Characteristic Classes, Lattice Points,
and Euler-MacLaurin Formulae
JULIUS L. SHANESON
Department of Mathematics,
University of Pennsylvania,
Philadelphia, PA, USA
1 Introduction
In this paper I would like to survey and strengthen some of the connections between topology and other areas of mathematics, pure and applied, and areas beyond mathematics as well. One type of problem that appears in many areas of
mathematics, applied mathematics, physics, economics, and probably other sciences concerns the summing of the values of a function over a discrete set of
points in a prescribed region of space. Slightly more precisely, let L be a lattice
in Euclidean space R n , such as the set of points with integral co-ordinates. Let
S C Mn be a region, and let / be a (reasonable) function. Then how can one write
(or approximate) the sum J2xeLns f(x) m terms of quantities that are continuous
or continuously computed from / and the (static) geometry of the region S? For
example, if / = 1, one is asking for a computation of # ( L D S), the number of
lattice points in S, in terms of its geometry.
The classical Euler-MacLaurin formula provides a solution to this problem
for S = [a, b] C M1, a and b integers:
b
fb
1
°°
R
r
£ / ( * ) = / /(s)+ 5 [/(a)+/(fc)]+D-^
a
Ja
i
(°)
2
for the case when / is a polynomial (or in general when the remainder term goes
to zero). Here Br is the rth Bernoulli number.
In dimension two, for A C M2 a convex polygon with integral vertices (a
"lattice polygon"), and / = 1, Pick's theorem says
# ( A n L) = vol(A) + "Perimeter" A + 1.
(1.2)
"Perimeter" is the sum of normalized lengths of the edges, where the normalization is to require that adjacent lattice points on each edge have distance one. In
dimension three, Mordell gave a formula, involving Dedekind sums, for counting
the lattice points of a certain lattice tetrahedron. As one consequence of his work
of singular toric varieties, Pommersheim obtained the corresponding result for all
tetrahedra.
Proceedings of the International Congress
of Mathematicians, Zürich, Switzerland 1994
© Birkhäuser Verlag, Basel, Switzerland 1995
Characteristic Classes, Lattice Points, ...
613
For certain special polytopes, Khovanski and Pukhlikov obtained an interesting result with a somewhat different flavor, by considering infinitesimal parallel
motions of the codimension one faces. Assume that minimal lattice vectors parallel to the edges meeting any given vertex of the convex lattice polytope A form
a lattice basis. (This is equivalent to the smoothness of the toric avriety X A ; see
Section 5.) Let h = (hi,... , hk) parametrize parallel motion of the k codimension
one faces of A, and let A/t be the resulting convex polytope. Let
^i
Then
YI
(L3)
^o = nr^==r-
f(x)=T(d/dhi,...,d/dhk)
[
f\
.
(1.4)
_ Cappell and I found a complete and explicit solution for convex lattice polytopes in any dimension. Our results have the following form:
1.5 There exist infinite order linear partial differential operators PE, E any face
of A, with constant rational coefficients, such that for any f,
XELnA
E<A **E
Here VE is Lesbesgue measure, normalized so that the fundamental domain
in the intersection of L with the plane of E has volume one, and the operators PE
are unaffected by parallel translation of facets ( "fan invariance" ). The collection
{PE I E < A} will be called an Euler-MacLaurin expansion for A.
Cappell and I give an explicit description of the operators PE in terms of the
geometry of A ; see Section 6. The coefficients of the operators reflect the geometry
of A in the same way that Bernoulli numbers reflect the geometry of intervals.
The constant terms in the operators PE yield formulae for the Hilbert-Ehr hart
polynomial PA(^) = # ( £ H A;A) in all dimensions, extending Pick's theorem for
dimension two and calculations of Mordell and Pommersheim's calculation dim
A = 3. Partial expansions with remainder will be discussed elsewhere.
Topologists seek to classify spaces up to homeomorphism or other suitable
notions of equivalence (e.g. piecewise linear homeomorphism or diffeomorphism),
usually by means of distinguishing invariants. For example, let
e(X) = ^ ( - 1 ) * #(cellsof dimensioni),
(1.6)
the Euler-Poincaré characteristic. Then e(X) turns out to be a topological invariant, and a compact orientable surface is determined by this invariant.
This classification endeavor, often called "surgery theory", had many successes in the study of manifolds, starting with the work of Kervaire and Milnor,
and then of Browder, Novikov, Sullivan, Wall, and many others. The L-classes
Li(M) G # n - 4 i ( M n ; Q ) , which will be described in Section 2, play a key role
throughout this theory, as illustrated by the celebrated early result of BrowderNovikov.
1.7 The diffeomorphism type of a closed smooth manifold of dimension at least
five is determined up to a finite number of jjossibilities by its homotopy type and
its L-classes.
614
Julius L. Shaneson
'
Their results also show:
1.8 In a homotopy type every possible L-class can arise, at least up to a finite
index and satisfying the Hirzebruch signature theorem [H].
Gradually, however, topologists were forced to ventine more and more outside the world of manifolds. For example, motivated by questions in transformation
groups, Browder and Quinn extended the manifold theory to the category of stratified spaces and isovariant transverse maps. In our work exhibiting topologically
conjugate but linearly distinct real representations of finite groups Cappell and
1 were implicitly working with a stratified situation, and this became explicit in
our further work with Steinberger, West, and Weinberger. Most relevant for the
present discussion, nontrivial stratifications also arose in my earlier work with
Cappell on piecewise linear submanifolds; see Section 3.
The introduction of intersection homology by Goresky and MacPherson gave
an enormous impetus to the study of stratified spaces. In particular they and
Cheeger defined L-classes for stratified spaces with even codimension strata. For
example, Cappell and Weinberger, proved the analogue of 1.7 for stratified spaces
with even codimension strata. In this paper I will describe how Cappell and I
studied these classes and how we applied our methods and results to invariants
and classes of algebraic geometry and to the problems mentioned above.
2 L-classes
Let £ be a vector bundle over a reasonable space (e.g. a finite complex) X. Factor
formally the total Chern class of the complexification of £,
1 + ci(£ ® C) + ... = (1 +
TI)
. . . (1 +
7 f c ).
(2.1)
Then
actually lies in H*(X; Q) and defines the Thorn-Hirzebruch L-classes of £. In fact,
these are universal polynomials L — £(pi,... ,pn,...),
where Pi(£) = (—l)*C2i(£ <8>
C). Recall
1
OO
j-,
If M is a manifold, we set C(M) = C(TM), TM the tangent bundle. For example
£(CPn) = (l-hc2)n+1,
(2.4)
where c is a generator of H2(CPn).
All this is beautifully explained in [H]. For M oriented, the signature a(M)
is the index of the intersection pairing on i?2/c(Af) and the famous Hirzebruch
signature theorem asserts:
a(M) = [ C(M).
JM
(2.5)
Characteristic Classes, Lattice Points, ...
615
For manifolds let L(M) = C(M) n [Af] be the Poincaré dual of C(M). Just
for a moment, let M be a complex 71-manifold whose stable tangent bundle splits
as a sum of line bundles with Chern classes ai,..., a^. Let Td(M) be the Poincaré
dual of the Todd class of M. Let M^...^ be a submanifold of M dual to G^1 ... a%£
(such a submanifold always exists). Let
T(M) = [M] + 2Td 2 n _ 2 (M) + 4Td 2 n _ 4 (M) + . . . .
Then (omitting inclusion induced maps from the notation)
T{M)=
£
L(Maii...aih).
(2.6)
7i<...<»fr
This result is a homological restatement of something Hirzebruch used to
derive the Riemann-Roch theorem from the signature theorem.
Let V1 C V x Mn_?: C Mn be a submanifold with trivial normal bundle, and
let ipH [M] = [V] E Hi(M) (we omit inclusion induced maps from the notation).
Then (2.5) implies
a{V) = j
<p.
(2.7)
JL(M)
Thorn showed that (2.7) can be used as a definition of L(M) G H*(M;Q) for M
a piecewise linear manifold, and for suitable spaces, e.g. algebraic varieties, (2.7)
can be viewed as the definition of an invariant L(X) — the Goresky-MacPhersonCheeger L-class — extending the above Thom-Hirzebruch class to stratified spaces.
For more sophisticated definitions, see [Si], [CSW].
One should note that these classes are homeomorphism invariants. For manifolds, this was proven by Novikov. For stratified spaces this essentially is a consequence of the local character of the sheaf-theoretic proof of Gorensky-MacPherson
of homeomorphism invariance of intersection homology (cf. [CSW]).
3 Mapping theorems for L-classes
This section will describe some results of Cappell and myself relating L(X) and
L(Y), when X and Y are stratified spaces and f'.X —> Y is a map. When X
and Y are manifolds and / a bundle map or a smooth embedding, there are many
classical results [H], [CHS], [At] but such maps are actually fairly rare.
One could work with various notions of stratified space. Although Quinn's
homotopy-theoretic definition is in many ways the most flexible and advantageous,
here I will stay closer to geometry. A stratification of a space is a filtration
cj) = X-i C XQ C ... C Xn = X such that, first of all, the open i-stratum
Xi — Xi-i is an i-manifold (or empty). Let Vx be the set of components of open
strata. Then, in addition, it is assumed that near x E V G Vx,X looks like
D% x cone(Ly), where i = dim V and Ly is a stratified space of dimension n — i — 1,
with x = {0} x {cone point}. It will also be assumed that n — dim V is even. For
example, a complex algebraic variety has such a stratification (even satisfying
Whitney's conditions), with the additional property that V is a subvariety; see
[GM3] for references.
616
Julius L. Shaneson
A (surjective) map / : X —> Y is stratified if for each V G V y , / - 1 ^ is a
union of components of open strata of X, and / | f~1 V is a locally trivial map. (In
Thorn's theory one assumes that the derivative of / is surjective on each stratum
in the inverse image and then uses the first isotopy lemma to prove local triviality.)
Again, algebraic morphisms are stratified.
Given / stratified, for each V in Vy, with dim V < climY, we define
Ny = NVJ = / - ^ c o n e L y ) U c o i i e t f " 1 ^ ) ) ,
a stratified space with even codimension strata. For dim y = dim y , let JVV =
f~1(v)i some v in V.
3.1 Assume each V G Vy is simply connected. Then f*(L(X))
=
}^L(V)a(Ny).
The terms on the right in (3.1), for singular strata, can be viewed as describing
in terms of the singular structure the difference between f*(L(X)) and what we
would expect — (L(Y)a(f~1 (pt)) — from the bundle case (see [CHS]).
Without simple connectivity, the terms on the right become L-classes of V
with coefficients in the nonsingular intersection pairing on IH™ (Ny), dim Ny =
2c, viewed as affording a form-preserving representation of TTIV. This situation can
sometimes be analyzed as in [At] for bundles, but a completely general description
in terms of some kind of product of characteristic classes of the stratum and of
the representation remains to be found.
Actually, the interest of Cappell and myself on these questions goes all the
way back to our work on piecewise linear embeddings in the early 1970s, in which
we discovered an at first somewhat surprising result (see [CS3], also Cappell's ICM
talk [C]).
3.2 Let Mn be a smooth of piecewise submanifold ofWn+2,n
> 4. Let h:Nn —>
M be a homotopy equivalence. Assume M is simply connected. Then the composite
g: N —> M (ZW is homotopie to a piecewise linear embedding.
This result was surprising because if g is a smooth or PL locally flat embedding (i.e. locally smoothable), then g*L(N) = L(M). On the other hand,
according to 1.8, there are many N in the homotopy type of M, with different
L-classes. Qualitatively, this means that the embedding N C W must fail to be
locally smoothable in a rather dramatic way: the set of non locally smoothable
points must carry a cycle representing L(N) — g~1L(M).
In fact, the degree on non-local-smoothability imposes a nontrivial stratification V on AT" (with connected strata), such that near each point of y G V, the
pair (W,N) looks like Dl x cone(Gy,Fv) with i = dimK Because W and N are
manifolds, Gy ** S71-^1 and Fv ^ S"-*- 1 .
Using a torsion pairing on intersection homology with local coefficients in
the ring Q[£,£ -1 ] of Laurent polynomials, we were able to recover all the usual
invariants, e.g. the signature, of knot cobordism for such very general singular p.l.
knots. For any p.l. embedding of a stratified space Xn in a manifold Wn+2, with
Characteristic Classes, Lattice Points, ...
617
even codimension strata (e.g. a hypersurface in a complex algebraic variety), let V'
be the set of components of singular strata, and assume each Y simply connected.
Then the relation between singularities and characteristic classes is made explicit
by the formula [CS9]
[X] n o*£ {P(W) U (1 + x2)'1}
- HX) = £ L ( V > ( G V ; Fy).
(3.3)
v
Again, without the simply connected hypothesis one must use L-classes for representations of mV preserving the appropriate torsion pairing. Actually, (3.3) is
a special case of a result valid for any characteristic class associated to a knot
cobordism invariant. The result (3.3) for the term of maximal dimension affected
by singularities, i.e. (Gy\Fy) a smooth knot, was proven, but not published, by
Cappell and myself in the 1970s. A similar argument proves the top dimensional
case of (3.3), at least for X a manifold.
The equivariant class AG(X) G K^(X;l4^0^)
defined by Cappell, Weinberger, and myself [CSW], where G is a finite group acting on X, also obeys an
equivariant version of the mapping formula.
4 Genera for algebraic varieties
The goal of this section is the study of a morphism of complex varieties. It will
be assumed that / : X —> Y is a projective morphism, Y is irreducible, and / has
been stratified so that the strata are subvarieties; this is always possible (see, e.g.
[GM3]). In principle, the results of the previous section are perfectly applicable to
this situation. However, (3.2) has the disadvantage that if X and Y are algebraic
and / an algebraic morphism, Ny, V G V, will usually not be a variety at all.
To remedy this problem, we use algebraic analogues of the topological notion of
neighborhood, and complete by projectivizing instead of coning. More precisely,
for Z an irreducible subvariety of a variety Y, defined by a sheaf of ideals £, the
normal cone is defined as
CzWr = Spec[0r i /r + 1 |
Let P(CzW © 1) be its projective completion. (See [Fl, Appendix B]). Let
Py}Y be the general fiber of the natural map P(Cy © 1) —> V, and let Pyj be
the general fiber of P(Cf-iy © 1) —• f~lV
H*(V]Q)by
L(V) = L(V)]T
-^
V. Define inductively L(V) G
o-(PViW)L(W),
w<v
where the sum is over W G Vy with W C V.
Let F be the general fiber of / . Then assuming either simple connectivity of
the strata of Y or some kind of triviality of monodromy, we obtain [CS10]:
f.L(X)
= a(F)L(Y)
+ £
{a(PVJ)
- a(F)a(Pv,Y)}
L(V).
(4.1)
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Julius L. Shaneson
This property of a characteristic class will be called the stratified multiplicative
property. When simple-connectivity of strata is not assumed, it must be rephrased
in terms of L-classes with coefficients in a representation of TTI, as in Section 3.
If one assumes that the normal cones above actually describe the topological
normal structure of strata and their inverse images, then (4.1) can be derived from
(3.2), additivity of signatures, and the fact that the signature of a suspension is
always zero. However, this assumption is usually not valid, and we prove (4.1)
using some much more algebraic geometric techniques.
Moreover, Cappell and I were able to use these methods to prove much more.
for X a projective variety over C, Saito [S] proved the existence of a pure Hodge
structure. Let Ihp,q(X) be the corresponding Hodge numbers. Let h?'q be the
Hodge numbers of Deligne's mixed Hodge structure on H%(X;C). Consider the
genera
%,(*) = £ $>ir^r*(x)
tf and IXy(X) = £
J2(-i)W(x)
VP-
z,q
For example, X-i = e i s the Euler-Poincaré characteristic, Ixi = cr is the
signature discussed above, and Ixo and xo are two possible extensions of the
arithmetic genus to singular varieties. These genera extend to characteristic classes
Ty and ITy whose values in dimension zero are the different genera. For example,
T-i is the total MacPherson-Chern class, ITi — L is the L-class, and To is the
image in homology of the Baum-Fiilton-MacPherson Todd class [BFM1,2] that
appears in the generalized Grothendieck-Riemann-Roch theorem.
4.2 The genera Xy>IXy>
tiplicative property.
an
d their characteristic classes satisfy the stratified mul-
For example, let X be obtained from Yn by blowing up a point y. Let D —
P(C{y}) = f~1(y) be the exceptional set. Then
e(X) = e(Y) + 2e(D) - e(P(C{y} © 1).
(4.3)
If y is a smooth point, D and P(C^yy © 1) are projective spaces, and this is the
well-known result
e(X) = e(Y) + n - 1
(4.4)
for blowing up a smooth point.
The result 4.2 for the Euler characteristic (in the version involving nontrivial monodromy) essentially includes results like the generalization of RiemannHurwitz given in [DK], [I], [K, (III, 32)].
5 Toric varieties
Toric varieties provide one way of relating the two problems discussed in the introduction. Let An C M71 be a lattice polytope. Let J7 be the set of codimension one faces of A. For each face E < A let TE = {F G T\E C F}, so that
E = f]{F\f G TE\- Let np G L be a minimal lattice vector orthogonal to L,
Characteristic Classes, Lattice Points, ...
619
pointing into A. Let Tn = Rn/L be the 7i-torus. Let V£ = 0 ^ RnF,E a face
of A, be the orthogonal subspace to E. Let L^ = Vjjr D L. Let TE C Tn be
the subtorus TE = V^/Lj^ C Rn/L. (This notation will also be needed in the
next section.) Then let XA bc defined as the quotient of A x Tn by the relations
(x,t) ~ (x, st) for x e E and s e TE.
From this definition it is clear that XA is compact and has an action of Tn
without quotient A. Let TT:XA —> A. Then XE = n~1(E) is also a toric variety,
and XA is stratified with strata X°E = 7r~1(E°). Usually XA is really singular,
even for polytopes in the plane. The condition needed in (1.4) is equivalent to
smoothness of XA.
Just from topological considerations, it is clear that invariants of XA are
related to properties of A. For example, the fixed set (XA)T
of the T"-action is
finite and
. _
e(XA) = # (XAfn]
= # o f vertices of A.
(5.1)
However, the important applications require the algebraic structure which is more
readily apparent from the definition of XA in terms of the dual fan E A • The cones
a E of this fan are spanned by rays R>QIIF, F G TE- For any cone a in a fan E, let
Ua = Spec {C [n G L | n • u > 0 for all u G a}} .
These affine pieces fit together to form a variety X^, and X^A = XA. The relation
between invariants of XA and counting lattice points in A is a consequence of the
Riemann-Roch theorem:
5.2 Let Td(X^) G H*(X&;Q) be the (image of the) Todd class, and suppose that
Td(X) = J2 *E[XE\E<A
Then for the Ehrhart-Hilbert polynomial PA(k) = #(&A H L) one has PA(k) =
ankn H
h aik + ao with am = J2dim E=m aE^(E), where v(E) = JE dvE is the
normalized volume as described above.
All this was well understood before 1980, and is clearly explained in many
references, e.g. [Kh], [D], [O], [F2], in chronological order. For ease of exposition
we will also assume that A is simple (77, edges meeting each vertex). In this case,
[D], [O], [F2] also include the explicit calculation of #*(ÀV,Q) = A* (X) ® Q ^
# * ( ^ A ; Q ) - (For the general case, see [FS]).
Cappell and I realized already in the mid-1980s that sufficiently powerful
mapping theorems for characteristic classes, together with what was already known
about toric varieties, would lead to a computation of their Todd classes, and hence
to the calculation of the Ehrhart-Hilbert polynomial. However, the first significant
result for singular toric varieties was obtained by Pommersheim in his 1991 Ph.D.
thesis. He calculated Td 2n _4 (denoted Td 2 in [P]), which is the highest dimensional
Todd class that can be affected by singularities. A resolution of X^ can be obtained
as Xw, where the fan E' is obtained by defining E by adding more rays to make
620
Julius L. Shaneson
the cones nonsingular. Pommersheim used the canonical and explicit procedure
for doing this through cones of dimension two described, for example, in [O],
and the fact, also in the above references, that 7r*(Td(Xs')) = Td(X^) to make
his calculation. He formulated his results by defining a "mock" class TD(XA) as
the Todd polynomial in the homology classes [XE] — which would equal Td in
the smooth case — and calculating Td 2n -4(-^A) — T D 2 T I _ 4 ( X A ) . In particular,
he obtained a formula for a n _ 2 ; because an = vol(A),a n _i = \^2pv(F),
and
a 0 = 1 were known, the computation of P A for tetrahedra follows. Pommersheim
also used his computation in dimension two to reprove and extend the reciprocity
formulae [Mo], [R] for classical Dedekind sums. The results of this section and
the next imply many reciprocity-type formulae for generalized Dedekind sums.
(Cappell and I are grateful to Pommersheim for explaining his work to us prior to
publication.)
In contrast to this direct attack, Cappell and I take advantage of the relation
of Todd classes and L-classes and use our mapping theorems. From (2.6) and the
above mapping theorems, even the topological versions, it follows that
T(XA)
= [XA] + 2Td 2 n _ 2 (X A ) + 4Td 2 n _ 4 (X A ) + •..
is a linear combination of the (images under the maps inclusions of the) classes
L(XE),
E < A. But by using the algebraic versions of Section 4 and taking advantage of some known facts about invariants of toric varieties (e.g. x ( ^ s ) = 1 from
[D] and others), we obtain
T(XA) = J2 L(XE).
(5-3)
E<A
Hence the problem is reduced to computing L-classes, which we do by considering branched coverings corresponding to certain sublattices of L of rank n
and applying appropriate versions of methods from transformation groups, such
as the G-signature theorem (see [H], [CSW]).
To describe our result, define mock L-classes and T-classes as
and
T^(XA)=
(5.5)
^LW(XE).
E<A
Then let G E = L^/Z{TIF\F
G TE}, a finite group whose order is by definition
the multiplicity m(E). (Note: XA is smooth iff m(E) = 1 for E < A.) Let mf G
L^, F G TE, be the minimal elements with rn^muF' = 0 for F ^ F( and raj^n^ >
0. For g G GE the coset of m G L, let ^(g) = (m • mp)/(nF • m^)_1, and let
G°E = {ge GE\^(g)
Let
.AA
£ ZforallF G TE}.
= 1 and for E < A let
^ = J ^ y E II c o t h
\^F(9)
+ l[XF}} •
(5.6)
Characteristic Classes, Lattice Points, ...
621
Then our general calculation of Todd classes in toric varieties is then given
by the formula:
T(XA)= ^T^(XE)AE-
(5.7)
E<A
Finally, note that Pommersheim's mock class T D ( X A ) and T^(XA)
not the same, even up to powers of two. In fact
TDi(XA) = (2)'"" £ mW^lJrHXB).
are
(5.8)
E<A
Using this, one recovers the results of [P] on Td 2 n _4 mentioned above. For
other valuable perspectives and results on Todd classes and toric varieties, see the
work of Brion [B] and Morelli [Mr].
6 Euler-MacLaurin formulae
Finally, here is the promised determination of the operators PE in (1.7). For /
constant; i.e., for counting lattice points, it would be enough to apply (5.7) and the
known calculation of H(XA\Q). In general, one has to consider an infinite family
of toric varieties lying over XA, associated to monomials in variables measuring
suitably normalized distances from facets, and show that the bits of information
obtained from each fit together compatibly to obtain the result that will now be
described. In effect one is defining and computing a Todd class in some type of
enhanced Chow ring and using it to derive an Euler-MacLaurin formula, but the
presentation here will use a language more analogous to the polytope algebra [Mc]
(see also [FS]). The role of ground ring will be played by V = Q [[di,..., dn]], the
ring of formal power series in n variables. For in = (mi,... ,mn) we set Dm =
midi + • • • m n S n .
Let HE = {F G T - TE\FnE
^ $}. Let P 0 (A) be the polynomial algebra
over T> on generators Up, F G T. Let S(A) be the quotient of this algebra by
the ideal generated by the elements C/^ UF2 •. • Upk for F\ f i . . . D F^ = cj), and the
elements
Dm + y^(m • UF)UF
T
for 771 G L. (Of course, it suffices to consider any n linearly independent elements
of L.) Let p be the quotient map, and let W =
PQ(UF).
Let V(A) C Po (A) be the graded £>-submodule generated in dimension k by
monomials UF1 • • • Upk with Fi,...,F/~ pairwise distinct and F\ f i . . . C\Fk ^ (ß. Let
p:V(A) —> Q(A) be the V-module map defined by setting
p(UFl ...UFk)=
6.1 The map p is surjective.
m(Fi n ... n Fk)WFl • • • WFk •
622
Julius L. Shaneson
T h e proof (see [CS 11]) actually provides an inductive procedure for lifting
elements. For E a face of A let
Note from (6.1) t h a t 0 ( A ) is closed under completion with respect to monomials in t h e WF. Let U(A) = 1 and for E < A let
u
w
=
^
ë
)
E
I
G
E
Then set
T(A)
c
I
o
t
h
{
*
*
£
(
*
)
+
\
W
A
•
FK
= ]T T(E)U(E).
E<A
6.2 Let d operate on polynomials
by partial differentiation
d/dxi.
Let
s=Yl PEUE e ^( A )
E<A
where PE G V and UE = YlrE UF
(UA = 1), with p(E) = T(A).
Then
dv EXELHS
E<AJE
It is not hard to see t h a t for A an interval, this result gives the classical
Euler-Maclaurin formula. For illustration in detail of t h e case of a polygon, see
[CS11].
References
[At]
[AB]
[AS]
[BFM1]
[BFM2]
[BBD]
[Bo]
[B]
M. F. Atiyah, The signature of fibre-bundles, in Global Analysis, Papers in
honour of K. Kodaira (D. C. Spencer and S. Iyanayaga, eds.), Princeton
Univ. Press, Princeton, NJ, (1969), 73-84.
M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes II: Applications, Ann. of Math. (2) 88 (1968), 451-491.
M. F. Atiyah and I. M. Singer, The index of elliptic operators, I, III, and IV,
Ann. of Math. (2) 93 (1971), 119-138.
P. Baum, W. Fulton, and R. MacPherson, Riemann-Roch for singular varieties, Pubi. Math. I.H.E.S. 45 (1975), 101-145.
P. Baum, W. Fulton, and R. MacPherson, Riemann-Roch and topological Ittheory for singular varieties, Acta Math. 143 (1979), 155-192.
A. A. Beilinson, J. Bernstein, and P. Deligne, Faisceaux pervers, analyse et
topologie sur les espaces singuliers, Astérisque 100 (1982), 1-171.
A. Borei et al., Intersection cohomology. Progr. Math 50, Birkhäuser, Boston,
MA, 1984.
M. Brion, Points entiers dans les polyèdres convexes, Ann. Sci. Ecole Norm.
Sup. (4) 21 (1988), 653-663.
Characteristic Classes, Lattice Points, . . .
[Br]
623
W. Browder, Surgery on Simply Connected Manifolds, Springer Verlag, Berlin
and Heidelberg, 1972.
[BQ]
W. Browder and F. Quinn, A surgery theory for G-manifolds and stratified
sets, in Manifolds (Tokyo 1973), University of Tokyo Press (1975), 27-36.
[C]
S. E. Cappell, Submanifolds of sm.all codim.ension, Proc. of Internat. Congress
Math., Helsinki (1978), 455-462.
[CSI]
S. E. Cappell and J. L. Shaneson, The codimension two placement problem
and homology equivalent manifolds, Ann. of Math. (2) 99 (1974), 277-348.
[CS2]
S. E. Cappell and J. L. Shaneson, An introduction to embeddings, immersions,
and singulanties in codimension two, Proc. Sympos. Pure Math. XXXII, part
2 (1978), 129-150.
[CS3]
S. E. Cappell and J. L. Shaneson, Piecewise linear embeddings and their singularities, Ann. of Math. (2) 103 (1976), 163-228.
[CS4]
S. E. Cappell and J. L. Shaneson, Immersions and Singularities, Ann. of
Math. (2) 105 (1977), 539-552.
[CS5]
S. E. Cappell and J. L. Shaneson, Non-linear similarity, Ann. of Math. (2) 113
(1981), 315-355.
[CS6]
S. E. Cappell and J. L. Shaneson, Fixed points of periodic differentiable maps,
Invent. Math. 68 (1982), 1-20.
[CS7]
S. E. Cappell and J. L. Shaneson, Charactenstic classes, singular embeddings,
and intersection homology, Proc. Nat. Acad. Sci. 84 (1987), 3954-3956.
[CS8]
S. E. Cappell and J. L. Shaneson, Stratifiable maps and topological invariants,
J. Amer. Math. Soc. 4 (1991), 521-551.
[CS9]
S. E. Cappell and J. L. Shaneson, Singular spaces, characteristic classes, and
intersection homology, Ann. of Math. (2) 134 (1991), 325-374.
[CS10]
S. E. Cappell and J. L. Shaneson, Genera of algebraic and counting of lattice
points, Bull. Amer. Math. Soc. 30 (1994), 62-69.
[CS11]
S. E. Cappell and J. L. Shaneson, Euler MacLaurin expansions for lattices
above dimension one, O R . Acad. Sci. Paris, to appear.
[CSSW]
S. E. Cappell, J. L. Shaneson, M. Steinberger, and J. West, Non-linear similarity begins in dimension six, Amer. J. Math., to appear.
[CSSWW] S. E. Cappell, J. L. Shaneson, M. Steinberger, J. West, and S. Weinberger,
Classifiying on-linear similarities over Z 2 , Bull. Amer. Math. Soc, to appear.
[CSW]
S. E. Cappell, J. L. Shaneson, and S. Weinberger, Classes caractéristiques
pour les action des groupes sur les espaces singulières, O R . Acad. Sci. Paris
313 (1991), 293-295.
[CW]
S. E. Cappell and S. Weinberger, Classification of certain stratified spaces, C.
R. Acad. Sci. Paris, to appear.
[Ch]
J. Cheeger, On Hodge theory of Riemannian pseudomanifolds, Proc. Sympos. Pure Math. 36 (1980), Amer. Math. Soc, Providence, RI, 91-146.
[CHS]
S. S. Chern, F. Hirzebruch, and J.-P. Serre, The index of a fibered manifold,
Proc. Amer. Math. Soc. 8 (1957), 587-596.
[D]
V. I. Danilov, The geometry of toric varieties, Russian Math. Surveys 33
(1978), 97-154.
[DK]
P. Deligne and N. Katz, Groupes de Monodromy en Géométrie Algébrique,
Springer Lecture Notes in Math. 340 (1973), Exposé XVI, 2, 268-271 (by
P. Deligne).
[FI]
W. Fulton, Intersection Theory, Springer Verlag, Berlin and New York, 1984.
[F2]
W. Fulton, Introduction to toric varieties, Princeton University Press, Princeton, NJ, 1993.
[FS]
W. Fulton and M. Sturmfels, Intersection theory on toric varieties, to appear.
624
[GM1]
[GM2]
[GM3]
[H]
[I]
[KS]
[Kh]
[KP]
[K]
[M]
[Mc]
[Mo]
[Mr]
[O]
[Pi]
[P]
[Q]
[R]
[S]
[Shi]
[Sh2]
[Si]
[V]
[Wl]
[We]
Julius L. Shaneson
M. Goresky and R. D. MacPherson, Intersection homology theory, Topology
19 (1980), 135-162.
M. Goresky and R. D. MacPherson, Intersection homology II, Invent. Math.
72 (1983), 77-130.
M. Goresky and R. D. MacPherson, Stratified Morse Theory, Ergebnisse der
Mathematik, Springer Verlag, Berlin and New York, 1980.
F. Hirzebruch, Topological Methods in Algebraic geometry (3rd ed.), Grundlehren Math. Wiss. 131, Springer Verlag, Berlin and New York, 1978.
B. Iversen, Critical points of an algebraic function, Invent. Math. 12 (1971),
210-224.
M. Kashiwara and P. Schapira, Sheaves on Manifolds, Grundlehren Math.
Wiss. 292, Springer Verlag, Berlin and New York, 1990.
G. Khovanski, Newton polyhedra and toric varieties, Functional Anal. Appi.
4 (1977), 56-67.
A. G. Khovanski and A. V. Pukhlikov, A Riemann-Roch theorem for integrals and sums of quasi polynomials over virtual polytopes, St. Petersburg
Math. Journal 4 (1993), 789-812.
S. Kleiman, The enumerative theory of singularities, in Real and Complex
Singularities (P. Holm, ed.), Sijthoff and Noordhoff, Alpehn aan den Rijn
(1976), 298-384.
R. D. MacPherson, Global Questions in the Topology of Singular Spaces,
Proc. Internat. Congress Math. (1983), North-Holland, New York, 213-235.
P. McMullen, The polytope algebra, Adv. in Math. 78 (1989), 76-130.
L. J. Mordell, Lattice points in a tetrahedron and generalized Dedekind sums,
J. Indian Math. 15 (1951), 41-46.
R. Morelli, Pick's Theorem and the Todd Classes of a Toric Variety, Adv. in
Math., to appear.
T. Oda, Convex Bodies and Algebraic Geometry, Springer Verlag, Berlin and
New York, 1987.
G. Pick, Geometrisches zur Zahlenlehre, Sitzungber. Lotos Prag, 2, 19 (1870),
311-319.
J. E. Pommersheim, Toric varieties, lattice points, and Dedekind sums, Math.
Ann. 296 (1993), 1-24.
F. Quinn, Homotopically stratified spaces, J. Amer. Math. Soc. 1 (1988),
449-499.
H. Rademacher, On Dedekind sums and lattice points in a tetrahedron. Stud.
Math. Mech., Academic Press, New York (1954), 49-53. (Reprinted in Collected Papers of H. Rademacher, Vol. II, E. Grosswald, editor, M.I.T. Press
1974, 391-398).
M. Saito, Modules de Hodge polarizables, Pubi. RIMS 24 (1988), 849-995.
J.L. Shaneson, Wall's surgery obstruction groups for Z x G , Ann. of Math. 99
(1969), 296-334.
J. L. Shaneson, Linear algebra, topology, and number theory, Proc. Internat.
Congr. Math., Warsaw (1983), 685-698.
P. Siegel, Witt spaces: A geometric cycle theory for KO-homology theory at
odd primes, Amer. J. Math. 105 (1983), 1067-1105.
J. L. Verdier, Specialisation de faisceaux et monodromie modérée, Astérisque 101-102 (1982), 332-364.
C. T. C. Wall, Surgery on Compact Manifolds, Academic Press, London, 1970.
S. Weinberger, The topological classification of stratified spaces, University
of Chicago Press, to appear.