THE LEFSCHETZ-HOPF THEOREM AND
AXIOMS FOR THE LEFSCHETZ NUMBER
Martin Arkowitz and Robert F. Brown
Dartmouth College, Hanover and University of California, Los Angeles
Abstract. The reduced Lefschetz number, that is, L(.) - 1 where L(.) deno*
*tes the
Lefschetz number, is proved to be the unique integer-valued function ~ on*
* selfmaps
of compact polyhedra which is constant on homotopy classes such that (1) *
*~(fg) =
~(gf), for f: X ! Y and g: Y ! X; (2) if (f1, f2, f3) is a map of a cofib*
*er sequence
into itself, then ~(f1) = ~(f1) + ~(f3); (3) ~(f) = -(deg(p1fe1) + . .+.d*
*eg(pkfek)),
where f is a selfmap of a wedge of k circles, er is the inclusion of a ci*
*rcle into the rth
summand and pr is the projection onto the rth summand. If f: X ! X is a s*
*elfmap
of a polyhedron and I(f) is the fixed point index of f on all of X, then *
*we show
that I(.) - 1 satisfies the above axioms. This gives a new proof of the N*
*ormalization
Theorem: If f: X ! X is a selfmap of a polyhedron, then I(f) equals the L*
*efschetz
number L(f) of f. This result is equivalent to the Lefschetz-Hopf Theore*
*m: If
f: X ! X is a selfmap of a finite simplicial complex with a finite number*
* of fixed
points, each lying in a maximal simplex, then the Lefschetz number of f i*
*s the sum
of the indices of all the fixed points of f.
1. Introduction.
Let X be a finite polyhedron and denote by He*(X) its reduced homology with
rational coefficients. Then the reduced Euler characteristic of X, denoted by ~*
*Ø(X),
is defined by X
~Ø(X) = (-1)j dim eHj(X).
j
Clearly, ~Ø(X) is just the Euler characteristic minus one. In 1962, Watts [13] *
*char-
acterized the reduced Euler characteristic as follows: Let ffl be a function fr*
*om the
set of finite polyhedra with base points to the integers such that (i) ffl(S0) *
*= 1,
where S0 is the 0-sphere, and (ii) ffl(X) = ffl(A) + ffl(X=A), where A a subpol*
*yhedron
of X. Then ffl(X) = ~Ø(X).
Let C be the collection of spaces X of the homotopy type of a finite, connec*
*ted
CW-complex. If X 2 C, we do not assume that X has a base point except when
X is a sphere or a wedge of spheres. It is not assumed that maps between spaces
with base points are based. A map f: X ! X, where X 2 C, induces trivial
homomorphisms fj: Hj(X) ! Hj(X) of rational homology vector spaces for all
j > dim X. The Lefschetz number L(f) of f is defined by
X
L(f) = (-1)jT r fj,
j
______________
1991 Mathematics Subject Classification. 55M20.
Typeset by AM S-T*
*EX
1
2 MARTIN ARKOWITZ AND ROBERT F. BROWN
where T r denotes the trace. The reduced Lefschetz number eLis given by eL(f) =
L(f) - 1 or, equivalently, by considering the rational, reduced homology homomo*
*r-
phism induced by f.
Since eL(id) = ~Ø(X), where id: X ! X is the identity map, Watts's Theorem
suggests an axiomatization for the reduced Lefschetz number which we state below
as Theorem 1.1. W
For k 1,Wdenote by kSn the wedge of k copies of the n-sphere Sn , n 1.
If we write kSn as Sn1_ Sn2_ . ._.Snk, where Snj= Sn , then we have inclusions
W k Wk
ej: Snj! Sn into the j-th summand and projections pj: Sn ! Snjonto the
Wk W k
j-th summand, for j = 1, . .,.k. If f: Sn ! Sn is a map, then fj: Snj! Snj
denotes the composition pjfej. The degree of a map f: Sn ! Sn is denoted by
deg (f).
We characterize the reduced Lefschetz number as follows.
Theorem 1.1. The reduced Lefschetz number eLis the unique function ~ from the
set of self-maps of spaces in C to the integers that satisfies the following co*
*nditions:
1. (Homotopy Axiom) If f, g: X ! X are homotopic maps, then ~(f) = ~(g).
2. (Cofibration Axiom) If A is a subpolyhedron of X, A ! X ! X=A is the
resulting cofiber sequence and there exists a commutative diagram
A ----! X ----! X=A
? ? ?
f0?y f?y f~?y
A ----! X ----! X=A,
then ~(f) = ~(f0) + ~(f~).
3. (Commutativity Axiom) If f: X ! Y and g: Y ! X are maps, then ~(gf) =
~(fg). W W
4. (Wedge of Circles Axiom) If f: k S1 ! kS1 is a map, k 1, then
~(f) = -(deg (f1) + . .+.deg(fk)),
where fj = pjfej.
In an unpublished dissertation [10], Hoang extended Watts's axioms to char-
acterize the reduced Lefschetz number for basepoint-preserving self-maps of fin*
*ite
polyhedra. His list of axioms is different from, but similar to, those in Theo*
*rem
1.1.
One of the classical results of fixed point theory is
Theorem 1.2 (Lefschetz-Hopf). If f: X ! X is a map of a finite polyhedron
with a finite set of fixed points, each of which lies in a maximal simplex of X*
*, then
L(f) is the sum of the indices of all the fixed points of f.
The history of this result is described in [3], see also [8, p. 458]. A pro*
*of that
depends on a delicate argument due to Dold [5] can be found in [2] and, in a mo*
*re
condensed form, in [4]. In an appendix to his dissertation [12], D. McCord outl*
*ined
a possibly more direct argument, but no details were published. The book of Gra*
*nas
LEFSCHETZ NUMBER 3
and Dugundji [8, pp. 441 - 450] presents an argument based on classical techniq*
*ues
of Hopf [11]. We use the characterization of the reduced Lefschetz number in
Theorem 1.1 to prove the Lefschetz-Hopf theorem in a quite natural manner by
showing that the fixed point index satisfies the axioms of Theorem 1.1. That is*
*, we
prove
Theorem 1.3 (Normalization Property). If f: X ! X is any map of a finite
polyhedron, then L(f) = i(X, f, X), the fixed point index of f on all of X.
The Lefschetz-Hopf Theorem follows from the Normalization Property by the
Additivity Property of the fixed point index. In fact these two statements are
equivalent. The Hopf Construction [2, p. 117] implies that a map f from a fini*
*te
polyhedron to itself is homotopic to a map that satisfies the hypotheses of the
Lefschetz-Hopf theorem. Thus the Homotopy and Additivity Properties of the fixed
point index imply that the Normalization Property follows from the Lefschetz-Ho*
*pf
Theorem.
2. Lefschetz numbers and exact sequences.
In this section, all vector spaces are over a fixed field F , which will not*
* be
mentioned, and are finite dimensional. A graded vector space V = {Vn} will alwa*
*ys
have the following properties: (1) each Vn is finite dimensional and (2) Vn = 0*
* for
n < 0 and for n > N, for some non-negative integer N. A map f: V ! W of graded
vector spaces V = {Vn} and W = {Wn} is a sequence of linear transformations
fn: Vn ! Wn. For a map f: V ! V , the Lefschetz number is defined by
X
L(f) = (-1)nT r fn.
n
The proof of the following lemma is straightforward, and hence omitted.
Lemma 2.1. Given a map of short exact sequences of vector spaces
0 ----! U ----! V ----! W ----! 0
? ? ?
f?y g?y h?y
0 ----! U ----! V ----! W ----! 0,
then T r g = T r f + T r h.
Theorem 2.2. Let A, B and C be graded vector spaces with maps ff: A ! B, fi: B*
* !
C and selfmaps f: A ! A, g: B ! B and h: C ! C. If for every n, there is a line*
*ar
transformation @n: Cn ! An-1 such that the following diagram is commutative and
has exact rows:
0 - ---! AN --ffN--!BN --fiN--!CN --@N--! AN-1 -ffN-1---!. . .
? ? ? ?
fN?y gN?y hN ?y fN-1?y
0 - ---! AN --ffN--!BN --fiN--!CN --@N--! AN-1 -ffN-1---!. . .
4 MARTIN ARKOWITZ AND ROBERT F. BROWN
. . .-@1---!A0 --ff0--!B0 ---fi0-!C0 ----! 0
?? ?? ??
f0y g0y h0y
. . .-@1---!A0 --ff0--!B0 ---fi0-!C0 ----! 0,
then
L(g) = L(f) + L(h).
Proof. Let Im denote the image of a linear transformation and consider the com-
mutative diagram
0 ----! Im fin ----! Cn - ---! Im @n ----! 0
? ? ?
hn|Im fin?y hn?y fn-1|Im @n?y
0 ----! Im fin ----! Cn - ---! Im @n ----! 0.
By Lemma 2.1, T r(hn) = T r(hn|Im fin) + T r(fn-1 |Im @n). Similarly, the commu-
tative diagram
0 ----! Im @n ----! An-1 ----! Im ffn-1 ----! 0
? ? ?
fn-1|Im @n?y fn-1?y gn-1|Im ffn-1?y
0 ----! Im @n ----! An-1 ----! Im ffn-1 ----! 0
yields T r(fn-1 |Im @n) = T r(fn-1 ) - T r(gn-1 |Im ffn-1 ). Therefore
T r(hn) = T r(hn|Im fin) + T r(fn-1 ) - T r(gn-1 |Im ffn-1 ).
Now consider
0 - ---! Im ffn-1 ----! Bn-1 ----! Im fin-1 ----! 0
? ? ?
gn-1|Im ffn-1?y gn-1?y hn-1|Im fin-1?y
0 - ---! Im ffn-1 ----! Bn-1 ----! Im fin-1 ----! 0,
so T r(gn-1 |Im ffn-1 ) = T r(gn-1 ) - T r(hn-1 |Im fin-1 ). Putting this all t*
*ogether,
we obtain
T r(hn) = T r(hn|Im fin) + T r(fn-1 ) - T r(gn-1 ) + T r(hn-1 |Im fin-1 ).
We next look at the left end of the original diagram and get
0 = T r(hN+1 ) = T r(fN ) - T r(gN ) + T r(hN |Im fiN )
and at the right end which gives
T r(h1) = T r(h1|Im fi1) + T r(f0) - T r(g0) + T r(h0).
LEFSCHETZ NUMBER 5
A simple calculation now yields
NX N+1X
(-1)nT r(hn) = = (-1)n(T r(hn|Im fin) + T r(fn-1 ) - T r(gn-1 )
n=0 n=0
+ T r(hn-1 |Im fin-1 ))
XN XN
= - (-1)nT r(fn) + = (-1)nT r(gn).
n=0 n=0
Therefore L(h) = -L(f) + L(g).
We next give some simple consequences of Theorem 2.2.
If f: (X, A) ! (X, A) is a selfmap of a pair, where X, A 2 C, then f determi*
*nes
fX : X ! X and fA : A ! A. The map f induces homomorphisms fj: Hj(X, A) !
Hj(X, A) of relative homology with coefficients in F . The relative Lefschetz n*
*umber
L(f; X, A) is defined by
X
L(f; X, A) = (-1)jT rfj.
j
Applying Theorem 2.2 to the homology exact sequence of the pair (X, A), we
obtain
Corollary 2.3. If f: (X, A) ! (X, A) is a map of pairs, where X, A 2 C, then
L(f; X, A) = L(fX ) - L(fA ).
This result was obtained by Bowszyc [1].
Corollary 2.4. Suppose X = P [ Q where X, P, Q 2 C and (X; P, Q) is an proper
triad [6, p. 34]. If f: X ! X is a map such that f(P ) P and f(Q) Q then, f*
*or
fP , fQ and fP\Q the restrictions of f to P, Q and P \ Q respectively, we have
L(f) = L(fP ) + L(fQ ) - L(fP\Q ).
Proof. The map f and its restrictions induce a map of the Mayer-Vietoris homolo*
*gy
sequence [6, p. 39] to itself so the result follows from Theorem 2.2.
A similar result was obtained by Ferrario [7, Theorem 3.2.1].
Our final consequence of Theorem 2.2 will be used in the characterization of*
* the
reduced Lefschetz number.
Corollary 2.5. If A is a subpolyhedron of X, A ! X ! X=A is the resulting
cofiber sequence of spaces in C and there exists a commutative diagram
A ----! X ----! X=A
? ? ?
f0?y f?y f~?y
A ----! X ----! X=A,
6 MARTIN ARKOWITZ AND ROBERT F. BROWN
then
L(f) = L(f0) + L(f~) - 1.
Proof. We apply Theorem 2.2 to the homology cofiber sequence. The `minus one'
on the right hand side arises because that sequence ends with
! H0(A) ! H0(X) ! ~H0(X=A) ! 0.
3. Characterization of the Lefschetz number.
Throughout this section, all spaces are assumed to lie in C.
We let ~ be a function from the set of self-maps of spaces in C to the integ*
*ers
that satisfies the Homotopy Axiom, Cofibration Axiom, Commutativity Axiom and
Wedge of Circles Axiom of Theorem 1.1 as stated in the Introduction.
We draw a few simple consequences of these axioms. From the Commutativity
Axiom, we obtain
Lemma 3.1. If f: X ! X is a map and h: X ! Y is a homotopy equivalence with
homotopy inverse k: Y ! X, then ~(f) = ~(hfk).
Lemma 3.2. If f: X ! X is homotopic to a constant map, then ~(f) = 0.
Proof. Let * be a one-point space and *: * ! * the unique map. From the map of
cofiber sequences
* ----! * ----! *
? ? ?
*?y *?y *?y
* ----! * ----! *
and the Cofibration Axiom, we have ~(*) = ~(*) + ~(*), and therefore ~(*) = 0.
Write any constant map c: X ! X as c(x) = * for some * 2 X, let e: * ! X be
inclusion and p: X ! * projection. Then c = ep and pe = *, and so ~(c) = 0 by
the Commutativity Axiom. The lemma follows from the Homotopy Axiom.
If X is a based space with base point *, i.e., a sphere or wedge of spheres,*
* then
the cone and suspension of X are defined by CX = X x I=(X x 1 [ * x I) and
X = CX=(X x 0), respectively.
Lemma 3.3. If X is a based space, f: X ! X is a based map and f: X ! X
is the suspension of f, then ~( f) = -~(f).
Proof. Consider the maps of cofiber sequences
X ----! CX ----! X
? ? ?
f?y Cf?y f?y
X ----! CX ----! X.
Since CX is contractible, Cf is homotopic to a constant map. Therefore, by Lemma
3.2 and the Cofibration Axiom,
0 = ~(Cf) = ~( f) + ~(f).
LEFSCHETZ NUMBER 7
Wk W k
Lemma 3.4. For any k 1 and n 1, if f: Sn ! Sn is a map, then
n(deg (f ) + . .+.deg(f )),
~(f) = (-1) 1 k
W k Wk
where er: Sn ! Sn and pr: Sn ! Sn for r = 1, . .,.k are the inclusions and
projections, respectively, and fr = prfer.
Proof. The proof is by induction on the dimension n of the spheres.W TheWcase
n = 1 is the Wedge of Circles Axiom.W If nW 2, then the map f: k Sn ! kSn
is homotopic to aWbased mapWf0: k Sn ! k Sn . Then f0 is homotopic to g,
for some map g: k Sn-1 ! kSn-1 . Note that if gj: Sn-1j ! Sn-1j, then gj is
homotopic to fj: Snj! Snj. Therefore by Lemma 3.3 and the induction hypothesis,
~(f) = ~(f0) = -~(g) = -(-1)n-1 ((deg (g1) + . .+.deg(gk))
= (-1)n(deg (f1) + . .+.deg(fk)).
Proof of Theorem 1.1.
Since ~L(f) = L(f)-1, Corollary 2.5 implies that ~Lsatisfies the Cofibration Ax*
*iom.
We nextLshow that ~LsatisfiesWthe Wedge of Circles Axiom. There is an isomorphi*
*sm
`: k H1(S1) ! H1( k S1) defined by `(x1,W. .,.xk)L= e1*(x1) + . . .+ ek*(xk),
where xi 2 H1(S1). The inverse `-1 : H1( k S1) ! k H1(S1) is given by `-1W(y*
*) =
(p1*(y), . .,.pk*(y)). If u 2 H1(S1) is a generator, thenWa basis forWH1( k S*
*1) is
e1*(u), . .,.ek*(u). By calculating the trace of f*: H1( k S1) ! H1( k S1) w*
*ith
respect to this basis, we obtain ~L(f) = -(deg (f1) + . .+.deg(fk)). The remain*
*ing
axioms are obviously satisfied by ~L. Thus ~Lsatisfies the axioms of Theorem 1.*
*1.
Now suppose ~ is a function from the self-maps of spaces in C to the integers
that satisfies the axioms. We regard X as a connected, finite CW-complex and
proceed by induction on the dimension of X. If X is 1-dimensional, then it is t*
*he
homotopyWtype of a wedge of circles. By Lemma 3.1, we can regard f as a self-map
of kS1, and so the Wedge of Circles Axiom gives
~(f) = -(deg (f1) + . .+.deg(fk)) = ~L(f).
Now suppose that X is n-dimensional and let Xn-1 denote the (n - 1)-skeleton of
X. Then f is homotopic to a cellular map g: X ! X by the Cellular Approxima-
tion Theorem [9, Theorem 4.8, p. 349]. Thus g(Xn-1 ) Xn-1 , and so we have a
commutative diagram
W k
Xn-1 ----! X ----! X=Xn-1 = Sn
? ? ?
g0?y g?y ~g?y
W k
Xn-1 ----! X ----! X=Xn-1 = Sn .
Then, by the Cofibration Axiom, ~(g) = ~(g0) + ~(~g). Lemma 3.4 implies that
~(~g) = ~L(~g) so, applying the induction hypothesis to g0, we have ~(g) = ~L(g*
*0) +
L~(~g). Since we have seen that the reduced Lefschetz number satisfies the Cofi*
*bration
Axiom, we conclude that ~(g) = ~L(g). By the Homotopy Axiom, ~(f) = ~L(f).
8 MARTIN ARKOWITZ AND ROBERT F. BROWN
4. The Normalization Property.
Let X be a finite polyhedron and f: X ! X a map. Denote by I(f) the fixed
point index of f on all of X, that is, I(f) = i(X, f, X) in the notation of [2]*
* and
let ~I(f) = I(f) - 1.
In this section we prove Theorem 1.3 by showing that, with rational coeffici*
*ents,
I(f) = L(f).
Proof of Theorem 1.3.
We will prove that ~Isatisfies the axioms and therefore, by Theorem 1.1, ~I(f) =
L~(f). The Homotopy and Commutativity Axioms are well-known properties of the
fixed point index (see [2, pp. 59 and 62]).
To show that ~Isatisfies the Cofibration Axiom, it suffices to consider A a *
*sub-
polyhedron of X and f(A) A. Let f0: A ! A denote the restriction of f and
f~: X=A ! X=A the map induced on quotient spaces. Let r: U ! A be a deforma-
tion retraction of a neighborhood of A in X onto A and let L be a subpolyhedron
of a barycentric subdivision of X such that A int L L U. By the Homotopy
Extension Theorem there is a homotopy H: X x I ! X such that H(x, 0) = f(x)
for all x 2 X, H(a, t) = f(a) for all a 2 A and H(x, 1) = fr(x) for all x 2 L. *
*If we
set g(x) = H(x, 1) then, since there are no fixed points of g on L-A, the Addit*
*ivity
Property implies that
(4.1) I(g) = i(X, g, int L) + i(X, g, X - L).
We discuss each summand of (4.1) separately. We begin with i(X, g, int L).
Since g(L) A L, it follows from the definition of the index ([2, p. 56]) th*
*at
i(X, g, int L) = i(L, g, int L). Moreover, i(L, g, int L) = i(L, g, L) since t*
*here are
no fixed points on L - int L (the Excision Property of the index). Let e: A ! L*
* be
inclusion then, by the Commutativity Property [2, p. 62] we have
i(L, g, L) = i(L, eg, L) = i(A, ge, A) = I(f0)
because f(a) = g(a) for all a 2 A.
Next we consider the summand i(X, g, X - L) of (4.1). Let ß: X ! X=A be
the quotient map, set ß(A) = * and note that ß-1 (*) = A. If ~g: X=A ! X=A
is induced by g, the restriction of ~gto the neighborhood ß(int L) of * in X=A *
*is
constant, so i(X=A, ~g, ß(int L)) = 1. If we denote the set of fixed points of *
*~gwith
* deleted by F ix*~g, then F ix*~gis in the open subset X=A - ß(L) of X=A. Let W
be an open subset of X=A such that F ix*~g W X=A - ß(L) with the property
~g(W ) \ ß(L) = ;. By the Additivity Property we have
I(~g) = i(X=A, ~g, ß(int L)) + i(X=A, ~g, W ) = 1 + i(X=A, ~g, W ).
Now, identifying X - L with the corresponding subset ß(X - L) of X=A and
identifying the restrictions of ~gand g to those subsets, we have i(X=A, ~g, W *
*) =
i(X, g, ß-1 (W )). The Excision Property of the index implies that i(X, g, ß-1 *
*(W )) =
i(X, g, X - L). Thus we have determined the second summand of (4.1): i(X, g, X -
L) = I(~g) - 1.
LEFSCHETZ NUMBER 9
Therefore from (4.1) we obtain I(g) = I(f0) + I(~g) - 1. The Homotopy Proper*
*ty
then tells us that
I(f) = I(f0) + I(f~) - 1
since f is homotopic to g and ~fis homotopic to ~g. We conclude that ~Isatisfie*
*s the
Cofibration Axiom. W
It remains to verify the Wedge of Circles Axiom. Let X = kS1 = S11_ . ._.S*
*1k
be a wedge of circles with basepoint * and f: X ! X a map. We first verify
the axiom in the case k = 1. We have f: S1 ! S1 and we denote its degree by
deg(f) = d. We regard S1 C, the complex numbers. Then f is homotopic to gd,
where gd(z) = zd has |d - 1| fixed points for d 6= 1. The fixed point index of *
*gd in a
neighborhood of a fixed point that contains no other fixed point of gd is -1 if*
* d 2
and is 1 if d 0. Since g1 is homotopic to a map without fixed points, we see *
*that
I(gd) = -d + 1 for all integers d. We have shown that I(f) = -deg(f) + 1.
Now suppose k 2. If f(*) = * then, by the Homotopy Extension Theorem, f
is homotopic to a map which does not fix *. Thus we may assume, without loss
of generality, that f(*) 2 S11- {*}. Let V be a neighborhood of f(*) in S11- {*
**}
such that there exists a neighborhood U of * in X disjoint from V with f(U~) *
*V .
Since ~Ucontains no fixed point of f and the open subsets S1j- ~Uof X are disjo*
*int,
the Additivity Property implies
Xk
(4.2) I(f) = i(X, f, S11- ~U) + i(X, f, S1j- ~U).
j=2
The Additivity Property also implies that
(4.3) I(fj) = i(S1j, fj, S1j- ~U) + i(S1j, fj, S1j\ U).
___
There is a neighborhood Wj of (F ix f) \ S1jin S1jsuch that f(W j) S1j. Thus
fj(x) = f(x) for x 2 Wj and therefore, by the Excision Property,
__ 1 1 __
(4.4) i(S1j, fj, S1j- U ) = i(Sj, fj, Wj) = i(X, f, Wj) = i(X, f, Sj - U ).
__ __
Since_f(U_) S11, then f1(x) = f(x) for all x 2 U \ S11. There are_no fixed*
* points
of f in U , so i(S11, f1, S11\ U) = 0 and thus I(f1) = i(X, f, S11- U ) by (4.3*
*) and
(4.4).
For j 2,_the_fact that fj(U) = * gives us i(S1j, fj, S1j\ U) = 1 so I(fj) =
i(X, f, S1j- U ) + 1 by (4.3) and (4.4). Since fj: S1j ! S1j, the k = 1 case of
the argument tells_us that I(fj) = -deg(fj) + 1 for j = 1, 2, . .k.. In partic*
*ular,_
i(X, f, S11- U ) = -deg(f1) + 1 whereas, for j 2, we have i(X, f, S1j- U ) =
-deg(fj). Therefore, by (4.2),
__ Xk 1 __ Xk
I(f) = i(X, f, S11- U ) + i(X, f, Sj - U ) = - deg(fj) + 1.
j=2 j=1
This completes the proof of Theorem 1.3.
10 MARTIN ARKOWITZ AND ROBERT F. BROWN
References
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*Sci. 16 (1968),
845 - 850.
[2] R. Brown, The Lefschetz Fixed Point Theorem, Scott, Foresman, 1971.
[3] R. Brown, Fixed Point Theory, in History of Topology, Elsevier, 1999, 271 -*
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Hanover, NH 03755-1890, USA
E-mail address: Martin.A.Arkowitz@Dartmouth.edu
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E-mail address: rfb@math.ucla.edu,