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\vskip 2truein
\centerline{\bf INTERSECTION NUMBERS, TRANSFERS, AND GROUP ACTIONS}
\medskip
\centerline{by}
\bigskip
\centerline{\bf Daniel H.\ Gottlieb and Murad Ozaydin}
\bigskip
\centerline{\bf Purdue University}
\centerline{\bf  University of Oklahoma}
\vskip .25truein
\noindent
{\bf Abstract}

We introduce a new definition of {\it intersection number}
by means of umkehr
maps. This definition agrees with an obvious definition up to sign. It
allows us to extend the concept parametrically to fibre bundles. For
fibre bundles we can then define {\it intersection number transfers}.
Fibre bundles arise naturally in equivariant situations, so
the {\it trace of the action} divides intersection numbers. We apply
this to
equivariant projective varieties and other examples. Also we study
actions
of non-connected groups and their traces.
\bigskip

\noindent
AMS subject classification 1991:
Primary: 55R12, 57S99, 55M25
Secondary: 14L30

\noindent Keywords: Poincare Duality, trace of action, Sylow
subgroups,
Euler class, projective variety.

\vfill\eject
\baselineskip=18pt
\noindent

\noindent {\bf 1. Introduction}

      When two manifolds with complementary dimensions inside a third manifold
intersect transversally, their intersection number is defined. This classical
topological invariant has played an important role in topology and its
applications. In this paper we introduce a new definition of intersection 
number.

The definition naturally extends to the case of parametrized
manifolds over a base $B$; that is to fibre bundles over $B$ .  Then we
can
define transfers for the fibre bundles related to the intersection
number.
This is our main objective , and it is done in Theorem 6. The method of
producing transfers depends upon the Key Lemma which
we prove here. Special cases of this
lemma played similar roles in the establishment of the Euler--Poincare
transfer, the Lefschetz number transfer, and the Nakaoka transfer.

Two other features of our definition are:
1)  We replace the idea of submanifolds with the idea of maps from
manifolds, and in effect we are defining the intersection numbers of the maps;
2) We are not restricted to only two maps, we can define the
intersection number for several maps, or submanifolds. 


We can apply Theorem 6 to equivariant topology. When the maps involved
are equivariant, or even if they are only {\it homotopy} equivariant,
the intersection number is an equivariant invariant. This is seen by
using the Borel construction to get the fibre bundle situation for which
theorem 6 provides transfers. Transfers tell us much about equivariant
topology. Much of this information can be concentrated in a single
integer called the {\it trace of the action}.

The trace of an action is an integer which is defined by looking at all the
possible transfers arising from the Borel construction. It was defined in [{\bf
G2}] and is very closely related to the exponent of an action as defined by W.
Browder in [{\bf B}].
In fact the trace equals the exponent for orientation preserving actions
on manifolds. However the trace, in contrast to the exponent, is always
defined in any equivariant setting. When the equivariant situation
allows
intersection numbers, the trace must divide the intersection number.
This
result is our main application of transfers to group actions and is
found
in Theorem 7 and is a consequence of Theorem 6.

In addition to showing that the trace divides
intersection numbers, we examine how the trace behaves when the acting
group is disconnected.  In particular, we show that the trace of an action
of a finite group $G$ on a space $X$ must be the product of the traces of
the Sylow $p$-subgroups acting on $X$.
This latter part is contained in the thesis of Murad \"Ozaydin at Purdue
University \cite{\bf O}.

Finally, we give some examples of trace in action. In the case of
nonsingular projective varieties, the degree of the variety is an
intersection
number. Thus in suitable equivariant situations the trace divides the
degree of
the variety. We obtain generalizations of a theorem of W. Browder and N.
Katz
[{\bf B-K}] . Also we look at examples of elementary $p$--groups acting
on products of spheres. In these situations we find that appropriate 
$G$--submanifolds imply that the trace divides associated determinants,
permanents, and Pfaffians.

We would like to thank L. Avramov for helpful discussions.
 
%\vfill\eject
\noindent {\bf 2. Multiple functional intersection number}

\noindent DEFINITION 1. Let $f_{i}: A_{i}\to M$ be a finite set of maps
between closed oriented manifolds $A_{i}$ and $M$.  We suppose the sum of
the co-dimensions of $A_{i}$ equals the dimension of $M$.  That means,
assuming there are $k$ maps $f_{i}$, that $\underset{i=1}\to{\overset
k\to\sum} (\dim M - \dim A_{i}) = \dim M$, or $\underset{i=1}\to{\overset
k\to\sum}\dim A_{i} = (k-1)\dim M$.

Next define $F: A_{1}\times\dots\times A_{k}\to M\times\dots\times M$ by $F
= f_{1}\times\dots\times f_{k}$.  Define the \underbar{multiple
intersection number}, denoted $f_{1}\bullet f_{2}\bullet\dots\bullet f_{k}$, as
the integer 
$$
[\Delta^{*}F^{!}(1)]\cap [M]\in H_{0}(M;\bz)\cong\bz.
$$
Here $\Delta: M\to M\times\dots\times M$ is the diagonal map, $F^{!}$
denotes the Umkehr map on cohomology and $1\in
H^{0}(A_{1}\times\dots\times A_{k})$.  We denote by $[M]$ the orientation
class of $M$.  We require $M$ to be connected, so that $H_{0}(M;\bz)\cong
\bz$, but the $A_{i}$ need not be connected.  So the $1\in
H^{0}(A_{1}\times\dots\times A_{k})$ is the sum of the generators\newline $1\in
H^{0}$ (connected components of $A_{1}\times\dots\times A_{n})\cong\bz$, and
each of those generators is equal to a product $1_{A}\times
1_{B}\times\dots$, where the $A, B,\dots$ are connected components of
$A_{1}, A_{2},\dots$ respectively.  The Umkehr map $F^{!}$ is defined as
the composition $D^{-1}_{M^{\times}} F D_{A^{\times}}$ where $D_{M}$
denotes the Poincare duality map
$H^{*}(M;\bz)\overset\cong\to\longrightarrow H_{*}(M;\bz)$ given by
$D_{M}(x) = x\cap [M]$.  Here $M^{\times} = M\times\dots\times M$ where
the product is taken $k$-times and $A^{\times} = A_{1}\times\dots\times
A_{k}$.

Obviously the multiple intersection number depends on the choices of
orientation on $A^{\times}$ and on $M$.  For the case of two connected
closed oriented submanifolds $A$ and $B$ of complimentary dimensions in
$M$, the usual intersection number $A\bullet B$ agrees with $f\bullet g$ (up to
sign) where $f$ and $g$ are the inclusion maps.  That is because the usual
intersection number ends up being given by the cup product of the Poincare
duals of the fundamental classes $[A]$ and $[B]$ in $H_{*}(M;\bz)$.  Using
the concept of the Umkehr map, this is just $f^{!}(1_{A})\cup g^{!}(1_{B})$
which the following lemma implies is equal to $\pm(f\bullet g)[M]$.

\noindent REMARK: The reader may ask why the multiple functional 
intersection number $f_{1}\bullet f_{2}\bullet\dots\bullet f_{k}$
is defined using Umkehr maps instead of the obvious definition
as the cup product of the Poincare duals of the homology classes 
$f_{i*}[A_i]$ capped with $[M]$? As one can see from Lemma 3, the
obvious
definition differs from our definition by a complicated sign. The 
reason we need Umkehr maps is to construct the transfer in Theorem 6.
Note that the Key Lemma cannot be stated without Umkehr maps.
In general, if one can express an invariant in terms of Umkehr maps,
then there is a good chance transfers can be defined. This is 
especially true since the Key Lemma holds. The sign difference in
Lemma 3 is not important; it is just there.

\proclaim{Lemma 2} $\Delta^{*}(f\times g)^{!}(1_{A}\times 1_{B}) =
(-1)^{nq}f^{!}(1_{A})\cup g^{!}(1_{B})$ where $q = n-\dim B$ and $n=\dim
M$.
\endproclaim 

\noindent Proof:
$$
\aligned
\Delta^{*} &(f\times g)^{!}(1_{A}\times 1_{B})\\ 
&= \Delta^{*}D^{-1}_{M\times M}(f_{*}([A])\times g_{*}([B]))\\
&= \Delta^{*}D^{-1}_{M\times M}((f^{!}(1_{A})\cap [M])\times
(g^{!}(1_{B})\cap [M])))\\
&= (-1)^{nq}\Delta^{*}D^{-1}_{M\times M}((f^{!}(1_{A})\times g^{!}(1_{B}))\cap
([M]\times [M]))\\
&= (-1)^{nq}\Delta^{*}(f^{!}(1)\times g^{!}(1)) = (-1)^{nq}f^{!}(1)\cup
g^{!}(1).\endaligned
$$
In fact, we have in general
\proclaim{Lemma 3} $\Delta^{*}F^{!}(1) =
(-1)^{a}f^{!}_{i}(1_{A_{1}})\cup\dots\cup f^{!}_{k}(1_{A_{k}})$ where\newline $a=
n[(k-1)q_{k} + (k-2)q_{k-1}+\dots + q_{2}]$ and $q_{i}= n- \dim A_{i}$, the
codimension of $A_{i}$.
\endproclaim

\noindent Proof: The same as Lemma 2, only the notation is much more
involved.  Counting the sign change is the only complication.

Next we extend the definition of multiple intersection number to the case
where $M$ is a connected oriented manifold with boundary $\partial M$ and
the $f_{i}: A_{i}\to M$ are maps of pairs $(A_{i},\partial A_{i})
\overset{f_{i}}\to\longrightarrow (M,\partial M)$.  We assume that at
least one of the $A_{i}$ has an empty boundary.  Also the $A_{i}$ are all
oriented and the sum of the codimension $(n-\dim A_{i})$ is equal to $n$ as
before.  We let $DM$ denote the double of $M$.  It is  a closed oriented
manifold.  We let $Df_{i}: DA_{i}\to DM$ be the double of the map $f_{i}:
(A_{i},\partial A_{i})\to (M,\partial M)$ in the case where $A_{i}$ has a
boundary, and if $\partial A_{i}= \phi$ we let $Df_{i}:
A_{i}\overset{f_{i}}\to\longrightarrow M\overset j\to\longrightarrow DM$ be
the composition of $f_{i}$ with the standard inclusion $j$ of $M$ into its
double.

\noindent DEFINITION 4. We define the multiple intersection number
$f_{1}\bullet f_{2}\bullet\dots\bullet f_{k}$ to be the intersection number
$Df_{1}\bullet Df_{2}\bullet\dots\bullet Df_{k}\ .$

\noindent REMARKS: a) We required at least one of the $A_{i}$ to have an
empty boundary.  We can define the intersection number as above in the case
when all the $A_{i}$ have non-empty boundaries, but in this case it will
always be zero.

b) The intersection number is obviously preserved under homotopy of the
$f_{i}$.

c) If $f_{1}\bullet\dots\bullet f_{k}\not= 0$, then the intersection of the
images of the $f_{i}$ is not empty, that is $\underset{i=1}\to{\overset
k\to\cap} f_{i}(A_{i})\not= \phi$.

d) A geometric interpretation of multiple intersection number for the case
when the $A_{i}$ are submanifolds of $M$ should go as follows: choose
orientations of the normal bundles $\nu_{i}$ consistent with the choices of
orientation of $M$ and $A_{i}$.  Assuming the $A_{i}$ are mutually
transversal at the points $x_{j}\in\underset{i=1}\to{\overset k\to\cap}
A_{i}$, assign to the point $x_{j}$ the number $+1$ if the orientations at
$x_{j}$ combine to give the orientation of $M$ at $x_{j}$, and assign
$x_{j}$ the number $-1$ otherwise.  The sum of these local multiple intersection numbers is the global intersection number.

e) Let $\nu^{k}$ be a $k$ bundle over the closed oriented manifold $M^{n}$.
If $n=kr$, the self-intersection number of the zero section $s$ is defined
to be $s\bullet\dots\bullet s$ \ .  Let $\chi(\nu)$ denote the Euler class of a bundle
$\nu$.  Then
$$
\chi(\nu\oplus\dots \oplus\nu) = [\chi(\nu)]^{r} = \pm(s\bullet\dots\bullet
s)[\overline M]\in H^{n}(M;\bz)
$$

An application of the idea of intersection number can be used in a dual
situation to calculate the degree of a map. 
Suppose that $M$ is a closed
smooth oriented connected manifold and suppose that we have a family of $k$
smooth maps $f_{i}: M\to A_{i}$ where the $A_{i}$ are connected smooth
closed oriented manifolds the sum of whose dimensions equals the dimension
of $M$.  The relevant invariant, a ``co-intersection number", is the degree of
the composition
$$
G: M\overset{\Delta}\to\longrightarrow M\times\dots\times
M\overset{f_{1}\times\dots\times f_{k}}\to\longrightarrow
A_{1}\times\dots\times A_{k}.
$$
\proclaim{Corollary 5} With the notation of the preceding paragraph,
$\pm\deg G = i_{1}\bullet\dots\bullet i_{k}$ where $i_{j}: f^{-1}_{j}(a_{j})\to
M$ and $a_{j}$ is a regular value of $f_{j}$ in $A_{j}$.  Thus the degree of
$G$ is the intersection number of the fibres of the $f_{i}$.
\endproclaim

\noindent Proof: For any smooth map $f: M\to N$ between closed oriented
manifolds and fibre $F = f^{-1}(b)$ where $b$ is a regular point, we have
$i_{*}[F] = f^{*}([\overline N])\cap [M]$ where
$i: F\to M$ is the inclusion, \ \cite{\bf{G1}}.  So, in other words, $i^{!}(1_{F}) =
f^{*}([\overline N])$.  Now  $(\deg G)[\overline M] = G^{*}([\overline
A_{1}]\times\dots\times [\overline A_{k}]) = f^{*}_{1}([\overline
A_{1}])\cup\dots\cup f^{*}_{k}([\overline A_{k}]) =
i^{!}_{1}(1_{F_{1}})\cup\dots\cup i^{!}_{k}(1_{F_{k}}) =
\pm(i_{1}\bullet\dots\bullet i_{k}) [\overline M]$.
\medskip
\noindent {\bf 3. Transfers and intersection numbers}

Consider the following commutative diagram
$$
\CD
F @>f>> F_{1}\\
@ViVV @Vi_{1}VV\\
E @>{\overline f}>> E_{1}\\
@VpVV @Vp_{1}VV\\
B @>> 1_{B}> B
\endCD
$$
Here $p$ and $p_{1}$ are fibre bundle projections and $i$ and $i_{1}$ are
inclusion of a fibre.  The map $\overline f$ is a fibre preserving map
which restricts to $f$ on the fibre.  In addition every space is a closed
oriented manifold
\proclaim{Key Lemma} $\displaystyle{f^{!}i^{*} = (-1)^{B(F_{1}-F)}i^{*}_{1}\overline
f^{!}}$ where $(-1)^{X}$ means $(-1)^{\dim X}$.
\endproclaim

\noindent Proof:  Let $U$ be a small ball in $B$ centered on the base point
$*$ where $p^{-1}(*) = F$.  Then $V = p^{-1}(U)$ and $V_{1} =
p^{-1}_{1}(U)$ are open sets of $E$ and $E_{1}$ respectively.

The following diagram commutes except for the top rectangle.  The
undecorated arrows are induced by inclusions.
\vfill\eject
$$
\CD
@. H^{*}(F) @.{}^{f^{!}} @.@.H^{*}(F)\\
@.@AAA @.@. @AAA\\
@.H^{*}(\overline U\times F) @>\cap[\overline U, \dot U]\times [F]>\cong > H_{*}((\overline U, \dot U)\times F) @> (1\times f)_{*} >> H_{*}((\overline U,\dot
U)\times F_{1}) @< \cap[\overline U, \dot U]\times [F_{1}]<\cong
<H^{*}(\overline U\times F_{1})\\
@.@AA i^{*}A @V\cong V ex V @V\cong V ex V  @AA i^{*}_{1}A\\ 
@.H^{*}(E) @> \cap[E,E-V] >> H_{*}(E,E-V) @> \overline f_{*} >>
H_{*}(E_{1},E_{1}-V_{1}) @< \cap[E_{1},E-V_{1}] << H^{*}(E_{1})\\
@.@A \cong A 1_{E}A @AAA @AAA @A\cong A 1_{E_{1}}A\\
\overline f^{!}: @. H^{*}(E) @>\cap[E]>\cong > H_{*}(E) @> \overline f_{*} >>
H_{*}(E_{1}) @< \cap[E_{1}] <\cong < H^{*}(E_{1})
\endCD
$$
\vfill\eject

Now the top rectangle commutes up to $(-1)^{B(F_{1}-F)}$, since the second
row gives the following
$$
\aligned
& 1\times x\mapsto (1\times x)\cap[\overline U,\dot U]\times [F] =
(-1)^{xB}[\overline U,\dot U]\times (x\cap [F])\\
&\mapsto (-1)^{xB}[\overline U,\dot U]\times f_{*}(x\cap [F]) =
(-1)^{xB}[\overline U,\dot U]\times (f^{!}(x)\cap [F_{1}])\\
&= (-1)^{xB+(x+B(F_{1}-F))B}(1\times f^{!}(x))\cap([\overline U,\dot
U]\times [F_{1}])\\
&\mapsto (-1)^{B(F_{1}-F)}1\times f^{!}(x)\endaligned
$$
This proves the lemma.

Suppose we have the following $k$ fibre bundle squares.
$$
\CD
A_{i} @> {f_{i}} >> M\\
@VVi_{i}V @VViV\\
E_{i} @> {\widetilde f_{i}} >> E\\
@VVp_{i}V @VVpV\\
B @>> {1_{B}} > B
\endCD
\qquad i=1,\dots, k
$$
where each space is a closed oriented manifold, $M$ and $B$ are connected
and the codimension of the $A_{i}$ sum to the dimension of $M$.  More
generally we may assume that $M$ and $B$ and hence $E$ have non-empty
boundaries and that some of the $E_{i}$ have non-empty boundaries and in
those cases $\widetilde f_{i}: (E_{i},\partial E_{i})\to (E,\partial E)$.
Of course all spaces are compact.
\proclaim{Theorem 6} There is a transfer associated to the fibre bundle $p$
with trace $(f_{1}\bullet\dots\bullet f_{k})$.  That is, there is a homomorphism
$\tau_{*}: H_{*}(B,G)\to H_{*}(E;G)$ so that $p_{*}\circ\tau_{*}$ is
multiplication by $(f_{1}\bullet\dots\bullet f_{k})$.  Similarly there is  a
homomorphism $\tau^{*}: H^{*}(E,G)\to H^{*}(B,G)$ so that $\tau^{*}\circ
p^{*}$ is multiplication by $(f_{1}\bullet\dots\bullet f_{k})$.
\endproclaim

\noindent Proof: Let $E_{1}\circ\dots\circ E_{k}$ denote the multiple fibre
product of the fibre bundles $p_{i}: E_{i}\to B$.  That is, 
$$
E_{1}\circ\dots\circ E_{k}= \{(e_{1},\dots, e_{k})\in
E_{1}\times\dots\times E_{k}|p_{1}(e_{1})  =\dots  = p_{k}(e_{k})\}
$$
Now the map $\overline  p: E_{1}\circ\dots\circ E_{k}\to B$ defined by
$p(e_{1},\dots, e_{k}) = p(e_{1})\in B$ is a fibre bundle projection with
fibre $A_{1}\times\dots\times A_{k}$.  Also we have the multiple fibre
product $E\circ\dots\circ E\to B$ with fibre $M\times\dots\times M$.
Define  
 $\widetilde F: E_{1}\circ\dots\circ E_{k}\to
E\circ\dots\circ E$ by $\widetilde F(e_{1},\dots, e_{k}) = (\widetilde
f_{1}(e_{1}),\dots,\widetilde f_{k}(e_{k}))$.  The diagonal $\widetilde\Delta:
E\to E\circ\dots\circ E$ given by $\widetilde\Delta(e) = (e,\dots, e)$.  All this
fits into a commutative diagram
$$
\CD
A_{1}\times\dots\times A_{k} @> F >> M\times\dots\times M @<
{\Delta}<< M\\
@VVkV @VVjV @VViV\\
E_{1}\circ\dots\circ E_{k} @>{\widetilde F}>> E\circ\dots\circ E @<
{\widetilde\Delta}<< E\\
@VVV @VVV @VVpV\\
B @> {1_{B}} >> B @< {1_{B}} << B
\endCD
$$
Now we can define the class $\widetilde\Delta^{*} \widetilde F^{!}(1)\in
H^{n}(E;G)$ in the case of closed $E$.  Then
$i^{*}(\widetilde\Delta^{*}\widetilde F^{!}(1)) =
\Delta^{*}j^{*}\widetilde F^{!}(1)=
\pm\Delta^{*}F^{!}k^{*}(1) = \pm\Delta^{*}F^{!}(1) =
\pm(f_{1}\bullet\dots\bullet f_{k})[\overline M]$.  The equalities are true by
commutativity and the key lemma and the last equality is the definition.
The transfer maps are defined as usual: $\tau^{*}(x)=p^{!}(x\cup\widetilde\Delta^{*}\widetilde F^{!}(1))$ and $\tau_{*}(x) = p_{!}(\widetilde\Delta^{*}\widetilde
F^{!}(1)\cap x)$.

In the case that $M$ has a non-empty boundary, and $B$ is closed, taking
the doubles $D\widetilde f_{i}: DE_{i}\to DE$ and $D\widetilde f_{j}:
E_{j_{i}}\to E\to DE$ gives a transfer for $DM\to DE\to B$.  Since $M\to
E\to B$ is a fibrewise retraction $DM\to DE\to B$, the transfer defined for
the latter restricts to the former with the same
trace, $f_{1}\bullet\dots\bullet f_{k}$ \ .  In the remaining case where $B$ has
non-empty boundary, we consider the pullback of $M\to E\overset
p\to\longrightarrow B$ by the retraction map $r: DB\to B$.  Suppose this
pullback is denoted $M\to DE\overset{Dp}\to\longrightarrow DB$.  Then $Dp$
admits a transfer with associated number $f_{1}\bullet\dots\bullet f_{k}$ , and
so $p$, which is a fibrewise retract of $Dp$, admits a transfer with
associated number $f_{1}\bullet\dots\bullet f_{k}$.
\medskip
\noindent {\bf 4. The trace of an action}

Suppose $G$ is a group acting on spaces $A_{i}$ and $M$ and suppose $f_{i}:
A_{i}\to M$ are homotopy $G$-maps.  
Let us say that $i:W\to V$ is a {\it homotopy $G$-map} if there is
a
fibre bundle square
$$
\CD
W @>>i> V\\
@VVV @VVV\\
E @>>f> V_{G}\\
@VVV @VVV\\
B_{G} @>1>> B_{G}
\endCD
$$
which extends $i$ to a map $f$ over $1_{B_{G}}$. Here $V_G$ is the Borel
construction $E_G {\times}_G V$

In \cite{\bf{G2}} and \cite{\bf {O}} we introduced the integer $\tr(p)$, read the
trace the fibration $p$. 
Let $\deg(p)$ be the
smallest positive integer $N$ so that there is a homomorphism $\tau$ so that 
$p_{*}\circ\tau =$
multiplication by $N$. If no such integer exists then set $\deg(p)$ 
equal to zero. In fact, $\deg(p)$ is the greatest common divisor
of these numbers.  Next the {\it trace} of $p$, denoted $\tr(p)$, is
the least upper bound of all the positive $\deg(p_{f})$ where the $p_{f}$ 
are the
pullbacks of $p$ by arbitrary maps $f: X\to B$.
If the least upper bound does not exist set $\tr(p)$ equal to zero. 
In fact the $\tr(p)$ is
the least common multiple of these numbers.  Then we define the trace of an
action, denoted $\tr(G,M)$, by $\tr(G,M) = \tr(p_{G})$ where $p_{G}:
E_{G}\times_{G}M\to B_{G}$ is the Borel construction of the universal
$G$-principal bundle $G\to E_{G}\to B_{G}$ with $M$.
\proclaim{Theorem 7} Suppose $G$ is a group so that the classifying space
$B_{G}$ has finite type and suppose that $M$ is an oriented compact
$G$-manifold
and $f_{i}: A_{i}\to M$ are homotopy $G$-maps from oriented compact 
manifolds $A_{i}$ which preserve boundaries.  Then
$\tr(G,M)|(f_{1}\bullet\dots\bullet f_{n})$.
\endproclaim

\noindent Proof: Let $B$ be a regular neighborhood of the  $(\Dim M+2)$-
skeleton embedded in some Euclidean space.  So $B$ is an oriented
manifold with boundary.  Let $M\to E\overset p\to\longrightarrow B$ be the
pullback of $M\to M_{G}\to B_{G}$ onto $B$ where the map $B\to B_{G}$,
which induces the pullback, is homotopic to the inclusion of the $(\Dim M +
2)$-skeleton.  By Proposition (6.6) of \cite{\bf{G2}} we have $\tr(G,M) =
\deg(p)$.  By theorem 6 and the definition of $\deg p$ \ , we see that
$f_{1}\bullet\dots\bullet f_{k}|\deg (p) = \tr(p) = \tr(G,M)$.

Theorem 7 allows us to add multiple intersection numbers to the list of
topological and group theoretical properties divided by the trace of an
action.  Most of this list so far may be found in Theorem (1.1) of {\cite{\bf
{G2}}}.

\noindent REMARK: By Theorem (1.5) of {\cite{\bf{G2}}}, if $G$ 
acts orientably on a closed manifold $M^n$ the trace $\tr(G,M)$
is the order of 
quotient group of the top dimensional integer cohomology group divided
by the image of fibre inclusion of the cohomology of 
$H^n(EG {\times}_G M)$, if this is finite, and zero otherwise. This is
just what Browder's exponent is in {\cite{\bf{B}}}. In this special case
the trace appears more tractible than in the general definition. But 
the general definition works for any action and therefore appears in
theorems which are not restricted to closed manifolds and orientation
preserving actions. The list of properties of the general trace is 
quite substantial and they work together well, so that despite its
strange looking definition it is actually quite easy to use the 
concept. This is exemplified by the arguments in the next section
which do not use the exponent definition above and so result in 
facts which are true in more generality and which yield even more
additions to the list of properties of the trace.
 

\medskip
\noindent {\bf 5. Traces and subgroups}

Let $G$ be a group, $X$ a $G$-space and let $H$ be a subgroup of $G$.  By
(\cite{\bf{G2}}, Theorem 6.2a), $\tr(H,X)$ divides $\tr(G,X)$.  We will now show
that the quotient divides $\tr(G,G/H)$ where $G/H$ is the $G$ space
consisting  of a single orbit of isotropy type $H$.
\proclaim{Proposition 8} Let $H$ be a subgroup of $G$, and let $X$ be a
$G$-space.  Then
$$
\tr(H,X)\big\vert\tr(G,X)\big\vert\tr(G,G/H)\tr(H,X).
$$
\endproclaim

\noindent Proof:  We already know the left division by \cite{\bf{G2}}, Theorem
6.2a.  For the second, given any map $f: Y\to B_{G}$ \ , let $\widetilde Y$ be
the pullback fitting into the diagram:
$$
\CD
G/H @= G/H\\
@VVV @VVV\\
\widetilde Y @> {\widetilde f} >> (G/H)_{G} = B_{H}\\
@VVqV @VVV\\
Y @> f>> B_{G}.
\endCD
$$

From the definition of $\tr(G,G/H)$ as the l.c.m. of the degrees of the
projection maps over all pullbacks of the Borel construction, it follows
that $\deg(\widetilde Y\to Y)$ divides $\tr(G,G/H)$.  Hence there is a
transfer $\tau: H_{*}(Y;\bz)\to H_{*}(\widetilde Y;\bz)$ so that
$q_{*}\circ\tau = \tr(G,G/H)$.

From the diagram
$$
\CD
@. X @= X\\
@. @VVV @VVV\\
G/H @>>> X_{H} @>>> X_{G}\\ 
@| @VVV @VVV\\
G/H @>>> B_{H} @>>> B_{G}
\endCD
$$  
we get the pullback diagram (via $f$ and $\widetilde f)$
$$
\CD
@. X @= X\\
@. @VVV @VVV\\
G/H @>>> \widetilde f^{*}(X_{H}) @>{\widetilde q} >> f^{*}(X_{G})\\
@| @VV{\widetilde p}V @VVpV\\
G/H @>>>{\widetilde Y} @> q>> Y
\endCD
$$
If 
$\sigma$ is a transfer realizing $\deg(\widetilde p)$ then $\rho = \widetilde
q_{*}\circ\sigma\circ \ \tau$ satisfies $p_{*}\rho = p_{*}\circ\widetilde
q_{*}\circ\sigma\circ\tau = q_{*}\circ\widetilde p_{*}\circ\sigma\circ\tau
= \tr(G,G/H) \deg(\widetilde p)$.  Hence $\deg(p)\big\vert\tr(G,G/H)\deg(\widetilde
p)$.  Now taking the lowest common multiple over all $f: Y\to B_{G}$, and
noting that $\tr(H,X)$ is defined as the l.c.m. of degrees over a set
containing all $\widetilde f: Y\to B_{H}$, we get $\tr(G,X)$ divides
$\tr(G,G/H)\tr(H,X)$.  $\square$

The $G$ space $G/H$ consisting of a single orbit is perhaps the simplest
case we should consider, also important in view of the proposition above.
In general, with some finiteness assumptions, we have $\tr(G,G/H)$ dividing
the Euler characteristic $\chi(G/H)$ (\cite{\bf{G2}}, Proposition 6.7b).  When
$G/H$ is finite we have $\tr(G,G/H)$ equalling $\chi(G/H) = [G:H]$, but
$\tr(G,G/H)$ does not equal $\chi(G/H)$ in general.
\proclaim{Lemma 9} Let $H$ be a subgroup of $G$ such that $G/H$ is finite
and discrete.  Then $\tr(G,G/H) = [G:H]$.
\endproclaim

\noindent Proof: We have $\tr(G,G/H)|\chi(G/H) = [G:H].$  Hence it will
suffice to find a pullback of the Borel construction $G/H\to B_{H} =
(G/H)_{G}\to B_{G}$ whose projection has degree $[G:H]$.  Let
$\sigma_{1},\dots,\sigma_{n}$ be loops representing classes in
$\pi_{1}(B_{G})$ mapping onto $\pi_{0}(G/H) =G/H$ in the homotopy exact
sequence
$$
1\to \pi_{1}(B_{H}) \to\pi_{1}(BG)\to G/H\to *.
$$

Let $r: M\to\overset i\to \vee S^{1}$ be a retraction of a closed, connected,
oriented manifold on a bouquet of $n$ circles.  Let $f$ be given by
$$
f: M\overset r\to\longrightarrow \overset i\to \vee S^{1}\overset
\vee_{\sigma}\to\longrightarrow B_{G}.
$$
Then $f^{*}(B_{H})$ is a closed, connected, oriented manifold, covering $M$
with fiber $G/H$.  Thus the projection $\rho: f^{*}(B_{H})\to M$ has degree
exactly $[G:H]$. $\square$

Combining Propositions 8 and 9 we obtain the following.
\proclaim{Corollary 10} If $G/H$ is finite then
$$
\tr(H,X)\big\vert\tr(G,X)\big\vert[G:H]\tr(H,X)
$$
\endproclaim
\proclaim{Theorem 11} Let $G$ be a finite group and let $X$ be a $G$ space.
Then
$$
\tr(G,X) = \Pi \ \tr(G_{p},X)
$$
where $G_{p}$ is a Sylow $p$-subgroup of $G$ and the product is over all
primes dividing the order of $G$.
\endproclaim

\noindent Proof: Since $\tr(G_{p},X)\big\vert \ |G_{p}|$ and $|G_{p}|$ is relatively
prime to $[G:G_{p}]$ we see that the divisions
$$
\tr(G_{p},X)\big\vert\tr(G,X)\big\vert[G:G_{p}]\tr(G_{p},X)
$$
implies that the $p$-primary factor of $\tr(G,X)$ is given by
$\tr(G_{p},X)$.

In the proof above, the relevant fact about a collection $G_{p}$ was that
their orders are relatively prime to each other and the order of $G$ is the
products of the orders of the $G_{p}$.  Hence if $\{H_{i}\}$ is a collection
of subgroups satisfying this condition, we have
$$
\tr(G,X) = \underset i\to\prod \tr(H_{i},X)
$$

Now we consider the consequences of Proposition 8 for a compact Lie group
$G$.  We will denote the connected component of the identity by $G_{0}$, a
maximal torus of $G_{0}$ by  $T$, the normalizer of $T$ in $G_{0}$ by
$N(T)$, and the Weyl group $N(T)/T$ by $W$.
\proclaim{Lemma 12} Let $X$ be a $G$-space ($G$ compact Lie).  Then
$\tr(G,X)$ divides\newline $[G:G_{0}]\tr(N(T),X)$. \  In particular if $G$ is
connected then $\tr(G,X)$ equals $\tr(N(T),X)$.
\endproclaim

\noindent Proof: Note that $\tr(G_{0}, G_{0}/N(T))\big\vert\chi(G_{0}/N(T))$ by
(\cite{\bf{G2}}, 6.7b).  But it is known that $\chi(G_{0}/N(T)) = 1$, hence
$\tr(G_{0}/N(T)) = 1$ thus Proposition 8 tells us that
$$
\tr(N(T),X)\big\vert\tr(G_{0},X)\big\vert\tr(N(T),X)
$$
Hence $\tr(N(T),X) = \tr(G_{0},X)$.  Now Corollary 10 gives us
$\tr(G,X)\big\vert [G:G_{0}]\tr(G_{0},X) = [G:G_{0}]\tr(N(T),X)$.

Now Lemma 12 gives us $\tr(N(T),X)\big\vert |W|\tr(T,X)$.  When $X$ is a compact
manifold $M$, we know from (\cite{\bf{G2}}, 6.11) that $\tr(T,M) = 1$, if the
action has a stationary point; or $\tr(T,M) = 0$, if the action does not
have a stationary point.  Thus we obtain
\proclaim{Corollary 13} Let $G$ be a compact Lie group acting on a compact
manifold $M$.  Then $\tr(G,M)$ is either zero, or it divides $[G,G_{0}]
|W|$ where $W$ is the Weyl group of $G_{0}$.
\endproclaim
\proclaim{Theorem 14} Let $G$ be a compact
Lie Group acting on a compact manifold $M$.  Let $W_{p}$ be
the Sylow $p$-subgroups of the Weyl group $W$, and let $\hat W_{p}$ is the
pre-image in $N(T)$ of $W_{p}$ in $W = N(T)/T$.  Then
$$
\tr(G,M) = \underset p\to\prod \tr(\hat W_{p},M)
$$
where the product is taken over all the $p$ dividing $|W|$.
\endproclaim

\noindent Proof: We may assume that $\tr(\hat W_{p},M)\not= 0$ because
otherwise $\tr(T,M) = 0$, which implies $\tr(G,M) = 0$; then both sides of
the equation  would be zero, proving the theorem.  So since $\tr(\hat
W_{p},M)\not= 0$, we have $\tr(T,M) = 1$, so from Proposition 8, $\tr(\hat
W_{p},M)\big\vert \ |\hat W_{p}|$.  Now $\tr(\hat W_{p},M)\big\vert\tr(N(T),M)\big\vert[W:
W_{p}]\tr(\hat W_{p},M)$ by (\cite{\bf{G2}}, 6.2.b) for the first division and by
Corollary 10 for the second division.  Hence $\tr(\hat W_{p},M)$ is a
common divisor of $|W_{p}|$ and $\tr(G,M) = \tr(N(T),M)$, and in fact is the
greatest common divisor since $\tr(G,M)/\tr(\hat W_{p},M)$ is not a
multiple of $p$.  Since $\tr(G,M)$ divides $|W| = \underset p\to\prod
W_{p}$, the theorem follows.

\noindent REMARKS: a) Theorem 14 can be generalized for a family
$\{H_{i}\}$ of subgroups of $W$ such that $\Pi|H_{i}| = |W|$ and the
$|H_{i}|$ are pairwise relatively prime.

b) The following example shows that $\tr(G,G/H)$ does not always equal
$\chi(G/H)$, in fact it shows that $\tr(G, G/T)\not= \chi(G/T) = |W|$.

(c) In fact $\tr(S^{3},M)$ is always zero or $1$ as $M$ is a compact
manifold.  This follows from the following example where $\tr(S^{3},
S^{3}/S^{1}) = 1$.  Hence $\tr(S^{3},M) = \tr(S^{1},M)$

\noindent EXAMPLE.  Let $G = S^{3}$ and $H = S^{1}$.  Then $\tr(S^{3},
S^{3}/S^{1}) = 1$ but $\chi(S^{3}/S^{1}) = 2$.

\noindent Proof: This can be seen by considering the fibration
$S^{2}\overset i\to\longrightarrow B_{S^{1}}\overset p\to\longrightarrow
B_{S^{3}}$.  This is a fibration which arises from the action of $S^{3}$ on
$S^{2} = S^{3}/S^{1}$ by the Borel construction.  Now
$H_{i}(B_{S^{3}})\cong 0$ for $0 < i < 4$, so $i^{*}$ is an isomorphism
$i^{*}: H^{2}(B_{S^{2}})\to H^{2}(S^{2})$.  Thus the generator of
$H^{2}(S^{2})$ is in the image of $i^{*}.$  Hence by theorem 1.5 of
\cite{\bf{G2}} we see that $\tr(S^{3}, S^{3}/S^{1}) = 1$.
\medskip
\noindent {\bf 6. Applications of the trace theorems}

An application of Theorem 11 gives an extension of a result of Browder and
Katz \cite{\bf{B-K}} to the nonfree case.  Recall that a nonsingular complex
projective variety $V$ of dimension $k$ is a closed, complex analytic
submanifold of a complex projective space $P^{n}$, given as the zero set of
a set of homogeneous polynomials (in $n+1$ variables).  The Chern class
of the canonical line bundle $c(\xi)$ generates the cohomology of $P^{n}$.
The hyperplane section $\alpha$ in $H^{2}(V;\bz)$ is the image of $c(\xi)$
under the map induced by inclusion $i^{*}: H^{2}(P^{n};\bz)\to
H^{2}(V;\bz)$.  We also have $\alpha^{k} = \deg(V) \ [\overline V]$ where
$[\overline V]$ is the fundamental class in cohomology.  The degree of the
variety $V, \ \deg(V)$ (not to be confused with the degree of a map),
depends on the particular embedding of $V$ in the projective space, since
the hyperplane section $\alpha$ depends on the embedding.
\proclaim{Proposition 15} Let $G$ be a finite group acting on a
nonsingular complex projective variety $V$.  Assume $H^{1}(V) = 0$ and the
hyperplane section $\alpha$ in $H^{2}(V)$ is left invariant by $G$.  Then
$$
\tr(G,V)^{2}\big\vert \ |G| \ (\deg V)^{2}.
$$
If all Sylow subgroups of $G$ are cyclic, then $\tr(G,V)$ divides $\deg V$.
\endproclaim

\noindent Proof: The $p$-primary part of $\tr(G,V)$ is exactly
$\tr(G_{p},V)$ for the Sylow $p$-subgroup $G_{p}$.  Hence we may assume that
$G$ is a $p$-group.  The action of $G$ is orientation preserving because
$\alpha$ and thus $\alpha^{k} =( \deg \ V)[\overline V]$ is invariant ($k =$
dimension of $V$).  In the cohomology Serre spectral sequence of the
Borel construction $V\to V_{G}\to B_{G}$ we have $E^{0,2}_{2} \cong
H^{2}(V)^{G}$ and $E_{2}^{*}\cong H^{*}(G)$.  These survive to $E_{3}$
since $H^{1}(V)$ is zero.  Let $d\alpha\epsilon H^{3}(G)$ be the image of
$\alpha$ under
$$
d: E_{3}^{0,2}\cong H^{2}(V)^{G}\to H^{3}(G)\cong E_{3}^{3,0}.
$$

Now there is a subgroup $K$ of $G$ such that $[G:K]^{2}$ divides $|G|$ and
$d \ \alpha$ maps to $0$ under the restriction map $H^{3}(G)\to H^{3}(K)$
(\cite{\bf{B-K}}, 2.1).  Restricting the action to $K$, from the functoriality
of the Serre spectral sequence we deduce that $\alpha$ survives to
$E_{\infty}$ (for $V\to V_{K}\to B_{K})$.  Hence $\alpha$ and $\alpha^{k} =
\deg V[\overline V]$ are in the image of $H^{2k}(V_{K})$ and thus
$\tr(K,V)$ divides $\deg V$.  Then $\tr(G,V)\big\vert[G:K]\tr(K,V)|[G:K]\deg V$,
and squaring the first and last terms, $\tr(G,V)^{2}\big\vert[G:K]^{2}(\deg
V)^{2}\big\vert \ |G|(\deg V)^{2}$.  When $G$ is cyclic, $H^{3}(G)$ is zero so $d
\alpha$ is zero.  We don't have to restrict to $K$, and we conclude
$\tr(G,V) \big\vert\deg V$.  $\square$

When $G$ acts freely, $\tr(G,V) = |G|$ and we have $|G|$ divides $(\deg
V)^{2}$ as in \cite{\bf{B-K}}.

Now the degree of $V$ is just the self intersection number of $W$ with
itself where $W = V\cap P^{k-1}$.  So if $G$ is acting on $V$ such that $W$
is left invariant, then by Theorem 7 we have $\tr(G,V)\big\vert\deg (V)$.  In fact
we can relax the condition that the inclusion $i: W\to V$ is a $G$-map.
Recall that $i:W\to V$ is a \underbar{homotopy $G$-map} if there is a
fibre bundle square
$$
\CD
W @>>i> V\\
@VVV @VVV\\
E @>>f> V_{G}\\
@VVV @VVV\\
B_{G} @>1>> B_{G}
\endCD
$$
which extends $i$ to a map $f$ over $1_{B_{G}}$.  Then we have shown the
following.
\proclaim{Proposition 16} Let $G$ be compact Lie group acting on a
non-singular projective variety $V$ so that the hyperplane section
inclusion $i: W\to V$ is a homotopy $G$-map.  Then $\tr(G,V)\big\vert\deg (V)$.
\endproclaim

Browder and Katz have an example (\cite{\bf{B-K}}) where $G$ acts freely on $V$
and satisfies the hypothesis of Proposition 15, so $|G| \ \big\vert \ (\deg
V)^{2}$.  But $|G|$ does not divide $\deg (V).$  Hence we can conclude that
in that example the inclusion of the hyperplane section is not a homotopy
$G$-map.

\noindent EXAMPLE: Suppose that $M= S^{2n}\times\dots\times S^{2n}$ 
is a product of $k$ spheres each of dimension $2n$.  
Suppose $G$ acts on $M$ leaving a $2n(k-1)$-dimensional submanifold $W$ 
invariant.  The Poincare dual of
$W$ is a class $\alpha\in H^{2n}(M)$ and so $\alpha = a_{1}\alpha_{1} +\dots
+ a_{k}\alpha_{k}$ where the $a_{i}$ are integers and $\alpha_{i}$ generate
$H^{2n}(S^{2n})$ of the $i$-th sphere in the product $M$.  Now
$\alpha^{2}_{i} = 0$ for all $i$ and $\alpha_{i}\alpha_{j} =
\alpha_{j}\alpha_{i}$ since the $\alpha_{i}$ are even dimensional.  Hence
$\alpha^{k} = k!(a_{1}\dots a_{k})\alpha_{1}\cup\dots\cup\alpha_{n} =
k!(a_{1}\dots a_{k})[\overline M]$.  Thus $\tr(G,M)$ divides $k!(a_{1}\dots a_{k})$.
More generally, if we have k $G$--invariant
 submanifolds ${W_{i}}$ (repetitions allowed), each having co-dimension
 2n, then
 their Poincare duals are of the form $\Sigma a_{ij}\alpha_{j}$. Their
 intersection number is the permanent of $(a_{ij})$, and it is divisible by
 $\tr(G,M)$.   

As a particular example, suppose $G$ is an elementary abelian $p$-group.
By (\cite{\bf{G2}}, 7.4) we have $\tr(G,M) =$ (the number of points in the
smallest orbit).  Thus (the number of points in the smallest
orbit) divides $k!(a_{1}\dots a_{k})$.  

In the odd dimensional situation, where $M = S^{2n+1}\times\dots\times
S^{2n+1}$ is a product of $k$ odd dimensional spheres each of dimension
$2n+1$, suppose we have a set $\{W_{i}\}$ of $k \ G$-invariant submanifolds
of co--dimension $2n+1$.  Then the Poincare dual of each manifold is given by
$\beta_{i}= \Sigma a_{ij}\alpha_{j}\in H^{2n+1}(M)$ where the $a_{ij}$ are integers
and each $\alpha_{j}$ generates $H^{2n+1}(S^{2n+1})$.  Now
$\alpha_{j}\cup\alpha_{k} = -\alpha_{k}\cup\alpha_{j}$.  Hence
$W_{1}\bullet\dots\bullet W_{k} = \det(a_{ij})$.  So if $G$ is an elementary
abelian $p$-group we see that (the number of points in the smallest orbit)
divides \ 
$\det(a_{ij})$.  In addition, suppose that $k = 2l$ and $W$ 
is an invariant oriented closed manifold of 
codimension $2(2n + 1)$. Suppose the Poincare dual of $W$ is given by 
$\beta = \Sigma a_{ij}\alpha_{i}\alpha_{j}$. Now $\beta^l = l!
\Pf(a_{ij})\alpha_1 \dots \alpha_k$. Thus $\tr(G,M)$ divides 
$l! \Pf(a_{ij})$. Here $\Pf(a_{ij})$ is the Pfaffian.

Theorem 7.4 of \cite{\bf{G2}} is greatly generalized by theorem 1.1 of 
\cite{\bf{B}}. In particular, Browder's
result implies that the trace of an action of
an elementary abelian p--group on a $G$--CW complex is equal to the
number of points in the smallest orbit. 
\vfill\eject
\Refs\nofrills{BIBLIOGRAPHY}
\ref
\no \bf{B}
\by W. Browder
\paper Actions of elementary abelian p--groups
\jour Topology 27,(1988) pp. 459-472
\endref
\ref
\no {\bf B-K}
\by W. Browder and N. Katz
\paper Free actions of finite groups on varieties I
\jour in Current Trends in Algebraic Topology, Vol. 2, part 2 (1982), pp. 23-33
\endref
\ref
\no \bf{G1}
\by D.H. Gottlieb
\paper Partial transfers 
\jour geometric applications of homotopy theory I (Proc. Evanston Conf. 1977), Lecture Notes in Math., Vol. 657,
Springer-Verlag, Berlin and New York
\yr 1977
\pages 255-266
\endref
\ref
\no \bf{G2}
\by D.H. Gottlieb
\paper The trace of an action and the degree of a map
\vol 293
\jour Transactions of the American Math. Soc. 
\yr 1986
\pages 381-410
\endref
\ref
\no \bf{O}
\by M. Ozaydin
\paper An integer invariant of a group action
\jour Thesis, Purdue University
\yr 1984
\endref
\vskip.75truein
\noindent Purdue University

\noindent University of Oklahoma
\enddocument








