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\topmatter
\title A de Moivre Like Formula for Fixed Point Theory \endtitle
\author Daniel H. Gottlieb \endauthor
\endtopmatter
\document
\heading
1. Introduction.\\
\endheading

Suppose $V$ is a vectorfield on a compact manifold $M$ with or without
boundary $\partial M$.  There is a formula, originally due to Marston
Morse [Mo] in 1929, which is not as widely known as it should be. It was
rediscovered in 1968 by C. Pugh [P] and by the author [G] in 1985.
The formula related the index of the vectorfield $V$ with the index of
a vectorfield on part of the boundary.  This formula, like de Moivre's
$e^{i\theta} = \cos\theta + i \sin\theta$, contains a large amount of
information.

So let $V$ be a continuous vectorfield on $M$ with no zeros on $\partial M$.
Then
$$
Ind(V) + Ind(\partial_{-}V) = \chi(M). \tag1
$$

\noindent Here $\chi(M)$ denotes the Euler-Poincare number of $M$, and $\partial_{-}V$
is a vectorfield defined on part of $\partial M$ as follows.  Let $\partial_{-} M$ be the subset of $\partial M$ containing all $m$ so that $V(m)$ points inward.
Then $\partial_{-}V$ is the vectorfield on $\partial_{-}M$ given by first 
restricting $V$ to $\partial M$, and then projecting $V\mid\partial M$, using
the outward pointing vectorfield $N$, onto its component field tangent to $\partial M$.  This is denoted $\partial V.$  Then $\partial_{-}V = \partial V
\mid\partial_{-}M$.

Observe that the vectorfield $\partial_{-}V$ is defined on a space of one
dimension lower than $V$.  Thus the formula may be thought of as an inductive
definition of index.  For example, if $M =I$ and $V$ is a vectorfield on $I$,
then $Ind\, V = 1 - Ind\,\partial_{-}V$.  Now $\partial_{-}M$ consists of two points, one point, or no points.  Correspondingly, we let the corresponding
$Ind\,(\partial_{-}V)$ equal 2, 1, or 0 respectively.  Then the respective
$Ind\,V$ equal -1, 0, or 1.  In the case $M$ is a circle we have $Ind\,(\partial_{-}V) = 0$ since $\partial S^{1} = \partial_{-}S^{1} =  \emptyset$, so $Ind\,V = 0$.
Thus we have defined the index for all connected 1-manifolds, and since the
sum of the indices over disjoint manifolds is the index of the union of the
vectorfields, we can calculate the index for one dimension.  Similarly, the
information in one dimension permits the calculation in two dimensions.  
There is a technical difficulty in that $\partial_{-}M$ is not a compact
manifold, but the formula (1) is morally a definition.

Thus one should expect that (1) should imply all formulas involving the
index of a vectorfield.  Also a systematic searching of vectorfield 
phenomena might yield new results when viewed in conjunction with (1).
In the rest of this report we will massage (1) and obtain some well known
results.  Also we will produce a formula relating the Euler-Poincare
number, the Lefschetz number and the coincidence number.  This leads to
an apparently unknown generalization to immersions of a famous theorem
of Hopf: If $\alpha : M^{n} \rightarrow {\Bbb R}^{n}$ is an immersion,
then the Gauss map on $\partial M$ must have degree equal to $\chi (M)$.
\heading
2. Massaging the formula.
\endheading

\noindent a) If $\partial_{-}M$ is empty, then $Ind\,(\partial_{-}V) = 0$
so $Ind\,(V) = \chi(M)$.  This is Hopf's theorem for closed manifolds
or when $V\mid\partial M$ always points outwards.

\noindent b) For nice enough vectorfields, the generic case, we get a
sequence of manifolds $M\supseteq \partial_{-}M \supseteq  \partial_{-}\partial_{-}M \supseteq \partial^{3}_{-}M \dots$ where $\partial_{-}\partial_{-}M$ is the set of points on the boundary of $\partial_{-}M$ on which $\partial V$ pointsinto $\partial_{-}M$, and etc.  Then we can eliminate the index of the
smaller dimensional vectorfields and show that
$$  
Ind\,V = \chi(M) - \chi(\partial_{-}M) + \chi(\partial^{2}_{-}M) - \dots \tag2 $$

\noindent Proving similar versions of (2) was the goal of [Mo] and [P]. 

\noindent c) On the other hand, we can try and eliminate the Euler-Poincare
number.  To do that, we first define $\partial_{+}M$ and $\partial_{0}M$
by letting $\partial _{+}M$ be the manifold where $V\mid\partial M$
points outward and $\partial_{0}M$ is the set where $V\mid\partial M$ is
tangent to $\partial M$.  Thus $\partial M = \partial_{-}M \bigcup \partial_{+}M \bigcup \partial_{0}M$.  Then let $\partial_{+}V = \partial V\mid\partial M_{+}$ and 
$\partial_{0}V = \partial V\mid\partial M_{0}$.  Now $Ind\,\partial_{0}V = 0$
since $\partial_{0}V$ has no zeros.  So
$$
Ind\,(\partial V) = \chi(\partial M) = Ind\,(\partial_{-}V) + Ind\,(\partial_{+}V) \tag3
$$
Thus for even dimensional $M$, since $\partial M$ is odd dimensional and so
$\chi(\partial M) = 0$, we have:
$$
Ind\,(\partial_{-}V) = -Ind\,(\partial_{+}V). \tag4
$$
For odd dimensional $M$ we have
$$
Ind\,(\partial_{-}V) = 2\chi(M) - Ind\,(\partial_{+}V). \tag5
$$
Combining (5) and (1) in the odd dimensional case we get
$$
Ind\,V = \frac{1}{2}(Ind\,(\partial_{+}V) - Ind\,(\partial_{-}V)). \tag6
$$
In the even dimensional case there is no such formula, so the theory is
different in even and odd dimensions.  This formula was used  by
Sue Goodman [Gd] in the case $V$ is transversal to $\partial M$.  Then
$Ind\,V = \frac{1}{2}(\chi(\partial_{+}M) - \chi(\partial_{-}M))$.

\noindent d) Now let us write (1) using $Ind\,(\partial_{+}V)$.

\noindent Using (4) and (5) we get
$$
Ind\,V = (-1)^{n}\chi(M^{n}) + Ind\,(\partial_{+}V). \tag7
$$
which is Morse's version of the formula, [Mo]

\noindent e) To prove that $Ind\,(-V) = (-1)^{n}Ind\,(V)$ is instructive.
Here $n$ denotes the dimension of $M$.  We assume that the formula is true
for vectorfields on $(n-1)$ - dimensional spaces.  Then $Ind\,(\partial_{-}(-V))
= Ind\,(-\partial_{+}V) = (-1)^{n-1}(\partial_{+}V)$.  Using this and (1)
and (3) we have
$$\align
Ind\,(-V) & = \chi(M) - Ind\,(\partial_{-}(-V))\\
          & = \chi(M) -(-1)^{n-1}Ind\,(\partial_{+}V)\\
          & = \chi(M) + (-1)^{n}(\chi(\partial M) - Ind\,(\partial_{-}V))
\endalign
$$
If $n$ is even then 
$$\align
Ind\,(-V) & = \chi(M) + (0 - Ind\,(\partial_{-}V)\\
          & = Ind\,V
\endalign
$$
If $n$ is odd  
$$\align
Ind\,(-V) & = \chi(M) - [2\chi(M) -Ind\,(\partial_{-}V)]\\
          & = -(\chi(M) - Ind\,(\partial_{-}V)) = - Ind\,V
\endalign
$$
f) Let $V$ be the vectorfield on ${\Bbb R}^{n}$ given by $V(\overrightarrow{x}) = A\overrightarrow{x}$ where $A$ is a linear transformation.  If $det A\not= 0$,
then $\overrightarrow{0}$ is the only zero of $V$.  We will show, using (1),
the well known fact
$$
Ind\,(V) = \frac{det A}{\mid det A\mid}. \tag8
$$
\demo
{Proof} We let $M = \{\overrightarrow{x}\parallel\overrightarrow{x}\parallel \leq 1\}.$  Then on $M$ we see that $Ind\,V + Ind\,(\partial_{-}V) = 1.$
Now $A$ is homotopic through linear transformations $A_{t}$ such that $det(A_{t}) \not= 0$, to a diagonal matrix $D$.  The corresponding vectorfields
$V_{t}$ all have a single zero at $\overrightarrow{0}$ and so the index of $V_{t}$ on $M$ does not change.  Now $D$ can be chosen to be either the identity $I$
or the diagonal matrix $J$ with  $-1$ in the $(1,1)$ position and $1$'s down
the main diagonal.  In the case of $det A > 0, I$ may be chosen for $D$ and the
vectorfield $V_{1}: V_{1}(\overrightarrow{x}) = \overrightarrow{x}$ always 
points outward on $\partial M$.  Thus $\partial_{-}M$ is empty and so
$ind\,(\partial_{-}V_{1}) =0$.  Hence $Ind\,V = 1$.  For the case 
$det(A) < 0$, the vectorfield is $V_{1}(x_{1},\dots,x_{n}) = (-x_{1},x_{2},\dots,x_{n}).$  Then $\partial_{-}M$ consists of the set of vectors $\overrightarrow{x}$ such that $\overrightarrow{x}\cdot J\overrightarrow{x} < 0$.  That is the
set $\{(x_{1},\dots,x_{n}) \in \partial M\mid x_{1}^{2} > \frac{1}{2}\}$.
The boundary of $\partial_{-}M$ are the spheres given by $x_{1} = \pm \frac{1}{\sqrt{2}}$.  Now if $\overrightarrow{x}$ is on the boundary of $\partial_{-}M$
we see that $\overrightarrow{x}\cdot J\overrightarrow{x} = 0.$  Now consider
for $\epsilon > 0$
$$
(\overrightarrow{x} + \epsilon J\overrightarrow{x})\cdot J(\overrightarrow{x}
+ \epsilon J\overrightarrow{x}) =
$$
$$
\overrightarrow{x}\cdot J\overrightarrow{x} + \epsilon\overrightarrow{x}\cdot(J^{2}\overrightarrow{x} + J\overrightarrow{x}\cdot J\overrightarrow{x}) + \epsilon^{2}J\overrightarrow{x}\cdot J^{2}\overrightarrow{x} =
$$
$$
\epsilon(\overrightarrow{x}\cdot\overrightarrow{x} + J\overrightarrow{x}\cdot J\overrightarrow{x}) > 0.
$$
Thus $J\overrightarrow{x}$ is tangent to the sphere at $\overrightarrow{x}$
and points out of $\partial_{-}M$.  Hence by (1), $Ind\,\partial_{-}V = \chi(\partial_{-}M)$.  But $\partial_{-}M$ can be deformed to $S^{0} = \{(1,0,\dots,0),
(-1,0,\dots,0)\}$.  So $\chi(\partial_{-}M) = 2$.  Hence from (1)
$$
Ind\,V_{1} + 2 =1
$$
so $Ind\,V = -1$.
\enddemo

\noindent g) Let $f: {\Bbb R}^{n}$ be given by $f(\overrightarrow{x}) = A\overrightarrow{x}$ where $A$ is a linear transformation.  If $A$ has no eigenvalue $\lambda = 1$
then $f$ has a single fixed point at $\overrightarrow{0}$ and the
Lefschetz number
$$
\Lambda_{f} = \frac{det(I - A)}{\mid det(I - A)\mid} \tag9
$$
\demo
{Proof} The fixed point index of $f$ at $\overrightarrow{0}$ is the index
of the vectorfield $V(\overrightarrow{x}) = \overrightarrow{x} - f(\overrightarrow{x}) = (I - A)\overrightarrow{x}$.

Comparing (f) and (g) we see that if the linear transformation $A$ has a
unique zero and fixed point, the vectorfield index and the fixed point index
agree if and only if $A$ has an {\it even} number of eigenvalues between zero
and one for {\it even} $n$, and in the case of {\it odd} $n$, they agree if and
only if there are an {\it odd} number of eigenvalues between zero and one.
\enddemo

\head 3. \ $e^{i\pi} + 1 = 0$ \endhead

De Moivre's formula has the famous special case which relates the five
most common constants of calculus.  Similarly formula (1) implies an
equation which relates the three most common algebraic invariants of fixed
point theory, the Euler-Poincare number, the Lefschetz number and the
coincidence number.

If $\partial M^{n} \subset{\Bbb R}^{n+1}$, recall that the Gauss map
$\hat N: \partial M \rightarrow S^{n}$ is given by translating the ``outward
pointing" unit normal vector at $m$, denoted $N(m)$ to the origin.  Similarly
for any nonzero vectorfield $V$ on $\partial M$ we define the Gauss map
$\hat V: \partial M \rightarrow S^{n}$ by translating the unit vector
$V(m)/\parallel V(m)\parallel$ to the origin.

Now suppose that $f: M^{n}\rightarrow M^{n}\subset{\Bbb  R}^{n+1}$ is a map
such that $f$ has no fixed points on $\partial M.$  Let $V$ be the
vectorfield
$$
V(m) = (m - f(m))_{m}
$$
Then
$$
\Lambda_{f} + \Lambda_{(-\hat V),\hat N} = \chi(M) \tag10
$$
where $\Lambda_{f}$ is the Lefschetz number of $f$ and $\Lambda_{(-\hat V),\hat N}$ is the coincidence number of the Gauss maps for $V$ and $N$.

Equation (10) follows from (1) and the following observation.  We use the
description of coincidence numbers found in the last chapter of Vick [V].
There we see that a coincidence number $\Lambda_{f,g}$ is defined which is
equal to the sum of all the coicidence numbers of all the coincidence points
of $f,g: M_{1}^{n}\rightarrow M_{1}^{n}$ where $M_{1}^{n}$ and $M_{i}^{n}$
are closed oriented manifolds.  Then Vick shows that
$$
\Lambda_{f,g} = \sum_{i}(-1)^{i} trace (g^{!}f^{*})_{i} \tag11
$$
where $g^{!}: H^{i}(M_{1},{\Bbb Z})\rightarrow H^{i}(M_{2},{\Bbb Z})$
is the Umkehr map.  Now it is not hard to show that
$$
Ind\,(\partial_{-}V) = \Lambda_{-\hat V,\hat N} \tag12
$$
Since $Ind\,V = \Lambda_{f}$, (12) and (1) imply (10).
\heading
4. Normal degree. 
\endheading
Let $\alpha: M^{n}\rightarrow{\Bbb R}^{n+1}$ be a smooth immersion of a
closed oriented manifold.  Let $V$ be a vectorfield on ${\Bbb R}^{n+1}$
with no zeros on $\alpha (M)$.  Then
$$
Ind\,(\partial_{-}V) = deg(\hat N) - deg(\hat V) \tag13
$$
where $\hat N$ is the Gauss map of the unit normal chosen so that the
orientation of $M^{n}$ and ${\Bbb R}^{n+1}$ agree and $\partial_{-}V$
consists of those vectors of $V$ such that $N(m)\cdot V(m) < 0.$  This
follows from (12) and (11).

We define the {\it winding number} $W(p,\alpha)$ at $p\in{\Bbb R}^{n+1}$
of $\alpha: M\rightarrow {\Bbb R}^{n+1}$ of $p\notin\alpha(M)$ as
follows.  Consider a ray $\rho$ starting at $p$. Assume that $\rho$
intersects $\alpha(M)$ transversally.  For each $m$ so that $\alpha(m)$
meets $\rho$, assign +1 if $\rho$ and $N(m)$ point to the same ``side"
at $m$ and assign $-1$ if they point to opposite sides.  Then $W(p,x)$
is defined to be the sum of these $\pm 1$'s.  This $W(p,x)$ is equal to
the degree of the map
$$
\pi \circ\alpha: M^{n} \overset\alpha\to\rightarrow {\Bbb R}^{n+1} - p\overset\phi\to\rightarrow S^{n}
$$

Now suppose that $V$ has a finite number of zeros at points $p_{i} \notin
\chi(M).$  For each $p_{i}$ there is a local index $Ind\,(p_{i},V)$.  Then
we have the following formula
$$
\sum_{i}W(p_{i},\alpha)\cdot Ind\,(p_{i},V) + Ind\,(\partial_{-}V) = deg(\hat N) \tag14
$$
\demo{Proof}  Since $M$ is a closed oriented manifold immersed in ${\Bbb R}^{n+1}$,
we see that $M$ must be the boundary of an oriented manifold $W$.  Now the
embedding $\alpha: \partial W\hookrightarrow {\Bbb R}^{n+1}$ extends to a
smooth map $f: W\rightarrow {\Bbb R}^{n+1}$ for which the $p_{i}$ are
regular values.  Then the vectorfield $V$ may be considered as a map
$V: {\Bbb R}^{n+1}\rightarrow {\Bbb R}^{n+1}$.  Then $deg(V\circ f) =
\sum_{i}W(p_{i},\alpha)\cdot Ind\,(p_{i},V)$ because this is the local
degree calculated using the preimage of $\overset\rightarrow\to 0$.  But
$deg(V\circ f) = deg(\hat V)$ where $\hat V: \partial W \overset\alpha\to \rightarrow ({\Bbb R}^{n+1} - p)\rightarrow {\Bbb R}^{n+1} - 0 \overset\phi\to\rightarrow S^{n}$.
Then (l4) follows from (13).
\enddemo

%Then (l4) follows from (13).

Equation (l4) affords a means of studying the degree Gauss map of an immersion This is called the normal degree and is a classical topic begun by Hopf in [H1]
and continued by Milnor in [Mi].  Equations (1) and (14) combine to give the
following result.
\proclaim{Theorem:} If $\alpha: M^{n+1}\rightarrow {\Bbb R}^{n+1}$ is an
immersion, then the degree of the Gauss map $\hat N: \partial M\rightarrow S^{n}$ is equal to the $\chi (M)$.
\endproclaim
\demo{Proof} Since $\alpha$ is an immersion, the vectorfield $V$ on ${\Bbb R}^{n}$
pulls back to a vectorfield $V*$ on $M$. Now $Ind\,V^{*} =\sum_{i} W(p_{i},\alpha) Ind\,(p_{i},V)$, so (1) and (l4) imply that $deg\hat N = \chi(M)$.
\enddemo

Hopf orginally proved this in [H2] for $\alpha$ an embedding.  This is a famous
and important theorem.  Also Hopf showed in [H2] that $deg\hat N = \frac{1}{2}\chi(\partial M)$ for even immersions.  The case of $\partial M = S^{1}$ is known
and the immersions are classified.
\vfil\eject
\centerline{Bibliography}
\item{[Go]} Goodman, Sue E., Closed leaves in foliation of codimension one,
Comment Math. Helv. {\it 50} (1975), 383-388.
\item{[G]} Gottlieb, Daniel Henry, A de Moivre formula for fixed point theory,
ATAS de 5o.  Encontro Brasiliero de Topologia, Universidade de Sa\~o Carlos,
S.P. Brazil (to appear).
\item{[H1]} Hopf, Heinz,  \"Uber die curvatura integra geschlossener Hyperfl\"achen,
Math Ann. {\it 95} (1925/26), 340-367.
\item{[H2]} Hopf, Heinz, Vectorfelder in $n$-dimensionalen Mannigfaltigkeiten,
Math. Ann. {\it 96} (1926/l927), 225-250.
\item{[Mi]} Milnor, John, On the immersion of $n$-manifolds in $(n+1)$-space,
Comment. Math. Helv. {\it 30} (1956), 275-284.
\item{[Mo]} Morse, Marston, Singular points of vectorfields under general
boundary conditions, American Journal of Mathematics {\it 51} (1929), 165-178.
\item{[P]} Pugh, Charles C., A generalized Poincare index formula, Topology
{\it 7} (1968), 217-226.
\item{[V]} Vick, James W., Homology Theory, Academic Press, New York.
\enddocument






