GENERALIZED ORBIFOLD EULER
CHARACTERISTIC OF SYMMETRIC PRODUCTS
AND EQUIVARIANT MORAVA K-THEORY
Hirotaka Tamanoi
Department of Mathematics
University of California, Santa Cruz
Abstract. We introduce the notion of generalized orbifold Euler character*
*istic as-
sociated to an arbitrary group, and study its properties. We then calcula*
*te generat-
ing functions of higher order (p-primary) orbifold Euler characteristic o*
*f symmetric
products of a G-manifold M. As a corollary, we obtain a formula for the *
*number
of conjugacy classes of d-tuples of mutually commuting elements (of order*
* powers
of p) in the wreath product G o Sn in terms of corresponding numbers of G*
*. As
a topological application, we present generating functions of Euler chara*
*cteristic of
equivariant Morava K-theories of symmetric products of a G-manifold M.
Contents
1.Introduction and summary of results 1
2.Generalized orbifold Euler characteristics 5
3.Centralizers of wreath products 12
4.Higher order orbifold Euler characteristics of symmetric prod-
ucts 15
5.Euler characteristic of equivariant Morava K-theory of sym-
metric products 18
References 22
x1. Introduction and summary of results
Let G be a finite group and let M be a smooth G-manifold. We study gener-
alized orbifold Euler characteristics of (M; G). These are integer-valued invar*
*iants
associated to any group K. (See (1-3) below.) The simplest of such invariants (*
*cor-
responding to the trivial group K = {e}) is the usual Euler characteristic O(M=*
*G)
of the orbit space. It is well known that O(M=G) can be calculated as the avera*
*ge
over g 2 G of Euler characteristic of corresponding fixed point submanifolds:
X
(1-1) O(M=G) = _1_|G| O(M);
g2G
______________
1991 Mathematics Subject Classification. 55N20, 55N91, 57S17, 57D15, 20E22.
Key words and phrases. Equivariant Morava K-theory, generating functions, G-*
*sets, M"obius
functions, orbifold Euler characteristics, q-series, second quantized manifolds*
*, symmetric products,
twisted iterated free loop space, twisted mapping space, wreath products, Riema*
*nn zeta function.
Typeset by AM S-T*
*EX
1
2 HIROTAKA TAMANOI
where G is the subgroup generated by g 2 G. See for example, [Sh, p.127].
In 1980s, string physicists proposed a notion of orbifold Euler characterist*
*ic of
(M; G) defined by
X
(1-2) Oorb(M; G) = _1_|G| O(M);
gh=hg
where the summation is over all commuting pairs of elements in G [DHVW]. The
orbifold Euler characteristic is always an integer, since (1-1) implies
X
Oorb(M; G) = O(M=CG (g)) 2 Z;
[g]
where the summation is over all the conjugacy classes of G, and CG (g) is the
centralizer of g in G. This formula is of the form O(M=G) + (correction terms).
Generalized orbifold Euler characteristic. Let K be any group. The gener-
alized orbifold Euler characteristic of (M; G) associated to K is an integer
X 1 X
(1-3) OK (M; G) def= O M=CG (OE) = ___ O(M):
[OE]2Hom (K;G)=G |G| OE2Hom (KxZ;G)
The first summation is over G-conjugacy classes of homomorphisms, and the second
equality is a consequence of (1-1). Thus, either expression can be taken as the
definition of OK (M; G). Here, CG (OE) is the centralizer in G of the image of OE.
Note that when K is the trivial group {e} or Z, our OK (M; G) specializes to (1*
*-1)
or (1-2). In x2, we describe its various properties including multiplicativity *
*and the
following formula for a product K x L of two groups:
X
(1-4) OKxL (M; G) = OL M; CG (OE) :
[OE]2Hom (K;G)=G
This formula, which is easy to prove, is crucial for inductive steps in the pro*
*ofs of
our main results, Theorems A and B below.
In this paper, we are mostly concerned with the cases K = Zd and K = Zdp,
where Zp denotes the ring of p-adic integers. We use the following notations:
(1-5) OZd(M; G) = O(d)(M; G); OZdp(M; G) = O(d)p(M; G):
We call these d-th order (p-primary) orbifold Euler characteristics.
Our definition (1-3) is partly motivated by consideration of a mapping space
Map (; M=G), where a manifold has fundamental group K. When is the
genus g orientable surface g with ss1(g) = g, we call the corresponding quantity
Og (M; G) genus g orbifold Euler characteristic of (M; G).
Higher order orbifold Euler characteristics for symmetric products. It
turns out that O(d)(M; G) admits a geometric interpretation in terms of the map-
ping space Map (T d; M=G), where T dis the d-dimensional torus. See x2 for more
details. And as such, it is very well behaved. We demonstrate this point by cal*
*cu-
lating O(d)for symmetric products.
When M is a G-manifold, the n-fold Cartesian product Mn admits an action of
a wreath product G o Sn. The orbit space Mn =(G o Sn) = SP n(M=G) is the n-th
symmetric product of M=G.
GENERALIZED ORBIFOLD EULER CHARACTERISTIC 3
Theorem A. For any d 0 and for any G-manifold M,
X hY di(-1)O(d)(M;G)
(1-6) qn O(d)(Mn ; G o Sn) = (1 - qr)jr(Z ) ;
n0 r1
P d-1
where jr(Zd) = r2r23. .r.d is the number of index r subgroups in Zd.
r1r2...rd=r
For the case of d = 1 (O(1)= Oorb), the above formula was calculated by Wang
[W]. By letting M to be a point, and using the notation Gn = G o Sn, we obtain
Corollary 1-1. For any d 0, we have
X fi fi hY di(-1)|Hom (Zd;G)=G|
(1-7) qn fiHom Zd; Gn =Gnfi= (1 - qr)jr(Z ) :
n0 r1
Special cases of our results are known. Macdonald [M1] calculated Euler char-
acteristic of symmetric products of any topological space X. His formula reads
X 1
(1-8) qn O SP n(X) = ___________O(X):
n0 (1 - q)
Hirzebruch-H"ofer [HH] calculated the orbifold Euler characteristic (1-2), whic*
*h is
our O(1)(M; G), of symmetric products. Their formula is
X hY i(-1)O(M)
(1-9) qn Oorb(Mn ; Sn) = (1 - qr) :
n0 r1
After completing this research, the author became aware of the paper [BF] in
which the formulae (1-6) and (1-7) with trivial G were calculated. (That the e*
*x-
ponent of their formula can be identified with jr(Zd) was pointed out by Allan
Edmonds in Math. Review.) We remark thatftheiformulaf(1-7)iwith trivial G is
straightforward once we observe that fiHom(Zd; Sn)=Snfiis the number of isomor-
phism classes of Zd-sets of order n, and jr(Zd) is the number of isomorphism cl*
*asses
of transitive (irreducible) Zd-sets of order r. The second fact is because the *
*isotropy
subgroup of transitive Zd-sets of order r is a sublattice of index r in Zd. See*
* also an
exercise and its solution in [St, p.76, p.113]. On the other hand, when G is no*
*ntriv-
ial, a geometric interpretation of elements in Hom (Zd; Gn)=Gn is more involve*
*d.
Our method of proving (1-6) is a systematic use of the formula (1-4) for genera*
*lized
orbifold Euler characteristic and the knowledge of centralizers of elements of *
*the
wreath product Gn described in detail in x3. Our method can also be applied to
more general context including p-primary orbifold Euler characteristic O(d)p(M;*
* G).
The integer jr(Zd) has very interesting number theoretic properties. It is *
*easy
to prove that the Dirichlet series whose coefficientsPare jr(Zd) can be express*
*ed as
a product of Riemann zeta functions i(s) = n1 1=ns with s 2 C:
X jn(Zd)
(1-10) _______s= i(s)i(s - 1) . .i.(s - d + 1):
n1 n
For the history of this result, see [So].
4 HIROTAKA TAMANOI
Euler characteristic of equivariant Morava K-theory of symmetric prod-
ucts. Let K(d)*(X) be the d-th Morava K-theory of X for d 0. Since K(d)* is
a graded field, we can count the dimension of K(d)*(X) over K(d)*, if it is fin*
*ite.
We are interested in computing Euler characteristic of equivariant d-th Morava
K-theory of a G-manifold M:
* even odd
(1-11) O K(d)G (M) = dim K(d) (EG xG M) - dim K(d) (EG xG M):
In [HKR], they calculate this number in terms of M"obius functions [HKR, Theo-
rem B]. It is a simple observation to identify (1-11) as the d-th order p-prima*
*ry
orbifold Euler characteristic O(d)p(M; G) (see a paragraph before Proposition 5*
*-1).
Our second main result is as follows.
Theorem B. Let d 0 and let M be a G-manifold. The Euler characteristic of
equivariant Morava K-theory is equal to the d-th order p-primary orbifold Euler
characteristic of (M; G):
* (d)
(1-12) O K(d)G (M) = Op (M; G):
The generating function of Euler characteristic of equivariant d-th Morava K-th*
*eory
of symmetric products is given by
X hY r di(-1)O(K(d)*G(M))
(1-13) qn O K(d)*Gn(Mn ) = (1 - qp )jpr(Zp) :
n0 r0
Here, Gn = G o Sn, and j`(Zdp) is the number of index ` subgroups in Zdpgiven by
X
(1-14) jpr(Zdp) = r2r23. .r.d-1d; and j`(Zdp) = 0 if ` is not a power o*
*f:p
r1r2...rd=pr
Let M be a point. The resulting formula is both topological and combinatoria*
*l:
Corollary 1-2. For any d 0, we have
X hY r di(-1)O(K(d)*BG)
qn O K(d)*(BGn) = (1 - qp )jpr(Zp)
n0 r0
(1-15) X fi fi hY i d
r j r(Zd)(-1)|Hom (Zp;G)=G|
qn fiHom(Zdp; Gn)=Gnfi= (1 - qp ) p p :
n0 r0
When G is a trivial group and hence Gn = Sn, the above formula is straightfo*
*r-
ward by an argument in terms of Zdp-sets.
Again the integers jpr(Zdp) have number theoretic properties and it is well *
*known
that the corresponding Dirichlet series can be expressed as a product of p-local
factors of Riemann zeta function. Namely, letting ip(s) = (1 - p-s )(-1) denote*
* the
p-local factor in the Euler decomposition of i(s), we have
X jpr(Zdp)
(1-16) ________rs= ip(s)ip(s - 1) . .i.p(s - d + 1):
r0 p
GENERALIZED ORBIFOLD EULER CHARACTERISTIC 5
In particular, we have the following Euler decomposition of Dirichlet series:
X jn(Zd) Y X jpr(Zdp)
(1-17) _______s= ________rs:
n1 n p:primer0 p
In [H], Hopkins shows that when G is a trivial group, the exponents in (1-15)
satisfy (1-16) by a method completely different from ours: by integrating a cer*
*tain
function over GL n (Qp). And he identifies these exponents as Gaussian binomial
coefficients [M3, p.26].
There is a physical reason why symmetric products of manifolds give rise to
very interesting generating functions. In physics, the process of quantization*
* of
the state space of particles or strings moving on a manifold produces a Hilbert
space of quantum states. The second quantization then corresponds to taking
the total symmetric products of this Hilbert space, describing quantum states of
many particles or strings. Reversing the order of these two procedures, we can
first apply the second quantization to the manifold, by taking the total symmet*
*ric
products of the manifold. This object quantizes well, and in [DMVV] they calcul*
*ate
complex elliptic genera of second quantized K"ahler manifolds and it is shown t*
*hat
they are genus 2 Siegel modular forms whose weight depend on holomorphic Euler
characteristic of K"ahler manifolds.
The organization of this paper is as follows. In section 2, we define gener-
alized orbifold Euler characteristics associated to arbitrary groups, and descr*
*ibe
their properties. We also express abelian orbifold Euler characteristics in ter*
*ms of
M"obius functions. We also describe a geometry behind our definition of general*
*ized
orbifold Euler characteristic. In section 3, we describe some properties of wr*
*eath
products including the structure of centralizers. Materials here are not new, h*
*ow-
ever detailed description on this topic seems to be rather hard to find, so we *
*worked
out details and we decided to include it. This section is purely group theoreti*
*c and
is independent from the rest of the paper. In sections 4 and 5, we compute high*
*er
order (p-primary) orbifold Euler characteristics of symmetric products in the f*
*orm
of generating functions. A relation to Euler characteristic of equivariant Mor*
*ava
K-theory is discussed in section 5.
Acknowledgement. The author thanks M. J. Hopkins for useful discussions duri*
*ng
the author's visit at MIT, and for making his preprint [H] available. Our Theor*
*em
B was worked out after seeing his paper. The author also thanks N. J. Kuhn
who, during the initial circulation of this paper, informed him that in [K] he *
*had
considered a fixed point functor FK on G-CW complexes which is formally related
to our generalized orbifold Euler characteristic.
x2. Generalized orbifold Euler characteristics
A generalization of physicists' orbifold Euler characteristic (1-2) was give*
*n in
the introduction in (1-3). Here, group K can be an arbitrary group. Properties
enjoyed by O(d)(M; G) and O(d)p(M; G) given in (1-5) become transparent in this
generality. If the group K is abelian, our generalized orbifold Euler character*
*istic
is better behaved and it admits an expression in terms of M"obius functions def*
*ined
on the family of abelian subgroups of G.
Later in this section, we will explain a geometric meaning of generalized or*
*bifold
Euler characteristics in terms of twisted mapping spaces.
6 HIROTAKA TAMANOI
Generalized orbifold Euler characteristics. We prove basic properties of gen-
eralized orbifold Euler characteristic. Recall from x1 that the generalized orb*
*ifold
Euler characteristic associated to a group K is given by
X X
(2-1) OK (M; G) = _1_|G| O(M) = O M=CG (OE)
OE2Hom (KxZ;G) [OE]2Hom (K;G)=G
See the last subsection of x2 for a geometric motivation of this definition. Le*
*tting
M be a point, we obtain a useful formula
fifi fi
Hom (K x Z; G)fi fi fi
(2-2) OK (pt; G) = _________________|G|= fiHom(K; G)=Gfi:
Now we give a proof of (1-4).
Proposition 2-1. The orbifold Euler characteristic OK is multiplicative. Namel*
*y,
for any Gi-manifolds Mi for i = 1; 2, we have
(2-3) OK (M1 x M2; G1 x G2) = OK (M1; G1) . OK (M2; G2):
Furthermore, for any two groups K and L, we have
X
(2-4) OKxL (M; G) = OL M; CG (OE) :
[OE]2Hom (K;G)=G
Proof. The first formula is straightforward from the definition of OK given in*
* (2-1).
For the second formula, first we observe that OL M; CG (OE) depends only o*
*n the
conjugacy class of OE, so the formula is well defined. Now
X 1 1 X
(R.H.S )= _____________ O M
OE2Hom (K;G)#[OE]|CG (OE)| 2Hom (LxZ;C(OE))
X
= _1_|G| O M = OKxL (M; G):
(OE; )2Hom (KxLxZ;G)
Here, #[OE] is the number of elements in the conjugacy class of OE. Since |CG (*
*OE)| is
the isotropy subgroup of the conjugation action of G on the homomorphism set at
OE, we have #[OE] . |CG (OE)| = |G|. This completes the proof.
Next, we rewrite our orbifold Euler characteristic in terms of M"obius funct*
*ions
H ( X ) defined for any subgroup H and any G-CW complex X. These are defined
by downward induction on P G by the formula
X
(2-5) H (X) = O(XP ):
PHG
It is known that any additive functions on G-CW complexes can be expressed as
a linear combination of H ( . )'s with Z[1=|G|]-coefficients [HKR, Proposition *
*4.6].
Our generalized orbifold Euler characteristic has the following expression in t*
*erms
of M"obius functions.
GENERALIZED ORBIFOLD EULER CHARACTERISTIC 7
Lemma 2-2. For any group K and G-CW complex M, we have
X |Hom (K x Z; H)| X |H| fi fi
OK (M; G) = _________________H (M) = ____. fiHom(K; H)=Hfi. H (M):
HG |G| HG |G|
Proof. In the definition of OK (M; G) in (2-1), we replace Mby (2-5). We ha*
*ve
X 1 X X
OK (M; G) = _1_|G| O(M) = ___ H (M)
OE |G| OEH
X X 1 X fi fi
= _1_|G| H (M) 1 = ___ fiHom(K x Z; H)fi. H (M):
HG OE:KxZ-! H |G| HG
Here in the second and third summation, OE runs over all homomorphisms in the s*
*et
Hom (K x Z; G).
The second equality of the statement is due to (2-2). This proves the Lemma.
Abelian orbifold Euler characteristics and abelian M"obius functions. We
recall some facts on complex oriented additive functions [HKR, x4.1,x4.2]. Let
: {G-CW complexes } -! Z be an integer-valued G-homotopy invariant function
on G-CW complexes. Then the function is called additive if it satisfies
(X [ Y ) + (X \ Y ) = (X) + (Y ); (;) = 0
for any G-CW complexes X; Y . For a G-equivariant complex vector bundle on
X, let F () be the associated bundle of complete flags in . An additive function
is called complex oriented if satisfies F () = n!(X) for any G-equivariant
complex n-dimensional bundle on X. It is known that any complex oriented
additive function on G-CW complexes is completely determined by its value on the
family of finite G-sets {G=A}, where A runs over all abelian subgroups of G. In
fact, the following formula holds [HKR, Proposition 4.10]:
X
(2-6) ( . ) = _1_|G| |A| . (G=A) . CA( . );
AG
A:abelian
where A ( . ) is a complex oriented additive function defined by downward induc*
*tion
on an abelian subgroup A by
X
(2-7) CB(X) = O(XA )
ABG
B:abelian
for any G-CW complex X.
For our generalized orbifold Euler characteristic, when the group K is an ab*
*elian
group E, then OE ( . ; G) is a complex oriented additive function, since the im*
*age
of any homomorphism OE : E x Z -! G is abelian. As such, OE ( . ; G) satisfies*
* a
formula of the form (2-6). We will explicitly derive this formula in Propositio*
*n 2-3.
On the other hand, we can also show that OE ( . ; G) can be written as a lin*
*ear
combination of complex oriented additive functions {OE ( . ; A)}A with Z[1=|G|*
*]-
coefficients, where A runs over all abelian subgroups of G. For this descriptio*
*n, we
8 HIROTAKA TAMANOI
need a functionA : {abelian subgroups of G } -! Z defined by downward induction
on an abelian subgroup A by
X
(2-8) A (B) = 1:
ABG
B:abelian
Note that when G is abelian, this relation implies thatA (G) = 1 andA (A) = 0 f*
*or
any proper subgroup A of G. Thus, (2-8) is of interest only when G is non-abeli*
*an.
We call CA( . ) and A ( . ) abelian M"obius functions. We rewrite the generaliz*
*ed
abelian orbifold Euler characteristic as follows. In (2-9) below, the first id*
*entity
can be proved easily using (2-6), but here we give a different and amusing proo*
*f:
we calculate a triple summation in three different orders.
Proposition 2-3. Let E be an abelian group. Then the corresponding orbifold
Euler characteristic OE (M; G) satisfies
X |B| fi fi X |A|
(2-9) OE (M; G) = ____. fiHom(E; B)fi. CB(M) = ___ . A (A) . OE (M; A):
BG |G| AG |G|
Here in the above summations, A and B run over all abelian subgroups of G.
Proof. We consider the following summation in three variables OE; A; B:
X
(*) = A (A) . CB(M);
OE;A;B
where OE : E x Z -! G and A; B are abelian subgroups satisfying A and
B. We compute this summation in three different ways:
X X X X X X X X X
(1) ; (2) ; (3) :
A OE B OE A B B OE A
For the case (1), the summation becomes
X X n X o X X
(*) = A (A) CB(M) = A (A) O M
A OE:ExZ-! A B A OE:ExZ-! A
X
= A (A) . |A| . OE (M; A):
A
Here, the summation over A is over all abelian subgroups of G, and (2-7) was us*
*ed
for the second equality. This is allowed since is abelian for any homomorp*
*hism
OE : E x Z -! A. The third equality is the definition of OE ( . ; A) in (2-1).
For the case (2), we have
X n X o n X o X
(*) = A (A) CB(M) = 1 . O M = |G| . OE (M; G):
OE:ExZ-! G A B OE:ExZ-! G
Note that if E is not abelian, then the image of OE : E x Z -! G can be no*
*n-
abelian and the second equality above may not be valid. This is where we need to
assume that E is abelian.
GENERALIZED ORBIFOLD EULER CHARACTERISTIC 9
For the third summation
X h X n X o i X fi fi
(*)= CB(M) A (A) = CB(M) . fiHom(E x Z; B)fi
B OE:ExZ-! B A B
X fi fi
= |B| . fiHom(E; B)fi. CB(M):
B
Here the summation over B is over all abelian subgroups of G. Since B is abelia*
*n,
Hom (E x Z; B) is a product of Hom (E; B) and B. This completes the proof.
By letting E be the trivial group, we get an interesting formula for O(M=G).
Corollary 2-4. For any G-manifold M, we have
X 1 X
O(M=G) = _1_|G| |A| .A (A) . O(M=A) = ___ |B| . CB(M):
A:abelian |G| B:abelian
It is interesting to compare this formula with (1-1). Note that the above fo*
*rmula
does not imply thatA (A) . O(M=A) is equal to CA(M). The second equality holds
only after summation over all abelian groups. A similar remark applies to (2-9).
Higher order orbifold Euler characteristic. We specialize our previous results
on generalized orbifold Euler characteristic to higher order orbifold Euler cha*
*rac-
teristic O(d)(M; G) = OZd(M; G). First, (2-1) specializes to
X X
(2-10) O(d)(M; G) = _1_|G| O(M) = O M=CG (OE) :
OE2Hom (Zd+1;G)[OE]2Hom (Zd;G)=G
Notice that the second equality above is also a consequence of (2-2) with K = Zd
and L = {e}. If we apply (2-2) with K = Z and L = Zd-1 , then we obtain the
following inductive formula.
Proposition 2-5. For any d 1, and for any G-manifold M, we have
X
(2-11) O(d)(M; G) = O(d-1) M; CG (g) ;
[g]
where the summation is over all conjugacy classes [g] 2 Hom (Z; G)=G of G.
This is the formula which allows us to prove Theorem A inductively on d 0.
Lastly, formula (2-9) specializes in our case to
X X
(2-12) |G| . O(d)(M; G) = |A| . A (A) . O(d)(M; A) = |B|d+1 . CB(M):
AG BG
Here the summations is over all abelian subgroups of G.
Higher order p-primary orbifold Euler characteristic. Recall that the basic
formula of this orbifold Euler characteristic O(d)p(M; G) = OZdp(M; G) is given*
* by
letting K = Zdpin (2-1):
X X
(2-13) O(d)p(M; G) = _1_|G| O(M) = O M=CG (OE) :
OE2Hom (ZdpxZ;G)[OE]2Hom (Zdp;G)=G
Now let K = Zp and L = Zd-1p in the formula (2-4). We obtain the following
inductive formula corresponding to (2-11) for the p-local case.
10 HIROTAKA TAMANOI
Proposition 2-6. For any d 1 and for any G-manifold M, we have
X
(2-14) O(d)p(M; G) = O(d-1)pM; CG (OE) :
[OE]2Hom (Zp;G)=G
Here [OE] runs over all G-conjugacy classes of elements of order powers of p.
The following formula, which is a specialization of (2-9) in our setting, wi*
*ll be
used later in x5 to compare O(d)p(M; G) with Euler characteristic of equivariant
Morava K-theory.
Proposition 2-7. For any d 0 and for any G-manifold M,
X |A| X |B|
(2-15) O(d)p(M; G) = ___.A (A) . O(d)p(M; A) = ____. |B(p)|d . CB(M);
AG |G| BG |G|
where the summation is over all abelian subgroups of G.
Generalized orbifold Euler characteristic and twisted mapping space. We
discuss a geometric origin of orbifold Euler characteristics. Physicists' orbif*
*old Eu-
ler characteristic (1-2) originates in string theory. Higher order (p-primary) *
*orbifold
Euler characteristics O(d)(M; G) and O(d)p(M; G) have similar geometric interpr*
*eta-
tions in terms of twisted mapping spaces. There is a very strong analogy between
this geometric situation and methods used in orbifold conformal field theory. *
*We
can predict results in orbifold conformal field theory, for example a descripti*
*on of
twisted sectors for the action of wreath products, simply by examining this geo-
metric situation of twisted mapping spaces.
To describe the geometry, first we consider the free loop space L(M=G) =
Map (S1; M=G) on the orbit space M=G. Our basic idea here is to study the orbit
space M=G by examining holonomies of loops passing through orbifold singulari-
ties of M=G. To be more precise, we consider lifting a loop __fl: S1 -! M=G to*
* a
map fl : R -! M, where S1 = R=Z. This lift may not close after moving 1 unit
along R and the difference between fl(t) and fl(t + 1) comes from the action of*
* an
element g 2 G, the holonomy of __fl. When the loop __fldoes not pass through o*
*rb-
ifold points of M=G, the conjugacy class of the holonomy is uniquely determined
by __fl. However, when the loop __flpasses through orbifold point, it can have*
* lifts
whose holonomies belong to different conjugacy classes. Furthermore, it can have
a lift whose holonomy depend on the unit segment of R on which the holonomy is
measured. To avoid this complication, we consider only g-periodic lifts. This i*
*s the
notion of g-twisted free loop space LgM defined by
(2-16) LgM = {fl : R -! M | fl(t + 1) = g-1 fl(t); t 2 R}:
Since any loop __fl2 L(M=G)`can be lifted to a g-periodic lift`for some g 2 G,
we have a surjective map g2G LgM -! L(M=G). On the space g2G LgM, the
~=
group G acts inducing a homeomorphism h. : LgM -! Lhgh-1M for any h; g 2 G.
Quotienting by this action, we get a surjective map
i a j . a onto
(2-17) LgM G ~= LgM=CG (g) --! L(M=G):
g2G [g]2G*
GENERALIZED ORBIFOLD EULER CHARACTERISTIC 11
Here G* is the set of all conjugacy classes of G. This map is 1 : 1 on the subs*
*et of
L(M=G) consisting of loops not passing through orbifold points. Thus, if the ac*
*tion
of G on M is free, then the above map is a homeomorphism. When a loop passes
through orbifold points, its inverse image is not unique, but finite, correspon*
*ding
to finitely many possibilities of different conjugacy classes of lifts. Thus, i*
*n a sense,
the above surjective map gives a mild resolution of orbifold singularities.
Since LgM=C(g) is again an orbifold space, we can apply the above procedure
again on its free loop space. In fact, we can iterate this procedure. To descri*
*be this
general case, for any d 1 and for any homomorphism OE : Zd -! G, let LOEM be
the space of twisted d-dimensional tori defined by
(2-18) LOEM = {fl : Rd -! M | fl(t + m) = OE(m)-1 fl(t); t 2 Rd; m 2 Zd }:
Here OE plays a role of holonomy of the map __fl: T d-! M=G, where T d= Rd=Zd.
Observe that any fl in LOEM factors through the torus TOE= Rd=Ker OE. As before,
~=
the action of any h 2 G induces a homeomorphism h. : LOEM -! LhOEh-1M. Let
ia j . a
(2-19) Ld(M; G) = LOEM G = LOEM=CG (OE) :
OE2Hom (Zd;G)[OE]2Hom (Zd;G)=G
Here, CG (OE) G is the centralizer of OE. We may call this space d-th order tw*
*isted
torus space for (M; G). Let T = Rd=\OEKerOE, where OE runs over all homomorphis*
*ms
Hom (Zd; G). Then T is a d-dimensional torus and it acts on Ld(M; G).
We have a canonical map Ld(M; G) -! Ld(M=G) from the above space to d-th
iterated free loop space on M=G. This map is no longer surjective nor injective
in general. Of course when the action G on M is free, the above map is still a
homeomorphism.
__ The space Ld(M; G) can be thought of as the space of pairs (__fl; [OE]), whe*
*re
fl: T d-! M=G is a d-torus in M=G, and [OE] is the conjugacy class of the holon*
*omy
of a periodic lift of __flto a map fl : Rd -! M.
We want to calculate ordinary Euler characteristic of the space Ld(M; G). Ho*
*w-
ever, since this space is infinite dimensional, it may have nonzero Betti numbe*
*rs in
arbitrarily high degrees. We recall that for a finite dimensional manifold admi*
*tting
a torus action, it is well known that Euler characteristic of the fixed point s*
*ub-
manifold is the same as the Euler characteristic of the original manifold. In f*
*act, a
formal application of Atiyah-Singer-Segal Fixed Point Index Theorem predicts th*
*at
the Euler characteristic of Ld(M; G) must be the same as the Euler characterist*
*ic
of T-fixed point subset. Thus, the Euler characteristic of O Ld(M; G) ought to*
* be
given by
d T X (d)
(2-20) O L (M; G) = O M =CG (OE) = O (M; G):
[OE]2Hom (Zd;G)=G
This is the geometric origin of our definition of higher order orbifold Euler c*
*harac-
teristic O(d)(M; G).
We can give a similar geometric interpretation of the higher order p-primary
orbifold Euler characteristic O(d)p(M; G) as Euler characteristic of an infinit*
*e di-
mensional twisted mapping space with a torus action. The Euler characteristic of
12 HIROTAKA TAMANOI
the fixed point subset under this torus action is precisely given by O(d)p(M; G*
*), as in
(2-20). A similar consideration applies to generalized orbifold Euler character*
*istic
OK (M; G) for a general group K: we replace the mapping space Map (; M=G),
where is a manifold with the fundamental group ss1() = K, by G-orbits of
twisted mapping spaces parametrized by Hom (K; G)=G, we then take the Euler
characteristic of constant maps.
x3. Centralizers of wreath products
This section reviews some facts on wreath products. In particular, we explic*
*itly
describe the structure of centralizers of elements in wreath products. This mat*
*erial
may be well known to experts. For example, the order of centralizers in wreath
product is discussed in Macdonald's book [M2, p.171]. However, since the precise
details on this topic seem to be rather hard to locate in literature, our expli*
*cit and
direct description will make this paper more self-contained and it may be useful
for readers from different expertise. The structure of centralizers is describ*
*ed in
Theorem 3-5. This section is independent from the rest of the paper.
Let G be a finite group. The n-th symmetric group Sn acts on the n-fold
Cartesian product Gn by s(g1; g2; : :;:gn) = (gs-1(1); gs-1(2); : :;:gs-1(n)),*
* where
s 2 Sn, gi 2 G. The semidirect product defined by this action is the wreath pro*
*duct
GoSn = Gn oSn. We use the notation Gn to denote this wreath product. Product
and inverse is given by (g; s)(h; t) = (g . s(h); st) and (g; s)-1 = (s-1 (g-1 *
*); s-1 ).
Q
Conjugacy classes in wreath products. Let s = isi be the cycle decompo-
sition of s. If s = (i1; i2; : :;:ir) is a linear representation of si, then t*
*he product
gir. .g.i2gi1 is called the cycle product of (g; s) corresponding to the above *
*repre-
sentation of the cycle si. For each cycle si, the conjugacy class of its cycle *
*product
is uniquely determined.
Corresponding to si, let gi be an element of Gn whose a-th component is given
by (gi)a = (g)a if a 2 {i1; . .;.ir}, and (gi)a = 1Qotherwise. Then for each i*
*; j,
(gi; si) and (gj; sj) commute and we have (g; s) = i(gi; si). We often write
gi = (gi1; gi2; : :;:gir) as if it is an element of Gr. The conjugacy class of *
*the cycle
product corresponding to si is denoted by [gi].
Let G* denote the set of all conjugacy classes of G, and we fix the represen*
*tatives
of conjugacy classes. ForQ[c] 2 G*, let mr(c) be the number of r-cycles in the
cycle decomposition s = isi whose cycle products belong to [c]. This yields a
partition-valued function ae : G* -! P, where P is thePtotality of partitions, *
*defined
by ae([c]) = (1m1(c)2m2(c). .r.mr(c). .).. Note that [c];rrmr(c) = n. The fun*
*ction
ae associated to (g; s) 2 Gn is called the type of (g; s). It is well known th*
*at the
conjugacy class of (g; s) in Gn is determined by its type. This can be explici*
*tly
seen using the conjugation formulae in Proposition 3-1 below.
To describe details of the structure of the wreath product Gn, we use the fo*
*llow-
ing notations. We express any element (g; s) as a product in two ways:
Y Y Y mr(c)Y
(3-1) (g; s) = (gi; si) = (r;c;i; oer;c;i):
i [c]2G*r1 i=1
In the second expression, the conjugacy class of the cycle product correspondin*
*g to
oer;c;iis [r;c;i] = [c].
GENERALIZED ORBIFOLD EULER CHARACTERISTIC 13
Suppose the conjugacy class of the cycle product [gi] = [gir. .g.i1] corresp*
*onding
to si is equal to [c] 2 G*. Choose and fix pi 2 G such that gir. .g.i1= picp-1i*
*for
all i. Let
*
* (i)n
(3-2) ffi(gi) = (gi1; gi2gi1; : :;:gir. .g.i1); i(pi) = (pi; : :;:pi) 2 G*
*r G :
(i)
Here by Gr Gn, we mean that the components of the above elements at the
position a =2 {i1; i2; : :;:ir} are 1 2 G. This convention applies throughout *
*this
section. In the above, i is the diagonal map along the components of si. Let
cr = (c; 1; : :;:1) 2 Gr and cr;i2 Gn is cr along the positions which appear as
components of oer;c;ior si. The following proposition can be checked by direct
calculation.
Q
Proposition 3-1. (1) Let (g; s) = i(gi; si). For a given i, suppose [gi] = [*
*c] and
(i)
|si| = r. Then ffii = ffi(gi) . i(pi) 2 Gr Gn has the property
-1
(3-3) (gi; si) = (ffii; 1) (c; 1; . .;.1); si (ffii; 1) :
For i 6= i, elements (ffii;Q1) and (ffij; 1) commute.
(2) Let (ffi; 1) = i(ffii; 1). Then
i Y Y mr(c)Y j
(3-4) (g; s) = (ffi; 1) (cr;i; oer;c;i) (ffi; 1)-1 :
[c]r1 i=1
From this, it is clear that the conjugacy class of (g; s) 2 Gn is determined*
* by its
type {mr(c)}r;[c].
Actions of wreath products. Let M be a G-manifold. The wreath product Gn
acts on the n-fold Cartesian product Mn by
(g1; g2; : :;:gn); s (x1; x2; : :;:xn) = (g1xs-1(1); g2xs-1(2); : :;:gnxs*
*-1(n)):
The above conjugation formula shows that the fixed point subset of Mn under the
action of (g; s) is completely determined by its type.
Proposition 3-2. Suppose an element (g; s) 2 Gn is of type {mr(c)}. Then
Y P mr(c)
(3-5) (Mn )<(g;s)>~= M r :
[c]
Proof.QWith respect to the decomposition (3-1) of (g; s), we have (Mn )<(g;s)>~=
i(Mri)<(gi;si)>, where ri = |si|, and Mri Mn corresponds to components of si.
If [gi] = [c], then we have the following isomorphisms
~= r <(c ;s )> ~= r <(g ;s )>
M- ! (M i) ri i- --! (M i) i i:
i (ffii;1)
Now in terms of the other decomposition of (g; s) in (3-1), since [r;c;i] = [c]*
*, the
above isomorphism implies
Y Y mr(c)Y Y P
(Mn )<(g;s)>~= (Mri)<(r;c;i;oer;c;i)>~=(M) rmr(c):
[c]r1 i=1 [c]
This completes the proof.
14 HIROTAKA TAMANOI
Centralizers in wreath products. Next, we describe the centralizer CGn (g; s)
in the wreath product Gn. Let (h; t) 2 CGn (g; s) . ThenQ(h; t)(g; s)(h; t)-1*
* =
(g; s). In terms of the cycle decomposition (g; s) = i(gi; si), we see that *
*for
each i there exits a unique j such that (h; t)(gi; si)(h; t)-1 = (gj; sj). Si*
*nce the
conjugation preserves the type, we must have [gi] = [gj] = [c] for some [c] 2 G*
**, and
|si| = |sj|. Thus with respect to the second decomposition in (3-1), the conjug*
*ation
by (h; t) permutes mr(c) elements {(r;c;i; oer;c;i)}i for each [c] 2 G* and r *
*1.
Thus we have a homomorphism
Y Y
(3-6) p : CGn (g; s) -----! Smr(c):
[c]r1
Lemma 3-3. The above homomorphism p is split surjective.
Q Q
Proof. First we construct a homomorphismQ : [c] rSmr(c) -! Sn as follows.
In the decomposition (g;_s)Q=Q i(gi; si), we write each cycle si starting with*
* the
smallest integer. If t 2 [c]_r Smr(c) sends the cycle si = (i1; i2; : :;:ir) *
*to sj =
(j1; j2; : :;:jr), then let (t) = t 2 Sn where t(i`) = j`, 1 ` r. It is clear*
* that
defines a homomorphism, and any element in the image of commutes with s,
because t permutes cycles preserving the smallest integers in cycles appearing *
*in the
cycle decomposition of s. Previously, we constructed an elementQffiQ2 Gn for e*
*ach
element (g; s) in Proposition 3-1. Define a homomorphism : [c] rSmr(c) -!
_ _ *
* _
CGn (g; s) by (t) = (ffi; 1)_1; (t) (ffi; 1)-1 . Using_(3-4), we can check th*
*at (t)
commutes with_(g; s) for any t. Since conjugation by (t) on (g; s) induces the
permutation tamong {(gi; si)}i, we have p O = identityand is a splitting of p.
This completes the proof.
Next we examine the kernel of the homomorphism p in (3-6). If (h; t) 2 Ker p,
then (h; t)(gi; si)(h; t)-1 = (gi; si) for all i. In particular, we have tsit-1*
* = si for
all i, andQconsequently t must be a product of powers of si's. Thus, we may wri*
*te
(h; t) = i (hi; 1)(gi; si)ki for some hi 2 G|si| Gn and 0 ki < |si|, where
(hi; 1) commutes with (gi; si) for any i. Let G(i)r= Gr o Sr Gn be a subgroup
of Gn isomorphic to Gr corresponding to positions appearing in si. Recall that *
*we
defined ffii in (3-3).
Lemma 3-4. For a given i, suppose the cycle product corresponding to si is such
that [gi] = [c]. Then
k
CG(i)r(gi; si)= {(hi; 1) . (gi; si) i | 0 ki < |si|; [(hi; 1); (gi; si*
*)] = 1}
ff
(3-7) = ffii. i(CG (c)) . ffi-1i; 1 . (gi; si)
~= CG (c) . ; where (ar;c)r = c and [ar;c; CG (c)]*
* = 1:
Here ar;c= cr; (12 . .r.) .
Proof. The first equality is obvious. For the second one, first observe that if*
* (h; 1)
with h 2 Gr commutes with cr; (12 . .r.) 2 Gr, then h must be of the form
(h; h; : :;:h) 2 Gr with h 2 CG (c). Conjugation by ffii gives the second descr*
*iption.
For the third description, we simply observe that cr; (12 . .r.) r = (h); 1 *
*2 Gr.
This completes the proof.
GENERALIZED ORBIFOLD EULER CHARACTERISTIC 15
Thus we have a split exact sequence
Y Y mr(c)Y p Y Y
(3-8) 1 -! CG(i)r(r;c;i; oer;c;i) -! CGn (g; s) -! Smr(c) -*
*! 1:
[c]r1 i=1 [c]r1
Since conjugation preserves the type of elements in Gn, the centralizer splits *
*into
(r; [c])-components
Y Y
CGn (g; s) = CGn (g; s) (r;[c]);
[c]r1
Q
where CGn (g; s) (r;[c])is the centralizer of the element i(r;c;i; oer;c;i) i*
*n the sub-
group Grmr(c)corresponding to positions appearing in oer;c;ifor 1 i mr(c). The
conjugation by (ffii; 1) maps the following split exact sequence
mr(c)Y p
(3-9) 1 -! CG(i)r(r;c;i; oer;c;i) -! CGn (g; s) (r;[c])r;c--!Smr(c) -*
*! 1
i=1
isomorphically into the following split exact sequence
mr(c)Y imr(c)Y j
1 -! i CG (c) . -! CGrmr(c) (cr;i; oer;c;i) -! Smr(c)*
* -! 1:
i=1 i=1
Here a(i)r;cis ar;calong components of oer;c;i. Direct calculation shows that *
*in the
second exact sequence, Smr(c) acts on the left side product by permuting factor*
*s.
Hence the semidirect product structure in (3-9) is indeed isomorphic to a wreath
product. Hence we obtain the following description of the centralizer of (g; s*
*) in
Gn.
Theorem 3-5. Let (g; s) 2 Gn have type {mr(c)}r;[c]. Then
Y Y
(3-10) CGn (g; s) ~= CG (c) . o Smr(c) ;
[c]r1
where (ar;c)r = c 2 CG (c). Here, the isomorphism is induced by conjugation by *
*ffi
in Proposition 3-1.
x4 Higher order orbifold Euler
characteristic of symmetric products
In this section, we prove Theorem A in the introduction. Explicitly writing *
*out
jr(Zd), the formula we prove is the following:
X h Y 2 d-1i(-1)O(d)(M;*
*G)
(4-1) qn O(d)(M; G o Sn) = (1 - qr1r2...rd)r2r3...rd :
n0 r1;r2;:::;rd1
When d = 0, this is Macdonald's formula applied to M=G:
X 1
(4-2) qn O SP n(M=G) = ______________O(M=G):
n0 (1 - q)
When d = 1, the formula was proved by Wang [W]. We prove formula (4-1) by
induction on d 0, using Macdonald's formula as the start of induction. For the
inductive step, we need the following Lemma.
16 HIROTAKA TAMANOI
Lemma 4-1. Let G . be a group generated by a finite group G and an element
a such that a commutes with any element of G and \ G = for some integer
r 1. Suppose the element a acts trivially on a G-manifold M. Then
(4-3) O(d)(M; G . ) = rdO(d)(M; G):
` r-1
Proof. First note that G . = i=0 G . ai and |G . | = r . |G|. Observe*
* that
two elements of the form gai and haj, where g; h 2 G, commute if and only if g;*
* h
commute, since a is in the center of G . . Now by definition,
X i1 i2 id+1
O(d)(M; G . ) = __1___r . |G|O M
(g1;:::;gd+1)
0i`. Hence summing ov*
*er
i`'s first, the above becomes
d+1 X
O(d)(M; G . ) = _r____r . |G|O M= rd . O(d)(M; G):
(g1;:::;gd+1)
This completes the proof.
Proof of formula (4-1). By induction on d 0. When d = 0, the formula is
Macdonald's formula (4-2) and hence it is valid.
Assume the formula is valid for O(d-1) for d 1. Let G* = {[c]} be the total*
*ity
of conjugacy classes of G. By Proposition 2-5, we have
X X X
(*) qn O(d)(M; Gn) = qn O(d-1) (Mn )<(g;s)>; CGn((g; s)) :
n0 n0 [(g;s)]
Let (g; s) 2 Gn have type {mr(c)}. Then by Proposition 3-2 and Theorem 3-5, we
have the following compatible isomorphisms:
Y Y
(Mn )<(g;s)>~= (M)mr(c);
[c]r1
Y Y
CGn (g; s) ~= (CG (c) . ) o Smr(c) ;
[c]r1
where (ar;c)r = c 2 CG (c) and ar;cacts trivially on M. The above isomorphis*
*ms
are compatible in the sense that the action of CGn (g; s) on (Mn )<(g;s)>trans*
*lates,
via conjugation by (ffi; 1) in Proposition 3-1, to the action of the wreath pro*
*duct
(CG (c) . ) o Smr(c) on (M)mr(c) for any [c] 2 G* and r 1. Since the
conjugacy classes of elements in Gn are determined by their types, the summation
over all conjugacy classes [(g; s)] corresponds to the summation over all mr(c)*
* 0
GENERALIZED ORBIFOLD EULER CHARACTERISTIC 17
P
for all [c] 2 G* and r 1 subject to [c];rrmr(c) = n. By the multiplicativity*
* of
generalized orbifold Euler characteristic (2-3), the formula (*) becomes
X X X Y
qn O(d)(Mn ; Gn)= qn O(d-1) (M)mr(c); (CG (c)) o Smr*
*(c)
n0 n0 P mr(c)0 [c];r
rmr(c)=n
X Y
= (qr)mr(c)O(d-1) (M)mr(c); (CG (c)) o Smr(*
*c)
mr(c)0 [c];r
Y X
= (qr)m O(d-1) (M)m ; (CG (c) . )m
[c];rm0
By inductive hypothesis, this is equal to
Y h Y r2r2...rd-2i-O(d-1)(M;CG (c)<*
*ar;c>)
= 1 - (qr)r1:::rd-1 3 d-1
[c];rr1;:::;rd-11
hY r r2...rd-2i- P[c]O(d-1)(M;CG (c))
= 1 - qrr1...rd-12 3 d-1 :
r;r1;:::;rd-1
By Lemma 4-1, O(d-1)(M; CG (c).) = rd-1 O(d-1)(M; CG (c)). Hence su*
*m-
ming over [c] 2 G* and using Proposition 2-5, we see that the exponent is equal*
* to
(-1)rd-1 O(d)(M; G). Thus, renaming r as rd, the above is equal to the right ha*
*nd
side of (4-1). This completes the inductive step and the proof is complete.
Now let M = pt. Using (2-2) with K = Zd and G replaced by G or Gn, we get
Corollary 4-2. For each d 0 and for any finite group G, we have
X fi fi hY 2 d-1i(-1)|Hom (Zd;G*
*)=G|
(4-4) qn fiHom Zd; Gn =Gnfi= (1 - qr1...rd)r2r3...rd :
n0 r1;:::;rd1
The above formula is the formula (1-7) in the introduction. Furthermore, let*
*ting
G be the trivial group, we get
X fi fi hY 2 d-1i(-1)
(4-5) qn fiHom Zd; Sn =Snfi= (1 - qr1...rd)r2r3...rd :
n0 r1;:::;rd1
fi
Here, as remarked in the introduction, we recognize |Hom Zd; Sn =Snfias the
number of isomorphism classes of Zd-sets of order n. Any finite Zd-set decompos*
*es
into a union of transitive Zd-sets, and any isomorphism class of transitive Zd-*
*set of
order r corresponds to a unique subgroup of Zd of index r, by taking the isotro*
*py
subgroup. Thus, letting jr(Zd) be the number of index r subgroups of Zd, we have
X fi fi hY di(-1)
(4-6) qn fiHom Zd; Sn =Snfi= (1 - qr)jr(Z ) :
n0 r1
By comparing (4-5) and (4-6), we get a formula for jr(Zd). However, we can easi*
*ly
directly calculate the number jr(Zd) as follows. This calculation is well known*
* (for
its history, see [So]) and gives an alternate proof of (4-5).
18 HIROTAKA TAMANOI
Lemma 4-4. For any r 1, d 1, we have
X X
(4-7) jr(Zd) = r2r23. .r.d-1d; and jr(Zd) = m . jm (Zd-1 ):
r1...rd=r m|r
Proof. Let e1; e2; : :;:ed be the standard basis of the lattice. It is easyPto *
*see that any
sublattice of index r has a unique basis {xi}di=1of the form xi = riei+ i
(5-3) O K(d)G (M) = Op (M; G) = ___|G| O(M ):
OE:ZdpxZ-! G
Our objective in this section is to calculate the Euler characteristic of eq*
*uivariant
Morava K-theory of symmetric products (Mn ; G o Sn) for n 1. By Proposition 5-
1, this homotopy theoretic number can be calculated as the higher order p-prima*
*ry
orbifold Euler characteristic. We prove
GENERALIZED ORBIFOLD EULER CHARACTERISTIC 19
Theorem 5-2. For any d 0 and G-manifold M,
X hY `1 `2 `d`2 2`3 (d-1)`di(-1)O*
*(d)p(M;G)
(5-4) qn O(d)p(Mn ; G o Sn) = (1 - qp p ...p)p p ...p *
* :
n0 `1;:::;`d0
The proof is very similar to the one for the formula (4-1). Note the similar*
*ities
of exponents. Since the formula (5-4) is p-primary, there are differences at m*
*any
parts of the proof, although the idea of the proof is the same. Thus, we belie*
*ve
that it is better to give a complete proof of the above formula (5-4) rather th*
*an
explaining differences of proofs between formulae (4-1) and (5-4).
Proof of Theorem 5-2. By induction on d 0. When d = 0, by (2-13) we have
O(0)p(Mn ; G o Sn) = O SP n(M=G) , and the formula (5-4) in this case asserts
X 1
qn O SP n(M=G) = ______________O(M=G);
n0 (1 - q)
which is valid due to Macdonald's formula (1-8).
Assume that the formula (5-4) is valid for O(d-1)pfor d 1. By Proposition 2*
*-6,
X X X
(*) qn O(d)p(Mn ; G o Sn) = qn O(d-1)pM; CGn(ff) :
n0 n0 [ff]
Here the second summation on the right hand side is over all Gn-conjugacy class*
*es
[ff] of elements of p-power order in Gn, that is [ff] 2 Hom (Zp; Gn)=Gn. Let *
*ff =
(g; s) 2 Gn. Since ff has order a power of p, the second component s 2 Sn must *
*have
order a power of p. Thus, the type of ff = (g; s) must be of the form {mpr(c)}r*
*;[c].
Here, [c] runs over all G-conjugacy classes of elements of order powers of p. *
*We
indicate this by the notation [c]p. So [c]p 2 Hom (Zp; G)=G. By Proposition 3-*
*1, ff
is conjugate to an element of the form
apr;c;i
Y Y mpr(c)Yz________"_________- r
ff ~ (c; 1; : :;:1); oepr;c;i); where (apr;c;i)p = (c; c; : *
*:;:c); 1 :
r0 [c]pi=1 _____-z____"pr _____-z__*
*__"pr
By Proposition 3-2 and Theorem 3-5, the fixed point subset of Mn under the acti*
*on
of ff, and the centralizer of ff in Gn are each isomorphic to
Y P
(Mn )~= (M) rmpr(c);
[c]p
Y Y r
CGn(ff) ~= (CG (c) . ) o Smpr(c) ; (apr;c)p = c 2 CG (c):
[c]pr0
The above isomorphisms are compatible with the action of the centralizer on the
fixed point subset. The summation over all conjugacy classes [ff]p can be repla*
*ced
20 HIROTAKA TAMANOI
by the summation over all the types {mpr(c)}r;[c]p. By multiplicativity of O(d-*
*1)p,
the right hand side of (*) becomes
X X Y
(*)= qn O(d-1)p(M)mpr(c); (CG (c) . ) o Smpr(c)
n0 P mpr(c)0 [c]p;r
prmpr(c)=n
X Y r
= (qp )mpr(c)O(d-1)p(M)mpr(c); (CG (c) . ) o Smpr(c)
mpr(c)0 [c]p;r
Y X r
= (qp )m O(d-1)p(M)m ; (CG (c) . ) o Sm
[c]p;rm0
By inductive hypothesis, the summation inside is given by
Y h Y r `1 `d-1`2 2`3 (d-2)`d-1i(-1)O(d-1)p(M;CG (c*
*).)
= (1 - (qp )p ...p )p p ...p
[c]p;r`1;:::;`d-10
h Y `1 `d-1 r `2 2`3 (d-2)`d-1i(-1) P[c]pO(d-1)p(M;CG (*
*c).)
= (1 - qp ...p p)p p ...p :
`1;:::;`d-1;r0
At this point, we need a sublemma which is completely analogous to Lemma 4-1.
Sublemma. Let G . be a group generated by a finite group G and an element a
of orderra power of p such that a commutes with any element in G and G \ =
2 G(p). Suppose acts trivially on M. Then
(5-5) O(d)p(M; G . ) = prdO(d)p(M; G):
The proof of this sublemmaris analogous to Lemma 4-1. Using this sublemma
and the fact that (apr;c)p = c 2 CG (c), we see that the exponent of the previo*
*us
expression is equal to
X X
O(d-1)p(M; CG (c) . )= pr(d-1) O(d-1)pM; CG (c)
[c]p [c]p
= pr(d-1)O(d)p(M; G):
Thus, renaming r as `d, the expression (*) finally becomes
h Y `1 `d-1 ` `2 2`3 (d-2)`d-1(d-1)`i(-1)O(d)p(M:G)
(*) = (1 - qp ...p p)dp p ...p p d ;
`1;:::;`d-1;`d0
which is the right hand side of formula (5-4). This completes the proof.
Now letting M be a point and using (2-2) with K = Zdp, we get
Corollary 5-3. Let Gn = G o Sn for n 0. For any d 0, we have
(5-6)
X fi fi hY `1 `d`2 2`3 (d-1)`di(-1)|Hom (Zd*
*p;G)=G|
qn fiHom(Zdp; Gn)=Gnfi= (1 - qp ...p)p p ...p *
* :
n0 `1;:::;`d0
In particular, letting G to be the trivial group, we get
GENERALIZED ORBIFOLD EULER CHARACTERISTIC 21
Corollary 5-4. For any d 0,
X fi fi hY `1 `2 `d`2 2`3 (d-1)`di(-*
*1)
(5-7) qn fiHom(Zdp; Sn)=Snfi= (1 - qp p ...p)p p ...p :
n0 `1;`2;:::;`d0
fi fi
Observe that fiHom(Zdp; Sn)=Snfiis the number of isomorphism classes of Zdp-*
*sets
of order n. Any finite Zdp-sets can be decomposed into transitive Zdp-sets whic*
*h must
have order powers of p. For any r 0, isomorphism classes of transitive Zdp-se*
*ts
of order pr are in 1 : 1 correspondence with index pr subgroup of Zdp, by taking
isotropy subgroups. Let jpr(Zdp) be the number of index pr subgroup of Zdp. Note
that j`(Zdp) is zero unless ` is a power of p. This consideration of decomposin*
*g finite
Zdp-sets into transitive ones immediately gives the following formula.
X fi fi hY r di(-1)
(5-8) qn fiHom(Zdp; Sn)=Snfi= (1 - qp )jpr(Zp) :
n0 r0
There is an easy way to calculate the number jpr(Zdp).
Lemma 5-5. For any r 0 and d 1, we have
X X
jpr(Zdp)= p`2p2`3. .p.(d-1)`d= n2n23. .n.d-1d;
P ` =r n1...nd=pr
(5-9) iXi
jpr(Zdp)= p` . jp`(Zd-1p):
0`r
Proof. Any subgroup H of Zdpof index pr for any r 0 is a closed subgroup and
hence it has a structure of a Zp-submodule, and as such it is a free module. L*
*et
the standard basis of Zdpbe e1; e2; : :;:ed. It isPeasy to see that H has a un*
*ique
Zp-module basis {xi}ni=1of the form xi = p`ieiP+ i