Functions and the Unity of Mathematics
Daniel Henry Gottlieb
Abstract. We give a definition of Mathematics. In the context of this defi*
*nition
we investigate the question: Why does Mathematics appear to have an underl*
*ying
unity? We suggest in large part it is because of the modern notion of fun*
*ction.
We give a brief history of the concept of functions. Then we examine the p*
*rinciple
that any general concept easily expressed in terms of functions must appea*
*r in many
branches of mathematics. We explain the importance of groups and other cen*
*tral
concepts by the means of this function principle, and we predict that othe*
*r new
concepts will also come to enjoy the same frequency of application. In par*
*ticular we
discuss the de Moivre-Euler formula from the standpoint of the function pr*
*inciple.
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 sho*
*w 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 sho*
*w how it
leads to an invariant of space-time. We use this invariant to describe a n*
*ew feature
of electo-magnetic fields in terms of the index of vector fields. An exami*
*nation of
classical electromagnetic fields invites us to consider the proposition th*
*at 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 field*
*s beyond
being divergence free.
1. The Definition of Mathematics
We take the following definition of Mathematics:
Definition. Mathematics is the study of well-defined concepts.
Well-defined concepts are those with no ambiguities. They are clear and disti*
*nct.
A well-defined concept need not necessarily be defined, however. An important
question is: Is the concept of well-defined itself well-defined? After consider*
*able
thought, I am inclined to say yes. Hence I take the above definition of Mathema*
*tics
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.
____________
Mathematics Subject Classification. 00A30 57R25 83C45.
Key words and phrases. Index, vector fields, magnetic monopoles,.
1
2 D. H. Gottlieb
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 imagina*
*tion.
So if one accepts the definition that Mathematics is the study of the well-defi*
*ned,
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:
Question. Why does Mathematics appear to have an underlying unity?
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 per-
suade us to believe that Mathematics is indeed unified; just as the various phe*
*nom-
ena of Physics seem to be explained by a few fundamental laws. If we can discov*
*er
these principles it would give us great insight into the development of Mathema*
*tics
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 character-
istic, appears over and over in interesting results; or an idea, such as that o*
*f a
group, is involved in many different fields and is used in Science to predict o*
*r make
phenomena precise; or an equation, like De Moivre's formula
ei = cos + i sin
yields numerous interesting relations among important concepts in several field*
*s in
a mechanical way.
Thus the appearance of underlying unity comes from the ubiquity of certain
concepts and objects, such as the numbers ss and e and concepts such as groups
and rings, and invariants such as the Euler characteristic and eigenvalues, whi*
*ch
continually appear in striking relationships and in diverse fields of Mathemati*
*cs
and Physics. We use the word broad to describe these concepts.
Compare broad concepts with deep concepts. The depth of an idea seems to be
a function of time. As our understanding of a field increases, deep concepts be*
*come
elementary concepts, deep theorems are transformed into definitions and so on. *
*But
something broad, like the Euler characteristic, remains broad, or becomes broad*
*er
as time goes on. The relationships a broad concept has with other concepts are
forever.
The Function Principle. Any concept which arises from a simple construction
of functions will appear over and over again throughout Mathematics.
We assert the principle that function is one of the broadest of all mathemati-
cal 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 a*
*ppar-
ent underlying unity of Mathematics at least partly stems from the breadth of t*
*he
concept of function. We will show how the breadth of category and functor and
equivalence and e and ss and de Moivre's formula and groups and rings and Euler
Characteristic all follow from this principle. We will subject this principle t*
*o the
rigorous test of a scientific theory: It must predict new broad concepts. We m*
*ake
such predictions and report on evidence that the predictions are correct.
Unity of Math. 3
2. The History of Functions
On page 13 of Howard Eves mathematical history [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. T*
*here
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 ap*
*pli-
cations. The chord tables of Ptolemy, and his maps of the world, are two exampl*
*es
from ancient times. Oresme around 1360 studied "latitudes of forms" which was a
graphical representation of functions [Bo]. The Renaissance artist-mathematicia*
*ns
and mapmaker-mathematicians introduced projections. Galileo groped at the con-
cept in describing the outcomes of his experiments and his physical laws. Leibn*
*iz
first used the word function in 1694 to describe quantities that varied along a
curve, [Kl]. Controversies arose between Euler and d'Alembert and others about
which functions could count as solutions to the wave equations or be represente*
*d by
a trigonometric series, [Ka]. Slowly piecewise continuous, and then discontinuo*
*us
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 18*
*17 for
functions between the real numbers , but his work was ignored. The ffl-ffi defi*
*nition
is due to Weierstass in 1859. The topological definition was given by Hausdorff*
* in
1914, [Kl].
Thus the concept of a function as a mapping f: X ! 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 t*
*he
measurements of the initial conditions of an experiment and the final results g*
*ives
implicitly a function from the initial states of an experiment to the final out*
*comes;
but historians say that the early physicists and mathematicians never thought t*
*his
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 imp*
*ulse
but the modern notion was adopted only in the Twentieth Century. See [ML] for a
good account of these ideas.
3. The Unity of Mathematics
The careful definition of function is necessary so that the definition of the*
* com-
position of two functions can be defined. Thus f O g is only defined when the t*
*arget
of g is the source of f. This composition is associative: (f O g) O h = f O (g*
* O h)
and f composed with the identity of either the source or the target is f again.*
* We
call a set of functions a category if it is closed under compositions and conta*
*ins 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 *
*char-
acter. Category theory became a subject in its own right, it's practitioners jo*
*yfully
4 D. H. Gottlieb
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 u*
*nder
composition. The functions are abstracted into things called morphisms and com-
position 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 suc*
*h a
homely source as functions closed under composition.
A functor is a function whose source and domain are categories and which pre-
serves 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 bre*
*ath.
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 ex*
*ist
in every category are the identity functions. We can say that f is an identity,*
* or
that f is a retraction by which we mean that there is a function g so that f O *
*g is an
identity, or that f is a cross-section by which we mean that there is a functio*
*n h so
that h O f is an identity, or finally that f is an equivalence by which we mean*
* that
f is both a retraction and a cross-section. In the case of equivalence the func*
*tion
h must equal the function g and it is called the inverse of f and it is unique.
Retraction and cross-section induce a partial ordering of the sources and tar*
*gets
of a category, hereafter called the objects of the category. Equivalences induc*
*e 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 function*
*s.
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 group. The definition of a g*
*roup
in general is just an abstraction, where the functions become undefined elements
and composition is the undefined operation which satisfies the group laws of as*
*so-
ciativity 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
homomorphism. The equivalences in the category of groups and homomorphisms
are called 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 t*
*he
equations of Physics should be invariant under the correct symmetry groups arou*
*nd
1900. Since then groups have played an increasingly important role in Mathemati*
*cs
and in Physics. The increasing appearance of this broad concept must have fed t*
*he
feeling of the underlying unity of Mathematics. Now we see how naturally it fol*
*lows
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) . g(a). Here a runs through all the element*
*s in X
and "." is the group multiplication in G. This multiplication can be easily sho*
*wn
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
Unity of Math. 5
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 a*
*nd
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 tha*
*t 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 ri*
*ngs to
rings; that is, these functions are closed under composition. We call these fun*
*ctions
linear transformations. They contain the ring homomorphisms as a subset. Study
the equivalences of this category. We obtain the concepts of 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,
Question. 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: A ! 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: A ! 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 o: B ! A so that h O o is multiplication by N, and
third that o O h is multiplication by N. So we can give to any homomorphism thr*
*ee
non-negative integers: The exponent, the cross-section degree, and the retract*
*ion
degree. The exponent is the smallest positive integer such that N O h is the tr*
*ivial
homomorphism 0. If there is no such N then the exponent is zero. Similarly the
cross-section degree is the smallest positive N such that there is a o, called *
*a cross-
section transfer, so that h O o is multiplication by N. Finally the retraction *
*degree
is the smallest positive N such that there is a o, called a retraction transfer*
*, so that
o O h is multiplication by N.
In accordance with the Function Principle, we predict that these three num-
bers 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 v*
*alid
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 cat*
*egory
of topological spaces and continuous maps we can say that any continuous map
f: X ! Y has exponent N or cross-section degree N or retraction degree N if the
induced homomorphism f*: H*(X) ! 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 h*
*o-
mology, cross-section transfers already play an important role for fibre bundle*
*s.
There are natural transfers associated with many of the important classical inv*
*ari-
ants such as the Euler characteristic and the index of fixed points and the ind*
*ex
of vector fields,[BG],[G1] and the Lefschetz number and coincidence number and
most recently the intersection number, [GO]. And a predicted surprise relations*
*hip
6 D. H. Gottlieb
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, t*
*he
cross-section degree is precisely the absolute value of the classical Brouwer d*
*egree.
The retraction degree also is the Brouwer degree for closed manifolds if we use
cohomology as our functor instead of homology, [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 val*
*ue of
the variables for which the two expressions equal. We can think of the expressi*
*ons
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 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 Dualit*
*y,
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 t*
*heo-
rem, the Lefschetz theorem holds essentially for the wider class of compact spa*
*ces.
Similar to the coincidence theorem, the Lefschetz theorem has locally defined f*
*ixed
point indices which add up to a globally defined Lefschetz number. This Lefsche*
*tz
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 theo-
rem 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 line*
*ar
operator gives rise to a map on the one dimensional subspaces. The fixed subspa*
*ces
are generated by eigenvectors. Eigenvectors and their associated eigenvalues pl*
*ay
an important role in Mathematics and Physics and are to be found in the most
surprising places.
Consider the category of C1 functions on the Real Line. The derivative is a
function from this category to itself taking any function f into f0. The deriva*
*tive
practically defines the subject of calculus. The fixed points of the derivativ*
*e are
multiples of ex. 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 ez. Now it is possible to relate the funct*
*ion ez
Unity of Math. 7
defined on a complex plane with real valued functions by
e(a+ib)= ea(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 ob-
jects will appear frequently in undisclosed relationships with important concep*
*ts
throughout Mathematics. However the de Moivre equation gives us methods for
discovering the precise forms of some of the relationships it predicts. For exa*
*mple,
the natural question "When does ez restrict to real valued functions?" leads to
the "discovery" of ss. From this we might predict that ss 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 poi*
*nts
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 e*
*ven
broader than those from the fixed point question. But they are very uninteresti*
*ng.
However, if we consider the fixed point question for functions which are equiva-
lent to the identity under some suitable equivalence relation in a suitable cat*
*egory
we may find very broad interesting things. A suitable situation involves the fi*
*xed
points of maps homotopic to the identity in the topological category. For essen*
*tially
compact spaces the 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 "pa*
*rent"
concept. possibly first encountered by Descartes,[St], then used by Euler to st*
*udy
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 con*
*stant
times the Euler characteristic. Poincare gave it its topological invariance by *
*show-
ing 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 iden-
tity, thus explaining, according to the Function Principle, its remarkable hist*
*ory 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 fixe*
*d 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, [M], will play a very active role in Mathematics, and by extensi*
*on
Physics. This equation, which we call the Law of Vector Fields, was discovered *
*in
8 D. H. Gottlieb
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 bound-
ary. 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 @- V denote t*
*he
vector field defined on this open set on the boundary which is given by project*
*ing
V tangent to the boundary. The Euler characteristic of M is denoted by O(M),
and Ind(V ) denotes the index of the vector field. Then the Law of Vector Field*
*s is
Ind(V ) + Ind(@- V ) = O(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.
4. The Law of Vector Fields
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 F*
*ields
will give mechanical methods which will yield precise relationships among broad
concepts in Mathematics and Physics.
Method one.
1.Choose an interesting vector field V and manifold M.
2.Adjust the vector field if need be to eliminate zeros on the boundary.
3.Identify the global and local Ind V .
4.Identify the global and local index Ind(@- V ).
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 different*
*ial
Geometry [G3], the Brouwer fixed point theorem of Topology [G3], and Hadwiger's
formulas of Integral Geometry, [G3], [Had], [Sa]. In category 2 we have the Jor*
*dan
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, [G2].
Of course we have more results in category 3, but it is not so easy to describe*
* them
with a few words. One snappy new result is the following: Consider any straight
line and smooth surface of genus greater than 1 in three dimensional Euclidean
space. Then the line must be contained in a plane which is tangent to the surfa*
*ce,
([G3], theorem 15).
Unity of Math. 9
We will discuss the Gauss-Bonnet theorem since that yields results in all thr*
*ee
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 ss. If we consider exterior angles we see th*
*ey add
up to 2ss . 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 2ss. The normal vector also sweeps out an angle of
2ss. This can be rephrased as The Gauss map has degree equal to one for a simple
closed curve.
Gauss showed for a triangle whose sides areRgeodesics on a surface M in three-
space that the sum of the angles equals ss + M KdM, where K is the Gaussian
curvature of the surface. Bonnet piecedRthese triangles together to prove that *
*for a
closedRsurface M the total curvature M KdM equals 2ssO(M). Hopf proved that
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 ^N: M2n ! S2n times the volume of the unit sphere. Then he proved
2 deg(N^) = O(M2n). (Morris Hirsch in [Hi] gives credit to Kronicker and Van
Dyck for Hopf's result in dimension 2.) According to [Br], Lefschetz then showed
that degN^ = O(M) where N^is the Gauss map for the boundary @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 hi-
jacked. 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 "gen-
eralized Gauss-Bonnet Theorem" no longer contains the case of the plane triangl*
*e.
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 degN^ in terms of the Riemannian curvat*
*ure
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 [Gr], pp. 89-72 or [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: Find a formula for degN^ for a manifold which is immers*
*ed
in one higher dimensional Euclidean space. Method one produces the answer.
The Gauss-Bonnet Theorem. Let f : M ! Rn be a smooth map from a com-
pact Riemannian manifold of dimension n to n-dimensional Euclidean space so that
f near the boundary @M is an immersion. And let @M be orientable. The index of
the gradient of x O f : M ! R, where x is the projection of Rn onto the x-axis,*
* is
equal to the difference between the Euler Characteristic and the degree of the *
*Gauss
map. Thus
Ind(grad(x O f)) = O(M) - deg^N:
10 D. H. Gottlieb
The proof runs as follows following method one. The interesting vector field *
*is
V = grad(xOf). The zeros of @- V are the coincidence points of the Gauss maps ^N
and -V^, where the Gauss map of -V is defined by making -V of unit length and
parallel translating it from @M to the unit sphere. Now coincidence theory tell*
*s us
that the total coincidence number is the difference between the two degrees. Si*
*nce
V has no zeros, its Gauss map has zero degree. Thus Ind@- V = degN^. 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 sid*
*e of
the equation remains the same. Thus O(M) equals the degree of the Gauss-map,
which is the total curvature over the volume of the standard n - 1 sphere. Now
2O(M) = O(@M), so we get Hopf's version of the Gauss-Bonnet theorem.
Note as a by-product we also get Ind(grad(x O f)) = 0 which is a new result
thus falling into category 3. Another consequence of the generalized Gauss-Bonn*
*et
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 ze*
*ro and so
again O(M) = degN^. This is Haefliger's theorem [Hae], a category 2 result. Ple*
*ase
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 G*
*auss-
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 follo*
*wing
vector field. Suppose M is an n-dimensional body in Rn and suppose that f : M !
Rn is a continuous map. Then let the vector field Vf on M be defined by drawing
a vector from m to the point f(m) in Rn. 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 @M at m, then f has a fixed point if O(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 Vf. In fact the Law of Vector Fields applied to Vf is the prop*
*er
generalization of the Brouwer fixed point theorem.
Method two. Make precise the statement that the Law defines the index of vector
fields.
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
otopy, not only allows the vector field to change under time, but also its doma*
*in
of definition changes under time. An otopy is what @- V undergoes when V is
undergoing a homotopy. A proper otopy is an otopy which has a compact set of
zeros.
More precisely: An otopy is a vector field V defined on the closure of an open
set T M x I so that V (m; t) is tangent to the slice M x t. The otopy is proper
if the set of defects D of V is compact and contained in T . The restriction of*
* V to
Unity of Math. 11
M x 0 and M x 1 are said to be properly otopic vector fields. Proper otopy is an
equivalence relation.
Index Classifies Otopy Classes of Vector Fields. Let M be a connected man-
ifold. 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 Ind V = Ind W . In general, V is properly homotopic to W if and only if
Ind(@- V ) = Ind(@- W ) on each connected component of the boundary @M. [GS]
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 classif*
*ied
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 fiel*
*ds 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 boundar*
*y 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 h*
*ave
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 f*
*or
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 preser*
*ved
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 o*
*ut
to infinity", i.e. loses its compactness, [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 wh*
*ich
satisfies a conservation law under collisions. The main ideas behind the constr*
*uction
involve dimension, continuity, and the concept of pointing inside. We suggest t*
*hat
those ideas might lie behind all the conservation laws of collisions in Physics.
5.Vector Bundles and Lorentzian Manifolds
The generality of the concept of otopy suggests we can consider even more gen*
*eral
settings for the Index of vector fields. We can study vector fields along the f*
*ibre
on fibre bundles. An otopy generalizes to a vector field along the fibre V rest*
*ricted
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, beca*
*use of
a transfer associated with the index of V restricted to a fibre.
12 D. H. Gottlieb
We make this precise. A useful generalization of proper otopy is that of a pr*
*oper
vector field along the fibres of a fibre bundle
M ! E ! 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 t*
*he
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 rest*
*ricts
the possible proper otopies. This occurs because of the following transfer theo*
*rem,
[BG].
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 o : B+ ! E+ so that in ordinary homology
p*O o* (cohomology o* O p* ) is multiplication by the Index of V restricted to *
*a fibre,
Ind(V |F ).
A second general setting for otopy and the index are Lorentzian manifolds. He*
*re
a space-like vector field V on an open set is a generalized otopy; and this oto*
*py
gives an equivalence relation between the vector fields which are projections o*
*f V
on space-like slices, [GS].
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 smooth*
*ly
deformed into each other. Suppose D, the set of defects of V , is compact in t*
*he
region of S where the deformation takes place. Then the index of V projected on*
*to
M is equal to the index of V projected onto N.
proof. We can set up a proper otopy between the two projected vector fields giv*
*en
the hypotheses of the theorem. First we consider the mapping F : M xI -! S. On
each slice F is a smooth embedding. On M x {0} F restricts to the identity and *
*on
M x {1}, F restricts to an embedding onto N. The vector field k on M x I tangent
to I will map onto time-like vectors in S, so that the pullback V 0of V onto M *
*x I
will be space-like on M x I. Then we project V 0to a space-like vector field W *
*by
subtracting the k component from V 0. So W is an otopy and on the slice M x {0}
it is exactly the projection of V on M. On the other hand, the restriction of W*
* to
M x {1} need not correspond to the projection of V on N (under the identificati*
*on
of N with M x {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.
6. Physics
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
Unity of Math. 13
components move around and collide with one another. There is a conservation la*
*w,
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 r*
*emain
compact and there are only a finite number of them, [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 con-
struct, which depends only on continuity and dimension and pointing inside, sug-
gests that the topological concept of index may have physical content.
There is another compelling reason to consider the index as a physical quanti*
*ty.
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 prope*
*r space-
like vector field V the index of V projected onto to a space-like slice (by mea*
*ns of a
normal time-like vector field to the slice) is constant. Thus the index is an i*
*nvariant
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 s*
*olar
system, and if we consider this as a space-like slice of space time, then the i*
*ndex of
the gravitational field, which is -1, remains -1 no matter which space-like sli*
*ce is
substituted.
Space-like vector fields arise in several situations. If u is a unit time-lik*
*e vector
field, then the covariant derivative of u is a space-like vector field. Thus th*
*e index
of the covariant derivative is an invariant. If u represents a continuous fami*
*ly
of observers, then the covariant derivative a represents the field of "gravitat*
*ional
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 ^Fdenote 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 associate*
*d to
u which is a space-like vector field given by E = ^Fu. Now consider the 2-form
*F . Here the * denotes the Hodge dual which depends on the choice-of!orientati*
*on
made on S. Now another vector field relative to u is given by - B = (*^F)u. Note
that -!Breverses direction if the orientation is changed.
Now both -!Eand -!Bare space-like vector fields orthogonal to u since F is sk*
*ew
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 IndE and IndB . T*
*hese
integers are independent of which-space-like!slice they are calculated on. Note
however that the sign of IndB 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 nev*
*er zero.
Thus Eu and Bu is homotopic to Ek and Bk. 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 E ; IndB ) depends only on F .
We can tell when the pair (Ind-!E; Ind-!B) is well-defined. If the set of poi*
*nts of S
so that ^Fhas a non-trivial kernel is compact, then any homotopy between differ*
*ent
14 D. H. Gottlieb
future pointing unit vector fields leads to a proper homotopy, because no new z*
*eros
can be created-on!the-points!where ^Fis nonsingular. Now ^Fis nonsingular if and
only if Eu o Bu 6= 0 for any choice of u. If E = B 6= 0, then the kernel of ^F*
*is
space-like, so no new zeros-are!created-at!those points. Let DE be the closure *
*of
the set of points where-Eu!o-Bu!= 0 and Bu > Eu and let DB be the closure of the
set of points where Eu o Bu = 0 and Bu < Eu. If DE and DB are compact we say
that F is proper. We have established the following theorem.
Index for proper 2-forms. Let F be a two-form-such!that-DE!and DB are
compact. Then the pair of integers (Ind E ; IndB ) is well-defined.
If F is an electro-magnetic 2-form, we can calculate (Ind-!E; Ind-!B) where n*
*ow
-!Eand -!Bare the electric and magnetic vector fields respectively. So these t*
*wo
integers are properties of any suitable F and it is reasonable to use them as p*
*art of
the description of electro-magnetism.
It is not difficult to get the index of a vector field using the Law of Vecto*
*r Fields.
The Coulomb electric vector field E of an electron or a proton has index -1 or 1
respectively. G. Samarayanake, in her thesis [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 fi*
*nite
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, Icosahed*
*ron,
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 c*
*lassical
electro-magnetic fields, they should be used to describe physical phenomena.
Here is a true and meaningful statement using index. E fields with nonzero
indices are more common than B fields with nonzero indices. In fact, it is temp*
*ting
to conjecture: There are no B fields with finite nonzero indices.
This conjecture implies that not every solution to Maxwell's equations is phy*
*sical,
since it is possible to find a divergence free vector field which has nonzero f*
*inite
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 aga*
*inst
the existence of pure B fields. The conjecture also implies that magnetic monop*
*oles
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 magne*
*tic
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 existen*
*ce
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: 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 o*
*ri-
entation. If the conjecture were false, there would be an infinite number of ot*
*opy
classes of B fields and they would be sensitive to choice of orientation.
Unity of Math. 15
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
IndE 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 [FLS] has nonzero index. If we try
the trick with surfaces of genus g made out of super conductors, we find that t*
*he 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 surfa*
*ce,
there is always a zero on the surface (unless g = 1). Thus IndB = 1 except when
the surface is a torus, in which case IndB = 0.
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Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
e-mail: gottlieb@math.purdue.edu
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