and
generalization of the Gauss--Bonnet Theorem. 
We show how it leads to the classification of vector fields up to homotopy.
We show how it leads to an
invariant of space--time. We use this invariant to describe a new feature of
electo--magnetic fields in terms of the index of vector fields. An examination
of classical electromagnetic fields invites us to consider the proposition that
the index of the $B$ field can never be nonzero. A consequence of this 
proposition is that there are no magnetic monopoles; and that there are 
restrictions on pure $B$ fields beyond being divergence free.

\endabstract
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\define\Ind{\operatorname{Ind}}
\define\tr{\operatorname{tr}}
\define\Pf{\operatorname{Pf}}
\define\Proj{\operatorname{Proj}}
\define\Diff{\operatorname{Diff}} 
\define\Dim{\operatorname{Dim}}
\define\grad{\operatorname{grad}}
\define\leaderfill{\leaders\hbox to 1em{\hss.\hss}\hfill}
\define\Chi{\chi}
\define\TI{\operatorname{Ind}}
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\head 1. The Definition of Mathematics
\endhead

We take the following definition of Mathematics:

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

Well--defined concepts are those with no ambiguities. They are clear and
distinct. A well-defined concept need not necessarily be defined, however. An
important question is: Is the concept of well--defined itself well--defined?
After considerable thought, I am inclined to say yes. Hence I take the above
definition of Mathematics as mathematical and not merely philosophical.

This definition seems to be novel. I have not found it written in any book or
paper. It seems obvious to me, and I believe most mathematicians would subscribe
to it. We should treat it seriously, reflect on it, explain it to the world at
large. Then perhaps we can begin to eliminate the destructive misconceptions
that the public at large entertains about mathematics. And we may learn
something.


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 just
such an 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 
apparent 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.

\head 2. The History of Functions
\endhead


On page 13 of Howard Eves mathematical history \cite{E} there is a picture of
the Ishango bone. Carved on the 8,000 year old bone are notches. It is obvious
that this bone was used for counting. Each notch was assigned to some object; a
day, an eclipse, a cow. This bone is the physical representation of some
function. There is evidence of this sort from 50,0000 years ago. Humans were
probably employing functions long before there were words for numbers.

Over the centuries function--like concepts entered into mathematics and its
applications. The chord tables of Ptolemy, and his 
maps of the world, are two examples from
ancient times. Oresme around 1360 studied ``latitudes of forms'' which was a
graphical representation of functions \cite{Bo}. The Renaissance
artist--mathematicians and mapmaker--mathematicians introduced projections.
Galileo groped at the concept in describing the outcomes of his experiments and
his physical laws. Leibniz first used the word function in 1694 to describe
quantities that varied along a curve, \cite{Kl}. Controversies arose between
Euler and d'Alembert and others about which functions could count as solutions
to the wave equations or be represented by a trigonometric series, \cite{Ka}.
Slowly piecewise continuous, and then discontinuous functions were accepted.
Complex functions and various symmetry transformations were accepted by 1870.
Then Cantor introduced the generality of sets and one to one correspondences.
Soon one could think about functions on arbitrary sets. As for continuous
functions, Bolzano essentially gave the correct definition in 1817 for functions
between the real numbers , but his work was ignored. The $\epsilon$--$\delta$
definition is due to Weierstass in 1859. The topological definition was given by
Hausdorff in 1914, \cite{Kl}.


Thus
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 function 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.

\head 3. The Unity of Mathematics
\endhead

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.

\proclaim{Question} What statements can be made about a function $f$
which would make sense in every possible category?
\endproclaim

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 1872 in his Erlanger Programm.
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, 

\proclaim{Question} What statements can be made about a homomorphism $f$ 
which would make sense
in every possible category of abelian groups?
\endproclaim

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{BG},\cite {G1}
and the Lefschetz number and coincidence number and
most recently the intersection number, \cite{GO}.
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}.

A 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 
interesting 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.
possibly first encountered by Descartes,\cite{St}, 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.
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.


\head 4. The Law of Vector Fields
\endhead
\proclaim {Prediction}
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 in Mathematics and Physics.
\endproclaim

\proclaim {Method one}
\endproclaim


\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 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:
{\it 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 $\pi$.
If we consider exterior angles we see they add up to $2 \pi$ . 
In fact, that is true
for any polygon. This in turn is the limiting case of the theorem which states
that the tangent vector on a simply closed curve in a plane sweeps out an angle
of $2 \pi$. The normal vector also sweeps out an angle of $2 \pi$. This can be
rephrased as {\it The Gauss map has degree equal to one for a simple closed
curve}.


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 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 in dimension 2.)
According to \cite{Br}, Lefschetz then showed that $deg \hat N = \chi (M)$ where
$\hat N$ is the Gauss map for the boundary $\partial M$ for any co--dimension
zero manifold $M$ in Euclidean space as a consequence of his fixed point
theorem. The degree of the Hopf map is what Hopf called the Curvatura Integrala,
and is proportional to the integral of the product of the principal curvatures.

At this point something amazing happened. The Gauss--Bonnet theorem was
hijacked. In order to prove an intrinsic version of the Hopf--Gauss--Bonnet
theorem, the theorem was hijacked. The odd dimensional case was excluded. Thus
the ``generalized Gauss--Bonnet Theorem" no longer contains the case of the
plane triangle. Allendorfer and Fenchel both proved the ``generalized
Gauss--Bonnet Theorem" in 1940 and Chern in 1944 gave an intrinsic proof to the
intrinsic "``generalized Gauss--Bonnet Theorem".

From our viewpoint today we can see what happened if we replace the question of
finding a statement and proof the Gauss--Bonnet Theorem for an abstract
$2n$--dimensional manifold $M$ by alternative questions. Allendorfer and
Fenchel solved the question of finding an equation for $deg \hat N$ in terms of
the Riemannian curvature of $M$ embedded in Euclidean space, and Chern answered
the question of finding a formula for the Euler characteristic of $M$ in terms
of Riemannian curvature of $M$.
For a history
of the Gauss-Bonnet theorem see \cite{Gr}, pp. 89-72 or \cite{Sp}, p. 385.

Now the questions above lead 
to a wonderful theorem, but it should not be called
the Gauss--Bonnet Theorem. From the topological point of view, a more reasonable
question is: {\it Find a formula for $deg \hat N$ for a manifold which is
immersed in one higher dimensional Euclidean space. } Method one produces the
answer.

\proclaim{ The Gauss--Bonnet Theorem}
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. And let $\partial M$ be orientable. 
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
$$
\Ind(\grad(x \circ f))= \chi(M)-\deg \hat N.
$$
\endproclaim
 
The proof runs as follows following method one.
The interesting vector field is $V = {\text grad}( x \circ f )$. 
The zeros of
$\partial_{-}V$ are the coincidence points of the Gauss maps $\hat N$ and
$-\hat V$, where the Gauss map of $-V$ is defined by making $-V$ of unit length
and parallel translating it from $\partial M$ to the unit sphere. Now
coincidence theory tells us that the total coincidence number is the difference
between the two degrees. Since $V$ has no zeros, its Gauss map has zero degree.
Thus $\Ind \partial_{-}V = \deg \hat N$. Plug this into the Law of Vector Fields.


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. 
Finally we mention that any 
closed oriented manifold which can be immersed in a co-dimension one Euclidean
space is a boundary, thus the conditions of the above theorem can always be 
obtained.

Just as the Gauss-Bonnet theorem followed 
from considering gradient 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 any $m$ on the
boundary of $M$ and $f(m)$ is not tangent to $\partial M$ at $m$, then $f$
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.

\proclaim{ Method two}
Make precise the statement that
the Law
defines the index of
vector fields.
\endproclaim 

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 proper otopy is an otopy which has a compact set of zeros.

More precisely:  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.

\proclaim {Index Classifies Otopy Classes of Vector Fields}
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
$\TI\ V=\TI\ W$. In general, $V$ is properly homotopic to $W$ if and
only if
$\Ind(\partial_-V) = \Ind(\partial_-W)$ on each connected component of the
boundary
$\partial M$. \cite{GS}
\endproclaim

Over one hundred years ago Poincare introduced the index of a vector
field. He
knew it was invariant under proper homotopy. Now we discover that it is
classified by proper otopy, a concept given to us by the Law of Vector
Fields.
Another victory for the Function Principle. But there is even more, we
can
classify the proper homotopy classes of a vector field. 

We repeat. The proper otopy classes of vector fields on a connected
manifold are
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.


The generality of otopy and the Law of Vector Fields suggests we can
extend the setting for the definition of index.
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, \cite{GS}.

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. 


\head {5.Vector Bundles and Lorentzian Manifolds}
\endhead

The generality of the concept of otopy suggests we can consider even more
general settings for the Index of vector fields.
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. For example the Hopf fibrations of spheres admit only the
index zero. These restrictions arise, as the Function Principle foretells,
because of a transfer associated with the index of $V$ restricted to a fibre.

We make this precise. A useful generalization of proper otopy is that of a
proper 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 (if $B$ is
connected). In this view the fibre bundle is like a collections of possible
proper otopies. We view an otopy as a vector field changing under time. A proper
vector field along the fibres restricts the possible proper otopies. This occurs
because of the following transfer theorem, \cite{BG}.

\proclaim {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,  $\TI (V|F)$.
  \endproclaim

  A second general setting for otopy and the index are Lorentzian manifolds. Here
  a space--like vector field $V$ on an open set is a generalized otopy; and this
  otopy gives an equivalence relation between the vector fields which are
  projections of $V$  on space--like slices, \cite{GS}.


\proclaim{Space--time invariance of Index} 
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 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

\demo{proof}
We can set up a proper otopy between the two
projected vector fields given the hypotheses of the theorem. First we consider
the mapping $F: M \times I \longrightarrow S$. On each slice $F$ is a smooth
embedding. On $M \times \{0\}$ $F$ restricts to the identity and on
$M \times \{1\}$, $F$ restricts to an embedding onto $N$. The vector field $k$
on  $M \times I$ tangent to $I$ will map onto time--like vectors in $S$, so that
the pullback $V'$ of $V$ onto $M \times I$ will be space--like on $M \times I$.
Then we project $V'$ to a space--like vector field $W$ by subtracting the $k$
component from $V'$. So $W$ is an otopy and on the slice $M \times \{0\}$ it is
exactly the projection of $V$ on $M$. On the other hand, the restriction of $W$
to $M \times \{1\}$ need not correspond to the projection of $V$ on $N$ (under
the identification of $N$ with $M \times \{1\}$) because $k$ need not map onto
the normal vectors $n$ of $N$. However $tn +(1-t)k$ is a homotopy between them,
so $W$ on $N$ is homotopic to the projection of $V$ on $N$. No new zeros will be
created by this process since $V$ is space--like, so the otopy is proper. Thus
the projection of $V$ on $M$ is properly homotopic to the  projection of $V$ on
$N$.
\qed
\enddemo



\head 6. Physics
\endhead

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, \cite{GS}.
Thus the index remains conserved unless some component ``radiates out to
infinity.''
This mathematical result is very reminiscent of the action of charged particles
with the index of the defect playing the role of the charged particle.

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 may have physical content.

There is another compelling reason to consider the index as a physical
quantity. It is an invariant of General Relativity. 


We can study space-like vector fields on a Lorentzian space-time. The
otopy generalizes to a space-like vector field restricted to an open set. For a
proper space-like vector field $V$ the index of $V$ projected onto to a
space-like
slice (by means of a normal time-like vector field to the slice) is constant.
Thus the index is an invariant of General Relativity. For example, if we
consider the Newtonian gravitational field for our solar system in a ball which
extends far out beyond the matter of the solar system, and if we consider this
as a space-like slice of space time, then the index of the gravitational field,
which is $-1$, remains $-1$ no matter which space-like slice is substituted.


Space-like vector fields arise in several situations. If $u$ is a unit time-like
vector field, then the covariant derivative of $u$ is a space-like vector field.
Thus the index of the covariant derivative is an invariant. If $u$ represents a
continuous family of observers, then the covariant derivative $a$ represents the
field of ''gravitational acceleration'', and the index is an invariant for all
space-like slices.

We may study 2--forms using the index.
Let $F$ be a 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 there is a vector field associated to $u$ which 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 another vector field
relative to $u$ is given by
$- \overrightarrow{B} = (\hat{*F})u$. Note that $\overrightarrow{B}$
reverses direction if the orientation is changed.

Now both $\overrightarrow{E}$ and $\overrightarrow{B}$ are space-like vector
fields orthogonal to $u$ since $F$ is skew symmetric with respect to the
Lorentzian metric of $S$. Thus for any 2--form $F$ and a choice $u$ of a
time-like vector field, we have two integers $\Ind \overrightarrow{E}$ and $\Ind
\overrightarrow{B}$. These integers are independent of which space-like slice
they are calculated on. Note however that the sign of $\Ind \overrightarrow{B}$
depends on the choice of orientation of $S$ .

Now suppose that $v$ is another unit time-like vector field on $S$. We assume
that $u$ and $v$ are both future pointing. Then the homotopy 
$tu + ( 1-t )v$ passes through future pointing 
time-like vector fields
which are never zero if $u$ and $v$ are never zero. Thus
$\overrightarrow{E_u}$ and $\overrightarrow{B_u}$ is homotopic to
$\overrightarrow{E_k}$ and $\overrightarrow{B_k}$. If this is a proper homotopy,
then the two indices for $F$ and $u$ agree with those for $F$ and $k$. If the
homotopy is proper for all $k$, then the pair of integers
$( \Ind \overrightarrow{E}, \Ind \overrightarrow{B} )$ depends only on
$F$. 

We can tell when the pair 
$( \Ind \overrightarrow{E}, \Ind \overrightarrow{B} )$
is well--defined. If the set of points of $S$ so that $\hat F$ has a non-trivial
kernel is compact, then any homotopy between different future pointing unit
vector fields leads to a proper homotopy, because no new zeros can be created on
the points where $\hat F$ is nonsingular. Now $\hat F$ is nonsingular if and
only if 
$\overrightarrow{E_u} \bullet  \overrightarrow{B_u} \neq 0$ for any choice of 
$u$.  
If $E = B \neq 0$, then the kernel of $\hat F$ is 
nowhere time-like, so no new zeros
are created at those points. Let $D_E$ be the closure of the set of points where
$\overrightarrow{E_u} \bullet \overrightarrow{B_u} = 0$ and $B_u > E_u$ and let
$D_B$ be the closure of the set of points where $\overrightarrow{E_u} \bullet
\overrightarrow{B_u} = 0$ and $B_u < E_u$. If $D_E$ and $D_B$ are compact we say
that $F$ is {\it proper}. We have established the following theorem.

\proclaim {Index for proper 2--forms}
Let $F$ be a two--form such that $D_E$ and $D_B$ are compact. Then the pair of
integers $( \Ind \overrightarrow{E}, \Ind \overrightarrow{B} )$ is
well--defined.
\endproclaim

If $F$ is an electro-magnetic 2--form, we can calculate
$( \Ind \overrightarrow{E}, \Ind \overrightarrow{B} )$ where now
$\overrightarrow{E}$ and $\overrightarrow{B}$ are the electric and magnetic
vector fields respectively. So these two integers are properties of any suitable
$F$ and it is reasonable to use them as part of the description of
electro-magnetism.

It is not difficult to get the index of a vector field using the Law of Vector
Fields. The Coulomb electric
vector field $E$ of an electron or a proton has index $-1$ or $1$ respectively.
G. Samarayanake, in her thesis \cite{Sam}, has a computer program  which
estimates the index of a zero using the Law of
Vector Fields.
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.

Since these indices are easy to find and since they describe some aspect of
classical electro-magnetic fields, they should be used to describe physical
phenomena.

Here is a true and meaningful statement using index. {\it $E$ fields with
nonzero indices
are more common than $B$ fields with nonzero indices.} In fact, it is tempting
to conjecture: {\it There are no $B$ fields with finite nonzero indices.}


This conjecture
implies that not every solution to Maxwell's equations is physical, since it is
possible to find a divergence free vector field which has nonzero finite index.
Thus certain pure $B$ fields should not exist according to our conjecture. Note
the
theory that electro-magnetic fields have point like sources also argues against
the existence of pure $B$ fields. The conjecture also implies that magnetic
monopoles do not exist. 

Now if two magnetic monopoles of index $1$  are placed near
each other then there is a zero of the $B$ field of index $-1$. 
So we should look for $B$ zeros with nonzero index. If this cannot be done, it
argues against
the existence of magnetic monopoles. If it can be done, maybe magnetic monopoles
will be present, since if magnetic monopoles were present, these zeros would be
present. Also the existence or non-existence of these $B$ zeros with nonzero
index would add evidence about whether pure $B$ fields can exist.


Mathematically the conjecture can be restated as follows: {\it Every proper $B$
field
is properly otopic to every other one and this is true regardless of the choice
of orientation of space-time used to represent either of the $B$ fields. This
means there is only one otopy class for $B$ fields and this is the same
whichever choice of orientation.} If the conjecture were false, there would be
an infinite number of otopy classes of $B$ fields and they would be sensitive to
choice of orientation.

Let us see how easy it is to get non-zero indices of $E$ fields. The proton has
index $1$. The electron has index $-1$. Put a surface of genus $g$ made out of a
conductor inside a large imaginary ball. Charge the conductor with a positive
charge. Then $\Ind \overrightarrow{E}$ inside the sphere is equal to $2 - 2g$.
For a negatively charged surface the index equals $2g - 2$.

On the other hand, none of the $B$ fields in \cite{FLS} has nonzero index. If we
try the trick with surfaces of genus $g$ made out of super conductors, we find
that the $B$ field is tangent to the surface. This is opposite to the above case
where the $E$ field is normal to the conducting surface. Since the $B$ field is
tangent to the surface, there is always a zero on the surface (unless $g =1$).
Thus $\Ind B = \infty$ except when the surface is a torus, in which case 
$\Ind B = 0$.



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