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\centerline{\bf Lectures on Vector Fields and the Unity of Mathematics}
\medskip
\centerline{by}
\bigskip
\centerline{\bf Daniel H. Gottlieb}
\bigskip
\centerline{Purdue University}
\centerline{West Lafayette, Indiana 47906}
\bigskip
\noindent
{\bf 1. Introduction}

Prediction:\ \ A theorem of Marston Morse, [M], which I call 
the Law of Vector
Fields, will come to occupy a position in Mathematics rivaling that of
DeMoivre's Formula or the Pythagorian Theorem. 
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.
The purpose of these
lectures is to convince you of that fact and to teach you to become
proficient in using the theorem.


We describe the equation, which we call 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
$$
\Ind(V) + \Ind(\partial_{-}V) = \chi(M)
$$

In section 2 we investigate the {\it unity of Mathematics}. This leads
us to propose the {\it Function Principle} which predicts that concepts
fashioned out of simple concepts involving functions are broad. This
leads us to the prediction that the Law of Vector Fields must be a
central result in mathematics.

In section 3 we propose two strategies to get mathematics from the Law
of Vector Fields and we report on their successes so far.

In section 4 we indicate how the Law of Vector Fields can be used to
define the Index of a vector field and we define the concept of otopy.

In section 5 we discuss vertical vector fields on a fibre bundle. This
concept is the proper generalization of otopy.

In section 6 we give an elementary discussion of vector fields in one
dimension. This section can be read independently of the rest of the
paper and it illustrates with pictures many of the issues arising in the
definition of index and proper vector fields.

In section 7 the main equations related to the index are listed for the
convenience of the reader and many exercises are given to develop the
skill involved in their use.

In section 8 we apply the index to evaluation subgroups to get a
relationship between the Euler--Poincare number and the center of a
group.

Portions of this exposition are taken verbatim from [G--S], [G7], and
[G8].


\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 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 function or mapping grew.
Cantor's set theory gave the notion a good impulse but the modern notion was
adopted only in the Twentieth Century.

The careful definition of function is necessary so that the definition of the
composition of two functions can be defined.
Thus $f o g$ is only defined when the target of $g$ is the source of $f$.
This composition is associative:\ \ $(f o g)oh=fo(goh)$ 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, its 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 o g)= F(f)o  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 $f o g$ is an identity, or that
$f$ is a {\it cross--section} by which we mean that there is a function $h$
so that $h o f$ 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 $f*g(a)=f(a)\cdot g(a)$ where $a$ 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 an abelian group 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 $N o h$ is the trivial homomorphism 0, second that there is a
homomorphism $\tau\colon B\rightarrow A$ so that $h o \tau$ is multiplication
by $N$, or third that $\tau o h$ 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 $N o h$ 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
$h o \tau$ 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 $\tau o h$ 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},
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{G2}.

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 it is the alternating sum of traces of the composition of the umkehr
map $f_!$, which is defined using Poincare Duality, and $g_*$, the homomorphism
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 at first glance they seem to be 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 prove there were only
five platonic solids, 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, according to our principle,
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.
Now we have Morse's formula, which we call the Law of Vector Fields:
$$
\hbox{Ind}(V)+\hbox{Ind}(\partial_{-}V)=\chi (M).
$$
\bigskip\noindent

\noindent Exercises: 

\item{a)}Consider the set of functions $f\colon G @>>> G$ where $G$ is a
group.
What are the relations between composition $f\circ g$ and multiplication
$f*g$?
What additional relation is there if the $f$'s are homomorphisms?
Abstract these two concepts to get two algebraic structures which generalize
rings.
Why haven't we heard of these concepts?

\item{b)}Consider the function $f(z)=k e^z$ for $k$ a constant.
This is a function from ${\Bbb C}$ to ${\Bbb C}$ which is a fixed point 
of the complex derivative.
Show $e^x$ has no fixed point for real $x$.
Show that $e^z$ has two fixed points.
What are they?
What are the 
two non-negative $k$ for which $f(z)=k e^z$ has only one fixed point?

\item{c)}Let $X$ be a set of self maps of $Y$.
Then the evaluation maps $\hat{\omega}\colon X \times Y @>>> Y
\colon \hat{\omega} (x,y)=x(y)$
and $\omega \colon X @>>> Y \colon \omega (x)=x(*)$ 
for fixed base point $*$ should be
very broad concepts.
The induced maps for homotopy group $\omega _* \colon \pi_* (Y) @>>>
\pi_* (Y)$, and homology $\omega _* \colon H_* (X) @>>> H_* (Y)$ and cohomology
$H^* (Y) @>>> H^* (X)$ should appear throughout topology.
Search through the Korean literature to find out who studies questions
related to $\omega$.
Search through the literature and find predicted relationships between
$\omega _*$ and the Euler--Poincare number and the Lefschetz numbers and
transfers.

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

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.

Up to now our predictions had a nebulous quality.
They are not specific, they only assert that concepts following from the
function principle are broad.
They do not tell us how they are broad.
And if a predicted broad concept is unknown, it may argue against the
function principle, or it may be that in time that the concept will make
its importance known.

The Law of Vector Fields is a fair test of the Function Principle.
It involves the index and the Euler--Poincare number, the oldest concepts
of Algebraic Topology tracing back to Poincare.
It was discovered in 1929 by one of the greatest mathematicians of the 20th
century, Marston Morse.
Morse never used the Law of Vector Fields and it fell into obscurity.
It was rediscovered by Charles Pugh [P], and later by the author [G3],[G4].

Given this undistinguished 50 year history and the tremendous advances of
Topology during that period, 
most mathematicians would say that there is
little chance that much new could come out of the Law of Vector Fields.
And some mathematicians firmly believe this is the case, that it is
{\it impossible} to find much of interest.
So if we can show the contrary, that the Law of Vector Fields yields
interesting new results in a mechanical way, then we will have powerful
evidence for the Function Principle.

We propose two methods of drawing Mathematics from the Law.

The following observation will suggest one method:
The Law of Vector Fields is practically an inductive definition of the
index of
a vector field given the Euler characteristic.
The induction is on the dimension of the underlying manifold.
We say that the index of an empty vector field is zero and the index of a vector
field on a finite set of points is the number of points.
Then the knowledge of the index on the boundary gives the index in the one
higher dimensional compact manifold.
And the index can be defined on open sets by setting it equal to the index of a
compact manifold with boundary which contains all the zeros of the vector
field.

The method follows:

\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{G6},\cite{G5}, the Brouwer fixed point
theorem of topology \cite{G6}, and
Hadwiger's formulas of integral geometry \cite{G6}
,\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{G7}.
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{G6}}, 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 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.

Given a map $f$ between Riemannian manifolds $M$ to $N$ and a vector
field
$V$ on $N$, we can define a vector field on $M$, denoted $f^{*}V$, which
is
the pullback of $V$ by $f$.  For a map $f$ from $M$ into the real line
$R$
the pullback vector field of the unit positive pointing $d/dt$ is just
the
gradient of $f$.

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.  Let $V$ be any vector field on
$R^{n}$ such that $V$ has no zeros on the image of the boundary
$f(\partial
M)$.  Consider the pullback vector field $f^{*}V$ on $M$.  The local
index
on $\partial_{-}V$ turns out to be the local coincidence number of two
different Gauss maps.  Substituting into the Law of Vector Fields
results
in a great generalization of the Gauss-Bonnet Theorem,
({\cite{G6}}, theorem 5).
A very special case is the following, which still is a
generalization
of the Gauss-Bonnet Theorem.  We quote the special case only in order to
avoid introducing notation.  The index of the
gradient of
$xof: M\to R$, 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
$$
\Ind(\grad(xof))= \chi(M)-\deg \hat N.
$$
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 $\Ind(\grad(xof)) = 0$ 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 $xof$ 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. 

Just as the Gauss-Bonnet theorem followed from 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 $R^{n}$.  Locally the index of $-V_{f}$  at a zero is a
coincidence index of the Gauss map $\hat N$ and the ``Gauss" map of the
vector field $V_{f}$ given by setting every vector to unit length and
translating to the origin.  If $f$ maps $M$ into itself with no fixed
points on the boundary, then applying the method gives
$$
\Lambda_{f} + \Lambda_{\hat V_f,\hat N} = \chi(M)
$$
This is a category 3 relationship among the main invariants of fixed
point
theory, coincidence theory and the Euler characteristic.  Using the fact
that the coincidence number $\Lambda_{\hat V_{f},\hat N}$ is equal to
$\deg
\hat N - \deg \hat V_{f}$ and the fact from the Gauss-Bonnet theorem
above
that $\deg \hat N= \chi(M)$, we see that the Lefschetz number must be
equal
to $\deg \hat V_{f}$.  If we drop the requirement that $f$ maps $M$ into
itself, we still have $\Ind V_{f} = \deg \hat V_{f}$.  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 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. The Definition of Index for Discontinuous Vector Field}

We assume we are in a smooth manifold $N$.  A vector field is an
assignment of tangent vectors to some, not necessarily all, of the
points
of $N$.  We make no assumptions about continuity.  We will call this $N$
the {\it arena} for our vector fields.  We consider the set of defects
of a
vector field $V$ in $N$, that is the set $D$ which is the closure of the
set of all zeros, discontinuities and undefined points of $V$.  That is
we
consider a {\it defect} to be a point of $N$ at which $V$ is either not
defined, or is discontinuous, or is the zero vector, or which contains
one
of those points in every neighborhood.


We are interested in the connected components of the defects and how
they
change in time. Those connected components of  $D$  which are compact we
will call topological {\it particles}. If we can find an open set about
a particle
which does not intersect any defect not in the particle itself, then we
say the particle is {\it isolated}. If  $C$  is an isolated particle
we
can assign an integer which we call the index of  $C$  in  $V$.
We denote this by
$\TI(C)$.


The key properties of $\TI(C)$ are that it is nontrivial, additive over
particles, easy to calculate
and is conserved under interactions with proper components as $V$ varies
under
time.  For example, let $V$ be the electric vector field generated by
one
electron in $R^3$.  Then the position of the electron $e$ is the only
defect
and $\TI(e)=-1$.  Now if $V$ changes under time in such a way that there
are only a finite number of particles at each time, all contained
in some large fixed sphere, then the sum of the indices of the particles
at
each time $t$ is equal to $-1$.  Thus the electron vector field can
change to
the proton vector field only if the set of defects changing under time
is unbounded, since the proton has index $+1$ which is different from
the index of the electron. In this case we will say that the
transformation of the electron to the proton involves `` topological
radiation".


Vector fields varying under time, and defect components
interacting with each other, can be made precise by introducing the
concept
of {\it otopy}, which is a generalization of the concept of homotopy.
An
{\it otopy} is a vector field on $N\times I$ so that each vector is
tangent
to a slice $N\times t$.  Thus an otopy is a vector field $W$ on $N\times
I$ so
that $W(n,t)$ is tangent to $N\times t$.  We say that $V_0$ is otopic to
$V_1$
if
$V_0(n)=W(n,0)$ and $V_1(n)=W(n,1)$. We say that a set of components
$C_i$
of defects on $V_0$ {\it transforms} into a set of components of
defects $D_j$
of
$V_1$ if there is a connected component $T$ of the defects of $W$ so
that
$T\cap (N\times 0)=\cup C_i$ and $T\cap(N\times 1)=\cup D_j$.  If $T$ is
a
compact connected component of defects of $W$, which transforms a set of
isolated particles $C_i$ into isolated particles  $D_j$,  then we say
there is no topological {\it radiation} and
$$
\sum \TI(C_i)=\sum \TI(D_j).
$$
If $T$ is not compact, we say there is topological radiation.


We define $\TI(C)$ as follows.  Since $C$ is an particle, there is an
open set
$U$ containing $C$ so that there are no
defects in the closure of $U$ except for $C$.  We can define an index
for
any vector field defined on the closure of an open set so that the set
of
defects is compact and there is no defect on the frontier of the open
set.
We say such a vector field is {\it proper} with {\it domain} the open
set.
In the case at hand, $V$ restricted to $\overline U$ is proper with domain
$U$.
Hence we can define $\TI(V|U)$.  We set $\TI(C)=\TI(V|U)$.


Next we define $\TI(V)$ with domain $U$ to be equal to the index of
$V|M$ where
$M\subset U$ is a smooth compact manifold with boundary containing the
defects
of $V$ in its interior.  We can find such an $M$ since the defects are a
compact set in $U$.


We call a vector field $V$ defined on a compact manifold $M$ {\it
proper with respect to the boundary}
if there are no defects on the boundary.  Consider the open set of the
boundary where $V$ points inside.  We denote that set by $\partial_-M$.
We define the vector field $\partial_-V$ with domain $\partial_-M$ in
the
arena $\partial M$ by letting $\partial_-V$ be the end product of first
restricting $V$ to the boundary and then projecting each vector so that
it
is tangent to $\partial M$ which results in a vector field $\partial V$
tangent
to $\partial M$, and then finally restricting $\partial V$ to
$\partial_-M$
to get $\partial_-V$. Then we define $\TI(V)$ by the equation
$$
\TI(V)=\chi(M)-\TI(\partial_-V)\tag*
$$
where $\chi(M)$ denotes the Euler-Poincare number of $M$.  We know that
$\partial_-V$ is a proper vector field with domain $\partial_-M$ since
the
set of defects is compact unless there is a defect at the the frontier
of
$\partial_-M$.  If there were such a defect, it would be a zero of $V$
tangent
to $\partial M$ and hence a zero of $V$ on the boundary, so $V$ would
not have
been proper.


Now $\partial_-V$ is a proper vector field with domain the open set
$\partial_-M$ which is one dimension lower than $M$. Then $\TI (
\partial_-V)$
is defined in turn by finding a compact manifold containing the defects
of
$\partial_-V$ and using equation (*). We continue this process until
either $\partial_-M$ is a zero dimensional manifold where every point is
a
defect and so $\TI(\partial_-V)$ is simply the number of points, or where
$\partial_-M$ empty in which case $\TI(\partial_-V)=0$.


To summarize, we define the index of a proper vector field $V$ with
domain $U$
assuming
that the index for vector fields is already defined for compact
manifolds
with boundary.  Then the index of $V$ is defined to be the index of $V$
restricted to a compact smooth manifold with boundary of codimension
zero
containing
all the defects of $V$ in $U$.  In [G--S] it is shown
that this definition is well-defined, that
is it does not depend on the chosen manifold with boundary, by showing
that a vector field with no defects defined on a compact manifold with
boundary has index zero.
\medskip
  
{\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 a 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 of $S$ 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 $F$
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.

In each of those three cases we can only define the index for certain
nice
situations:\ \ The component of defects must be compact and isolated
from
the other components; the boundary $\partial M$ of the compact manifold
$M$
must contain no defects; the open set $U$ has no defects on its frontier
and the
set of defects is compact in $U$.
We use the term {\it proper vector field with respect to the boundary}
in the compact manifold case and
{\it proper vector field with respect to $U$} in
the open set case to mean vector fields satisfying the appropriate
conditions
so that the appropriate index can be defined.

Now in each of these situations there is a notion of {\it proper otopy}
under which the three forms of the index are preserved.
These naturally generalize to the concepts of Proper Vertical Vector
Fields.

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?

We can phrase the question, ``Are vector fields $V_0$ and $V_1$
homotopic?''
in terms of the extension question.
Let $V_1$ and $V_2$ be vector fields on a manifold $F$.
Let $\pi \colon S^1 \times F @>p\times 1 >> S^1 \times F
@>Pr>> S^1$ be the composition where $p\colon S^{1} @>>> S^1$
is the double covering.
So $\pi \colon S^1 \times F @>>> S^1$ is a fibre bundle with fibre
$F\cup F$, two disjoint copies of $F$.
Consider the vector field on $F\cup F$ given by $V_0$ on one copy of $F$
and
$V_1$ on the second copy of $F$.
Then any homotopy $V_t$ gives rise to a vertical vector $W$ which
extends
$V_0 \cup V_1$, and conversely any vertical vector field gives rise to a
homotopy from $V_0$ to $V_1$.

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.

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 $\Ind (V|F)$.}



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



Vertical vector fields can be constructed from equivariant vector fields
on a manifold with a group action.  Thus if $V$ is a $G$-vector field on a
manifold $M$,
then the trace of the  action $(G,M)$, denoted by tr$(G,M)$ and defined in
[G2], must divide $\Ind V$.  In symbols,
$$
\text{tr} (G,M)|\Ind (V).
$$




\bigskip
\noindent
{\bf 6. One Dimensional Examples of Vector Fields}

An 0--dimensional manifold consists of a discrete set of points.
A vector field on this manifold consists of zero vectors.
To each zero we assign the index 1.

Now consider the real line ${\Bbb R}$.
This is a 1--dimensional manifold.
We will look at vector fields on ${\Bbb R}$ carefully even though the index
is fairly easy.
There are many issues which can be explained clearly in one dimension.

Let $V$ be a continuous vector field defined on all of ${\Bbb R}$.
We would draw a picture as follows.

\ %
\vskip 1truein

\leftline{\bf Figure 0 }

\vskip 1truein

Now this is a cluttered picture, so let us adopt the convention that a vector
pointing upward represents a vector of the same
length pointing to the right and a vector pointing downwards represents
a vector of the same length pointing to the left.
Thus if $V$ is the vector field so that $V(x)$ is a one dimensional vector
field $x^2-1$ based at the point $x$ on ${\Bbb R}$, then the pictorial 
representation described above is

\ % 
\vskip 1truein

\leftline{\bf Figure 1}

\vskip 1truein
Now notice that there is a zero of $V$ at $x=-1$ and $x=1$.
We assign the index $-1$ to the zero at $x=-1$ and the index $+1$ to the
zero at $x=1$.
We say the index of $V$ is $-1+1=0$.

The rule for assigning indices is the following.
Find a small interval about the zero containing no other zeros.
If both vectors on the boundary of the interval are pointing inside, then
assign the index $-1$, if both vectors are pointing outside, assign the
value 1, and if one vector is pointing inside and one is pointing outside
assign the index 0.

\ %
\vskip 1truein

\leftline{\bf Figure 2}

\vskip 1truein

\noindent Exercise: Relate this definition to the definition in section 4.

Note that the definition of index based on pointing inside or outside is
independent of orientation.
Now the reader will be tempted to define the index by looking at the 
graph and
noticing that it is $-1$ at a decreasing zero and $+1$ at an increasing zero
and 0 at a tangent zero.
But suppose we change our conventions, quite literally changing the
orientation of ${\Bbb R}$, by representing left pointing vectors by upward
pointing vector and right pointing by downward pointing.
In this case, figure 1 becomes

\ %
\vskip 1truein

\leftline{\bf Figure 3}

\vskip 1truein

Now the official definition still gives the zero at $x=-1$ the index $-1$
and the zero at $x=1$ the index $+1$, whereas the tempted reader's
definition changes the sign of the indices.

There is a lesson to be learned here. The definition which leads to the
more easily computable concept is not necessarily the best definition.
The fact that the official definition is invariant under choices of
orientation will more than make up for the fact that it is harder for
humans to compute with. There is another thing we should mention here.
Why couldn't we have chosen the index to be $+1$ when the vectors point
inside? Keep this question in mind when you look at the properties of
the index in section 7.

Now consider what happens if $V$ changes in time.
This is called a {\it homotopy} and is frequently denoted $V_t$.
Consider the homotopy $V_t(x)=x^2+t$ where $t$ runs from $-1$ to 1.
Then $V_{-1}(x)$ is our example, and $V_0(x)=x^2$ has only one zero at the
origin of index 0, and $V_1(x)=x^2+1$ has no zeros whatsoever.

\ %
\vskip 1truein

\leftline{\bf Figure 4}

\vskip 1truein

As $t$ runs from $-1$ to 1, the zeros at $x=-1$ and $x=1$ move toward each
and finally collide in a zero of index zero at the origin at $t=0$.
The zero then disappears for $0 < t$.
It is as if an electron of charge $-1$ and a positron of charge $+1$
collide and annihilate each other.

\noindent We can represent this movement of zeros by another type of
diagram

\ %
\vskip 1truein


\leftline{\bf Figure 5}

\vskip 1truein

The vertical lines represent the vector fields $V_{-1},V_0,V_1$ on
${\Bbb R}$.
If we graph the paths of the zeros we get

{\vbox{
\vskip 1truein

\leftline{\bf Figure 6}

\vskip 1truein
}}

We say the zeros at $-1$ and 1 {\it interact} under the homotopy $V_t$ since the
set of zeros in $[-1,1]\times {\Bbb R}$ is connected.

Now consider the projection $[-1,1]\times {\Bbb R} @>p>> [-1,1]$.
Here $p$ is a trivial example of a fibre bundle with fibre ${\Bbb R}$.
The homotopy $V_t$ can be thought of as a vector field on the total space
$E=[-1,1]\times {\Bbb R}$ which is tangent to the fibres $F={\Bbb R}$.
Such a vector field will be called a {\it vertical vector field} on $E$.
Vertical vector fields are a generalization of homotopy.


Consider again the
example $V(x)=x^2-1$ on ${\Bbb R}$.
If $I=[a,b]$ is a closed interval, we define the $\text{Ind}_I V$ on $[a,b]$ by
the same rules we used to define the index of a zero.
Namely:\ \ $\text{Ind}_I V=-1$ if the vectors at $a$ and $b$ are pointing 
outward, $\text{Ind}_I V=-1$ if the vectors are pointing inward and 
$\text{Ind}_I V=0$ if one
vector is pointing outward and one vector is pointing inward.
If there is a zero at either $a$ or $b$, then $\text{Ind}_I V$ is not defined.

\noindent
Exercise:\ \ What is $\text{Ind}_I V$ when $I=[-2,2]\text{?},\ [-2,0]\text{?},\
[-2,-1]\text{?}$

From the exercise you may note that $\text{Ind}_I V$ is the sum of the 
indices of the zeros contained in $I$.
This is true in general.
This sum equation forces the conservation of indices under a homotopy.
For example, notice that $\text{Ind}_I V_t=0$ for all $t\in [-1,1]$.
Hence the sum of the indices of the zeros of $V_t$ inside $[-2,2]$ must
equal zero.
But $\text{Ind}_I V_t=-1$ when $I=[-2,0]$ until $t=0$ when it is undefined, and
then for $t > 0$ we see that $\text{Ind}_I V_t=0$.
So the change of the index occurred when a zero migrated across the
boundary of $I$.

We say a homotopy $V_t$ on a closed interval is {\it proper} if 
$\text{Ind}_I V_t$ is always defined for each $t$.
That is, no zeros appear on the boundary of $I$.
Thus $\text{Ind}_I V_t$ is constant during a proper homotopy.

\noindent
Exercise:\ \ Show that if $\text{Ind}_I V_0=\text{Ind}_I V_1 \not= 0$,
then $V_0$ is properly homotopic, with respect to the boundary, to $V_1$.
Show this is not the case for $\text{Ind}_I V_0=\text{Ind}_I V_1=0$.

Now suppose we have a vector field on ${\Bbb R}$ whose zeros are not isolated.
An example is given in Figure 7.

\ %
\vskip 1truein

\leftline{\bf Figure 7}

\vskip 1truein

The indices of the connected components are written under the component.
The rule is the following.
Find a closed interval $I$ around a connected component of the zeros so that
there are no other zeros in $I$ and no zeros on the boundary of $I$.
Then define the index of the zero component to be $\text{Ind}_I V$.

Everything above applies to these connected sets of zeros, except for two
difficulties.
There are two possible ways for $\text{Ind}(C)$, where $C$ is a connected 
set of zeros, not to be defined.
First and by far the most important, $\text{Ind}(C)$ is not defined if $C$ 
is not compact.
For then it is impossible to find a compact interval $I$ which contains $C$.
The second way is if $C$ is not isolated.
That is, it is impossible to find a closed interval $I$ which contains $C$
and no other zero.

The following is an example of such a $C$.
Let $V(x)=x\sin {1\over x}$ for $x\not= 0$ and $V(0)=0$.

\ %
\vskip 1truein

\leftline{\bf Figure 8}

\vskip 1truein

Then $\text{Ind}(0)$ is not defined.
Prove that $\text{Ind}_I V=0$ if $I=[-2,2]$.
So although $\text{Ind} (0)$ is not defined and there are an infinite number of
zeros in $I$, any small change of $V$ which results in a finite number of
zeros will have the sum of their indices equal to 0.

Now the reader might think that the index of a zero is an intersection
number.
In fact, this is a fruitful way to look at Index for continuous vector
fields everywhere defined.
But we can make the same definitions for vector fields which are not defined
everywhere, or are discontinuous.
We say that a {\it defect} of a vector field $V$ is a point where the vector
field has a zero, or is not defined, or is discontinuous, or is a limit
point of those kinds of points.
Thus the set of defects $D$ is a closed set.

A one--dimensional example with the indices written under the defects is
shown in the next figure.

\ %
\vskip 1truein

\leftline{\bf Figure 9}

\vskip 1truein

The indices of a compact connected component $C$ of $D$ are defined as before.
Find a closed interval $I$ containing $C$ so that there are no defects on
the boundary and no other defects of $V$ inside $I$.
Then the index $\text{Ind}(C)=\text{Ind}_I V$.

Consider the vector field, for example, $V(x)=\ln (x)$.
This is represented in the following figure.

\ %
\vskip 1truein

\leftline{\bf Figure 10}

\vskip 1truein

Then $V(x)$ is not defined for $x\leq 0$.
Also there is a zero of index 1 at $x=1$.
So there are two connected components of the defect set $D$.
The component $(-\infty,1]$ has undefined index since it is not compact and
the zero at $\{1\}$ has index 1.

Consider the vector homotopy $V_t(x)=\ln x + \ln (t-x)$.
Then for any $t$ the components of $D_t$ are $(-\infty,0] \cup [t,\infty)
\cup \left\{{t-\sqrt{t^2-4}\over 2}\right\} \cup \left\{{t+\sqrt{t^2-4}\over 2}
\right\}$.
The index is undefined for the first two components, it is $+1$ for
$\left\{{t-\sqrt{t^2-4}\over 2}\right\}$ and $-1$ for 
$\left\{{t+\sqrt{t^2-4}\over 2}\right\}$.
At $t=2$ the two zeros coalesce to a zero 
at $x=1$ of index 0.
For $t < 2$ there are only the two rays.
They coalesce at $t=0$.

The following picture, drawn with the same conventions as in Figure 6, shows
how the zeros interact.

{\vbox{
\vskip 1truein

\leftline{\bf Figure 11}

\vskip 1truein
}}

Thus as $t$ runs from 0 to infinity we see that the undefined component
which is all of ${\Bbb R}$ at $t=0$ splits into two undefined components.
At $t=2$, a zero at $x=1$ appears with index zero and then splits into a zero
of index $-1$, which travels in the positive direction, and a zero of index
$+1$ which travels in the negative direction.
In this homotopy at $t=4$ there are 4--path components.
The two zeros interact and the two undefined components interact, but the
zeros do not interact with undefined components.

Note that the homotopy is not a homotopy since the vector field is not defined
everywhere.
We will call it an {\it otopy}.
Similarly as before, if we regard Figure 11 as a fibre bundle, then the
otopy is a vertical vector field which is not necessarily defined
everywhere on the fibre.


Note the vertical vector field described in Figure 11 is not proper 
since the set of defects is not compact.
Let $U$ be the open set formed by the wedge $\{(t,x)|0 < x < t\}$.
Then $V$ with domain $U$ is not a proper vertical vector field since $V$
cannot extend continuously to the point, of the wedge at $t=0$.
If $U^\prime=U-(0,{1\over 10}]\times {\Bbb R}$, then $V$ with domain 
$U^\prime$ is a proper vertical vector field with respect to
$U^\prime$ .

Consider the open set $U^{\prime\prime}=U-(0,2]\times {\Bbb R}$. Now
$V$ with respect to $U^{\prime\prime}$ is not proper since the defects inside
$U^{\prime\prime}$ are not compact over $[2,3]$.
Note that for the subfibre bundle $(2,\infty)\times {\Bbb R} @>>> (2,\infty)$
that $V$ with respect to $U^{\prime\prime}$ is proper.
For the fibre bundle $[2,\infty)\times {\Bbb R} @>>> [2,\infty)$,
we see that $V$ with
respect to $U^{\prime\prime}$ is not proper, but $V$ with respect to
$U^{\prime\prime\prime}=U-(0,2]\times {\Bbb R}$ is proper.

A useful example to keep in mind of a vertical vector field which is not
proper is the following.  Let $E = I\times{\Bbb R}$ and $B=I$ with $p:
I\times {\Bbb R}\to {I}$ the projection, so $F= {\Bbb R}$.  Let $V$
be the constant vector field of unit length pointing in the positive
direction on ${\Bbb R}$.  Let $V(t,s)= t\times V$ if $0\leq t\leq
\frac{1}{2}$ and $V(t,s)= t\times (-V)$ if $\frac{1}{2} < t\leq 1$.
Then the set of defects $D$ is $\{(\frac{1}{2},s) | \text { all } s\in {\Bbb
R}\}$.
Note that $V$ has no defects when restricted to the fibre over $t=
\frac{1}{2}$.

\ %
\vskip 1truein

{\bf figure 12}

\vskip 1truein

Now let us consider the non-trivial 1--dimensional bundle
$I \rightarrow M \rightarrow S^1$ where $M$ is the Mobius Band. As an
exercise, show that there is a proper vertical vector field with respect to
the boundary which restricts to the fibre with index $+1$ and index $-1$, but 
there is none which restricts to index $0$. On the other hand, show that there
is a vertical vector field proper with respect to some open set which does 
restrict to a fibre with index $0$.

\bigskip

\noindent {\bf 7. 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 $\P_-M$, 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${}_3$-G${}_6$}, \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 zeros $Z$ 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)}
$\TI(-V)=(-1)^{\text{dim } M}(V)$
\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
$Z$ is the union of a finite number of compact connected components.
\bigskip
For certain vector fields the index is equal to classical invariants. Suppose
$f:{\Bbb R}^n\to{\Bbb R}^n$.  Let $M$ be a compact $n$ submanifold.  Define
$V^f$ by $V^f(m)=f(m)$.  If $f:\partial M\to{\Bbb R}^n-\vec 0$, then
\medskip

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

Suppose $f:U\to{\Bbb R}^n$ where $U$ is an open set of ${\Bbb R}^n$.  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 $f:M\to{\Bbb R}$ and $V=\frac d{dt}$, we have $f^*V=
\text{ gradient } f$.

Now suppose that $f:M^n\to {\Bbb R}^n$ where $M^n$ is compact and $V$ is a
vector field on ${\Bbb R}^n$ 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 ${\Bbb R}^n$ 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 $f:\partial M\to{\Bbb R}^n$.  
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.

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}$. Then there is a transfer so that on ordinary 
homology we have
$$
p_{*}\circ\tau^{}_{V_{*}}={\text Ind}(V|F) \tag11
$$

\bigskip

\noindent
Exercise: The Euler--Poincare number.

\noindent
Using the property that the Euler--Poincare number is an invariant of
homotopy type and the above properties of the index, prove the following
useful properties of the Euler--Poincare number.

\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 even
dimensional.

\noindent
Exercises:\ \ Calculating the Index

\noindent
Answer the three questions below for each problem and the given $N$, $V$,
$M$, $U$.

\noindent
Let $V$ be a vector field on $N$.

\item{a)}What are the defects of $V$ and what are their indices?

\item{b)}What is $\text{Ind}_U V$ for the open set $U$?

\item{c)}What is $\text{Ind}_M V$ for the compact manifold $M$?

\item{1.}Let $V=\nabla f,\ N={\Bbb R}^3$ and $f\colon {\Bbb R}^3 @>>>
{\Bbb R}$ where $f(x,y,z)=x^2+2 y^2+z^2$.
Suppose $U={\Bbb R}^3$ and $M=f^{-1} ([0,1])$

\item{2.}Let $V(m)=m\sin {1\over m}$ if $m\not= 0$ and 0 if $m=0$.
Then $V$ is a vector field on $N={\Bbb R}^1$,  $U=(-1,1)$,
$M=[-2 \pi, 2\pi]$.

\item{3.}Let $N={\Bbb R}^2$ and let $V$ be the velocity vector field given
by rotating ${\Bbb R}^2$ around origin.
$U={\Bbb R}^3,\ M=D^2$.

\item{4.}Let $N={\Bbb R}^3$, let $V$ be the velocity vector field given by
rotating about the $z$--axis.
\item{}$U={\Bbb R}^3,\ M=D^3$.

\item{5.}$\partial V$ from problem 4 tangent to unit sphere $S^2=N$.
\item{}$U=S^2,\ M$ = Northern hemisphere.

\item{6.}$N={\Bbb R}^3,\ V(m)=(-x^3,\ \sin y,\ z)$.
\item{}$U={\Bbb R}^3,\ M=D^3$.

\item{7.}$N={\Bbb R}^3,\ V$ = electrostatic vector field of two electrons,
i.e.\ $V(m)={\tilde{r}_1-\tilde{m}\over \|\tilde{r}_1-\tilde{m}\|^3}+
{\tilde{r}_2-\tilde{m}\over \|\tilde{r}_2-\tilde{m}\|^3}$.
\item{}$U={\Bbb R}^3,\ M$ = Ball of radius ${\|\tilde{r}_1-\tilde{r}_2\|
\over 2}$ with center at $\tilde{r}_1$.

\item{8.}Same as problem 7
except $V(m)={\tilde{r}_1-\tilde{m}\over \|\tilde{r}_1-\tilde{m}\|^3}-
{\tilde{r}_2-\tilde{m}\over \|\tilde{r}_2-\tilde{m}\|^3}$.
    
\noindent {\bf 9.  The Evaluation Subgroup}

\proclaim {Theorem}
Let $\pi$ be a group with a finite free resolution. Then if the Euler-Poincare
number of the group is non-zero, the center of the group must be trivial.
\endproclaim

This theorem, named Gottlieb's 
Theorem by Stallings in \cite{St}, has been considerably
generalized, first by Rossett and then by Cheeger and Gromov.  
Define $G(X)={\text {Image }} \omega_* $ where $\omega_*$ is induced
by the evaluation
map on the fundamental group.

The key lemma in the original proof is:  If $X$ is a compact
$CW$-complex
so that $\chi(X)\not=0$, then $G(X)$ is trivial.  Then if $X$ is a
$K(\pi,1)$ we know that $G(X)=$ center of $\pi$.  Thus we must show
that if $F:X\times S^1\to X$ so that $F|X\times *$ is the identity and
if
$\chi(X)\not=0$, then $F|x\times S^1$ is homotopically trivial.  The
original
proof used Nielsen-Wecken fixed point classes \cite{G${}1$}.  These
were
transformed by Stallings \cite{St} into an algebraic setting which is of
importance algebraically.  The present proof is more elementary and does
not need Nielsen-Wecken fixed point theory.

Suppose $M$ is a regular neighborhood of an embedding of $X$ in some
${\Bbb R}^n$.  Then we have a map $F:M\times I\to M$ so that $F(m,0)=m$
and $F(m,1)=m$ for all $m\in M$.  We may adjust $F$ so that
$F(m,t)\not= m$ for all $m\in \partial M$ and all $t$, and so that $F_0$
and $F_1$ are the identity outside a small collar neighborhood of the
boundary. Define the vector field $T$ so that
$T(m,t)=(\vec m-\displaystyle{\overrightarrow{F(m,t)}},t)$. 
Let $Z$ be the zeros of $T$ on $M\times I$.
Now
$Z$ is compact and $Z\cap(\partial M\times I)$ is empty.  Let $U$ be the
open set of $M\times I$ so that $\Vert T(m,t)\Vert<\epsilon$ where
$\epsilon$ is so small that two paths $\alpha(t)$ and $\beta(t)$ on $M$
are
homotopic if the distance between $\alpha(t)$ and $\beta(t)$ is always
less
than $\epsilon$. 
Such an $\epsilon$ exists since $M$ is compact.
Let $W$ be a path component of $U$ containing $M\times
0$.
Then $T|W$ is a proper otopy from $T_0$ to $T_1$.  Now $\TI\ T_0=
\TI\ T_1$ and $\TI\ T_0=\chi(M)$ and $\TI\ T_1=\chi(M)$.  So the path
connected $W$ contains $M\times 0$ and $M\times 1$.

We may find a path $\gamma:I\to U$ so that $\gamma(0)=*\times 0$ and
$\gamma(1)=*\times 1$.  We can homotopy $F:M\times S^1\to M$ to a $G$
such that
$G(\gamma(t),t)=\gamma(t)$.  Then
$\gamma\sim\alpha\cdot\gamma$ where $\alpha(t)=G(*,t)$.  Hence
$\alpha\sim 0$,
which was to be shown.
\bigskip
\noindent
Exercises: Prove the following results using properties of the index.
See [G7] for solutions.

{\item{1.}} Intermediate Value Theorem

{\item{2.}} Fundamental Theorem of Algebra

{\item{3.}} Jordan Curve Theorem

{\item{4.}} Any straight line is contained in a plane tangent at some
{\item{}}  point to a given surface of genus greater than 1.

{\item{5.}} Show that every vertical vector field proper with respect to
{\item{}}  some open set on the Hopf fibration
 $S^3\overset{i}\to\longrightarrow S^7\overset{p}\to\longrightarrow S^4$
{\item{}} has index equal to zero when restricted to a fibre.

\vfill\eject
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\vfill\eject
\enddocument    
