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CALCULATION OF HIRZEBRUCH GENERA FOR MANIFOLDS ACTED ON BY THE GROUP Z/p VIA INVARIANTS OF THE ACTION

Taras E. Panov
Abstract. We obtain general formulae expressing Hirzebruch genera of a manifold with Z/p-action in terms of invariants of this action (the sets of weights of fixed points). As an illustration, we consider numerous particular cases of well-known genera, in particular, the elliptic genus. We also describe the connection with the so-called Conner­Floyd equations for the weights of fixed points.

Introduction In this paper we obtain general formulae expressing Hirzebruch genera of a manifold acted on by Z/p with finitely many fixed points or fixed submanifolds with trivial normal bundle via invariants of this action. We also describe the connection with the so-called Conner­Floyd equations for the weights of fixed points. Actions of Z/p were studied in [12], [13], [11], [8], where the so-called Conner­ Floyd equations were deduced within cobordism theory (see formulae (31), (32)). These equations form necessary and sufficient conditions for a sets of elements of Z/p to be the set of weights of some Z/p-action (see § 3 for the definition). Two approaches for the calculation of Hirzebruch genera of a stably complex manifold with a Z/p-action were proposed in [5]. The first approach is based on the application of the Atiyah­Bott­Lefschetz fixed point formula [1], and so for its realization it is necessary to have an elliptic complex of bundles that are associated to the tangent bundle of the manifold. The Atiyah­ Bott­Lefschetz formula obtained in [1] generalizes the classical Lefschetz formula for the number of fixed points and enables us to calculate the equivariant index of an elliptic complex of bundles over a manifold by means of certain contribution functions of the fixed submanifolds (see the details below). In particular, if an operator acts on a manifold with finitely many fixed points, the corresponding equivariant index can be expressed in terms of the fixed point weights. It was shown in [5] how to express the Todd genus, which is the index of an elliptic complex (namely, the Dolbeault complex) over a manifold with Z/p-action, via the equivariant index of the same complex for the action of the generator of Z/p. This equivariant index enters into the Atiyah­Bott­Lefschetz formula. In this way one deduces the formulae expressing the Todd genus in terms of the weights of
1991 Mathematics Subject Classification. 57R20, 57S25 (Primary) 58G10 (Secondary). Partially supported by the Russian Foundation for Fundamental Research, grant no. 96-0101414. Typeset by AMS-TEX

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TARAS E. PANOV

fixed points for this Z/p-action. The formulae obtained by this method contain the number-theoretical trace of a certain algebraic extension of fields of degree (p - 1). In this paper, we use the same approach to obtain the formulae for other genera of manifolds with Z/p-action: the signature (or the L-genus), the Euler number, ^ the A-genus, the general y -genus, and the elliptic genus. By this method we also obtain some general equations (see. § 5) for an arbitrary Hirzebruch genus having the property to be the index of a certain elliptic complex of bundles associated to the tangent bundle of a manifold with Z/p-action. In this paper we use the generalized Lefschetz formula in the somewhat different formulation, stated in [3]. This formula and especially the "recipe" it suggests for calculating the equivariant index of a complex via contribution functions of fixed points (see. § 5) are more convenient for applications than the formula from [1] used in [5]. The generalized Lefschetz formula was deduced in [3] from the cohomological form of the Atiyah­Singer index theorem, which was also proved there. We apply both formulae: some of the results (see § 4) we obtain are based on the "old" Lefschetz formula of [1], while others use the "recipe" in § 5 based on the formula from [3]. Another approach to the equations for Hirzebruch genera, also proposed in [5], is based on an application of cobordism theory in the same way as in the derivation of the Conner­Floyd equations in [12], [13]. In [5], the authors give a formula expressing the mod p cobordism class of a stably complex manifold with a Z/paction in terms of invariants of the action. Then they show that the difference between the two formulae for the Todd genus (obtained by the first and second methods) is exactly the sum of the Conner­Floyd equations for the Todd genus. In this paper we show (see Theorem 7.1) that the difference between the two seemingly different formulae deduced by these two methods for an arbitrary genus is a weighted sum (with integer coefficients) of the Conner­Floyd equations for this genus. A case of particular interest is that of the so-called elliptic genus. In Witten's papers, a certain invariant was assigned to each oriented 2n-dimensional manifold M 2n . This invariant is the equivariant index of the Dirac-like operator for the canonical action of circle S 1 on manifold's loop space. S. Ochanine [14] showed that this index is a Hirzebruch genus corresponding to the elliptic sine; which led to the term "elliptic genus". In [2], [9] and other papers, the rigidity theorem for the elliptic genus of manifolds with S 1 -action was proved. This theorem states that if we regard the equivariant elliptic genus S 1 (M ) of such a manifold as a character of the group S 1 , then S 1 (M ) is the trivial character and is equal to the elliptic genus (M ). At the same time, the elliptic genus takes its values in the ring Z 1 [, ], and its value on any manifold M 2n is a modular form of weight n 2 on the subgroup 0 (2) SL2 (Z) (cf. [7]). In this paper we obtain formulae for the elliptic genus of a manifold with a Z/p-action having finitely many fixed points. A summary of results of this part of the paper has already been published in [15]. As an application we deduce certain relations between the Legendre polynomials by applying our formulae to a special action of Z/p on CP n (see § 8). In the remaining part of this article (see § 9) we generalize our constructions to the case of Z/p-actions with fixed submanifolds whose normal bundles are trivial. § 1. Necessary information about Hirzebruch genera Let M
2n

be a manifold with a complex structure in its stable tangent bundle,


CALCULATION OF HIRZEBRUCH GENERA FOR MANIFOLDS

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that is, there is k such that T M 2k is a complex bundle. We write the total Chern class of the tangent bundle T M as c(T M ) = 1 + c1 (M ) + c2 (M ) + · · · + cn (M ) = (1 + x1 )(1 + x2 ) · · · (1 + xn ). This means that the total Chern class of M is written as the product of the Chern classes of "virtual" line bundles whose sum gives T M . Therefore, ci (M ) is the ith elementary symmetric function in x1 , . . . , xn . To each series of the form Q(x) = 1 + · · · with coefficients in a certain ring n 2n there corresponds the Hirzebruch genus Q (M 2n ) = ] (see [6]). i=1 Q(xi ) [M -1 Along with Q(x), we introduce f (x) = x/Q(x) and g (u) = f (u). Each Hirze- bruch genus gives rise to a formal group law F (u, v ) = g 1 g (u) + g (v ) with - logarithm g (u) (see [5]). The corresponding power system [u] = g 1 ng (u) n (the nth power in the formal group law F ). Hirzebruch genera for real orientable manifolds M 4n are defined similarly. Here we replace the Chern classes ci by the Pontryagin classes pi , and xi by x2 , that is, i p(T M ) = 1 + p1 (M ) + p2 (M ) + · · · + pn (M ) = (1 + x2 )(1 + x2 ) · · · (1 + x2 ). 1 2 n Hence we now have Q(x2 ) instead of Q(x). The formulae obtained in this paper refer mainly to the following Hirzebruch genera. 1. The universal genus corresponds to the identity homomorphism id : U Q U Q. The corresponding formal group law of "geometric cobordisms" F (u, v ) (see [12]) is universal, that is, for any formal group law F (x, y ) over a ring there is unique ring homomorphism : U such that F (x, y ) = F (u, v ) (see [5]). The logarithm of F (u, v ) is


g (u) =
n=0

[CP n ] n u n+1

+1

=
n=0

CP n n+1 u . n+1

Therefore, for any Hirzebruch genus we have


g (u) =
n=0

[CP n ] n+1 u . n+1

(1)

2. The Todd genus td(M ) corresponds to the following series: Qtd (x) = x 1-e
-x

,

ftd (w) = 1 - e

-w

,

gtd (u) = - ln(1 - u).

These formulae could also be deduced from the fact that td(CP n ) = 1. Indeed, using the identity (1), we obtain


gtd (u) =
n=0

un+1 = - ln(1 - u). n+1

The corresponding power system is
- [u]td = gtd1 ngtd (u) = 1 - e n n ln(1-u)

= 1 - (1 - u)n .


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TARAS E. PANOV

Thus, for the Todd genus ftd (w) = 1 - e
-w

,

gtd (u) = - ln(1 - u),

[u]td = 1 - (1 - u)n . n

(2td )

3. The Euler number e(M ) (of the tangent bundle). Since e(CP n ) = n + 1, we deduce (see [13]) that ge (u) = u , 1-u
- fe (w) = ge 1 (w) =

w . 1+w

So Q(x) = 1 + x and, as might be expected,
n

e[M ] =
i=1

Q(xi ) [M ] = (x1 x2 . . . xn )[M

2n

] = cn [M

2n

].

Next,
- [u]e = ge 1 nge (u) = n

nu . 1 + (n - 1)u nu . 1 + (n - 1)u

Thus, for the Euler number, fe (w) = w , 1+w ge (u) = u , 1-u [u]e = n (2e )

4. The signature (the L-genus) corresponds to the following series: QL (x) = x x(ex + e-x ) = , fL (w) = tanh(w), tanh(x) ex - e- x 1+u 1 . gL (u) = arctanh(u) = ln 2 1-u

This is in accordance with Hirzebruch's theorem (the L-genus equals the signature, see [6]), from which we deduce that L(CP 2n ) = 1, L(CP 2n+1 ) = 0. The corresponding power system is [u ]L n =g
-1 L

ngL (u) =

e e

n 2 n 2

ln ln

1+u 1- u 1+u 1- u

-e +e

- -

n 2 n 2

ln ln

1+u 1-u 1+u 1-u

(1 + u)n - (1 - u)n = . (1 + u)n + (1 - u)n

Thus, for the L-genus, fL (w) = tanh(w), gL (u) = 1 ln 2 1+u 1-u , [u]L = n (1 + u)n - (1 - u)n . (2L ) (1 + u)n + (1 - u)n

5. There is a one-parametric genus which generalizes three previous examples, namely, the y -genus. This is the Hirzebruch genus that corresponds to the following series: x(1 + y e-x(1+y) ) , Qy = 1 - e-x(1+y) w 1 - e-w(1+y) 1 + yu 1 fy (w) = = ln . , gy (u) = -w(1+y ) Qy (w) 1+y 1-u 1 + ye


CALCULATION OF HIRZEBRUCH GENERA FOR MANIFOLDS

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Neglecting the normalizing condition fy (w) = w + · · · and replacing w(1 + y ) by w, we obtain 1 + yu 1 - e-w ^ , gy (u) = ln ^ fy (w) = . -w 1 + ye 1-u The expression for the power system associated to the y -genus is the same in both cases: n 1- 1 - 1+yu n n u (1 + y u) - (1 - u) [u]y = = (2y ) n. n n + y (1 - u)n (1 + y u) 1- 1 + y 1+yu u For y = 0, -1, 1 we get respectively the Todd genus, the Euler number and the Lgenus. ^ 6. The A-genus corresponds to the following series: QA (x) = fA (w) = 2 sinh w , 2 x/2 = sinh(x/2) e x -e , u + 2 1+ u2 4 .

x/2

-x/2

gA (u) = 2 arcsinh

u = ln 2

The corresponding power system is
- [u]A = gA 1 ngA (u) = 2 sinh n arcsinh(u/2) n

= ^ Thus, for the A-genus, fA (w) = 2 sinh(w/2),

u + 2

u2 1+ 4

n

-

u + 2

u2 1+ 4

-n

.

gA (u) = 2 arcsinh(u/2),

[u]A = 2 sinh n arcsinh(u/2) . n (2A )

§ 2. Calculation of Hirzebruch genera by means of the Atiyah­Singer index theorem We deal with the following elliptic complexes: the de Rham complex , (0 ) - (1 ) - · · · - (n ), and the Dolbeault complex p, , (
p,0 d d d

) - (p,1 ) - · · · - (p,n ),

d

d

d

where i = i (T M ) is the bundle of differential i-forms on M nd p,i = p (T M ) i ( T M ) p (T M ) i (T M ) is the bundle of differential forms of type (p, i). (If the manifold M is endowed with an Hermitian metric, then T M T M .) The following is the Atiyah­Singer index theorem in cohomological form (see [3]).


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TARAS E. PANOV

Theorem 2.1. Let M be a compact, oriented, differentiable manifold of dimension 2n and let E = {di : Ei Ei+1 } be an el liptic complex (i = 0, . . . , m - 1) associated to the tangent bund le (that is, al l bund les Ei are associated to T M ). Then the index of this complex is determined by the fol lowing formula : ind(E ) = (-1)
n

1 e(T M )

m

(-1)i ch(Ei ) td(T M C) [M ],
i=0

where ch(Ei ) is the Chern character of bund le Ei . Remark. Formally factoring the Euler class e(T M ) = x1 . . . xn out of td(T M C) = -xj xj n in the previous expression, we get the following formula: j =1 1-e-xj · 1-exj
m n

ind(E ) =
i=0

(-1)i ch(Ei )
j =1

1 xj · 1 - e-xj 1 - e
n

x

j

[M ].

(3)

Let us consider the elliptic complex Xy = p=0 y p p,i . (We note that this i ob ject becomes a true elliptic complex only after replacing y by actual integers. But its index is obviously defined for arbitrary y as a polynomial in y . In what follows we regard such ob jects as elliptic complexes.) Using the Atiyah­Singer index theorem, we easily prove the following fact, which was initially proved by Hirzebruch (see [6]). Theorem 2.2. The index of the el liptic complex {Xy } equals the y -genus (defined i above) of the manifold M , that is,
n

ind(X ) = y [M ] =
j =1

y

xj (1 + y e-xj ) [M ]. 1 - e-xj

Proof. Using formula (3), we have:
n n

ind(Xy ) =
i=0 n

(-1)i ch(Xy ) i
j =1 n i i

xj 1 - e-
p p

xj

1 1-e

x

j

[M ] xj 1 - e-
xj

n

=

ch
i=0 n

(-1) T M
n x

ch
p=0 -xj

yTM
j =1 n



xj

1 1 - ex

j

[M ]

= =
j =1

(1 - e )
j

(1 + y e
j =1

)
j =1

j =1 n

xj 1 - e-

x

j

1 1-e

[M ]

xj (1 + y e-xj ) [M ] = y [M ]. 1 - e-xj

Here we have used the following fact (see, for example, [7]).


CALCULATION OF HIRZEBRUCH GENERA FOR MANIFOLDS

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Lemma 2.3. Let E be a complex n-dimensional bund le over a differentiable manifold X , and let c(E ) = 1 + c1 (E ) + · · · + cn (E ) = (1 + x1 ) . . . . . . (1 + xn ) be the formal factorization of the total Chern class. We define


t E :=
k=0

( E )t ,

k

k

St E :=

(S k E )tk .
k=0

Then
n n

ch(t E ) =

(1 + te ),
i

x

ch(St E ) =
i=1

i=1

1 . 1 - texi

(4)

In particular, if we consider the complexes {0,i },
n

Ei =

n p=0

(-1)p p,i ,

Li = p=0 p,i , we see that their indexes are respectively td[M ] = ind(0, ), e[M ] = ind(E ), and L[M ] = ind(L). Now we construct an elliptic complex whose index, under some additional as^ sumptions, equals the A-genus of the manifold M . Suppose that c1 (M ) 0 mod 2. (This is equivalent to the condition w2 (M ) = 0, and hence to the existence of a spinor structure on M .) Then there is a line bundle L over M such that LL = n T M , that is, for c(M ) = (1+x1 ) . . . (1+xn ) we have c(L) = 1- x1 +···+xn 2 and ch(L) = exp - x1 +···+xn . We introduce the complex {Ai = 0,i L}. 2 ^ Theorem 2.4. The index of the above el liptic complex A equals the A-genus of M :
n

^ ind(A) = A[M ] =
j =1

xj [M ]. 2 sinh(xj /2)

Proof. We again use formulae (3) and (4):
n n

ind(A) =

ch(L) ch
i=0

(-1)i

0,i j =1 n

xj 1 - e-
i i

xj

1 1 - ex
n

j

[M ]

=

x1 + · · · + xn exp - 2
n n

ch
i=0 n

(-1) T M
j =1

xj 1 -xj 1 - e 1-e [M ]

xj

[M ]

=
j =1 n

e-

x j /2 j =1

(1 - exj )
/2 xj j =1 n

xj 1 - e-

xj

1 1-e

x

j

=
j =1

xj e-xj 1 - e-

[M ] =
j =1

xj ^ [M ] = A(M ). 2 sinh(xj /2)

Thus we have constructed elliptic complexes associated to the tangent bundle of M that enable us to calculate all the Hirzebruch genera in § 1.


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TARAS E. PANOV

§ 3. The problem of calculating Hirzebruch genera for manifolds with Z/p-action in terms of invariants of the action Let g be a points) acting action of Z/p JPj (g ) of the transversal endomorphism (that is, g has only finitely many fixed on a manifold M 2n such that g p = 1 for some prime p. (Hence an is given.) Let P1 , . . . , Pq be the fixed points, and the Jacobi matrix map g at the point Pj (j = 1, . . . , q ) has eigenvalues
(j ) k



2 ix = exp p

(j ) k

,

x

(j ) k

=0

mod p,

k = 1, . . . , n.

Suppose also that there is given an elliptic complex E on the manifold M . Let us give the following definition, which is taken from [1]. Definition 3.1. A lifting of g to the components of the el liptic complex E is a set of linear differential operators i : (g Ei ) (Ei ). Here g Ei is the pullback of the bundle Ei under the map g , and (Ei ) denotes the linear space of sections of the bundle Ei . Using i , one can define the "geometric" endomorphisms Ti (g , ) : (Ei ) (Ei ) to be the composite of and g , Ti (g , ) = i g , where g : (Ei ) (g Ei ) is the natural map from the sections of the bundle Ei into the sections of the pullback g Ei . Under these assumptions we have the following general Atiyah­ Bott­Lefschetz theorem (see [1]). Theorem 3.2. Suppose that g : M M is a transversal endomorphism of a compact oriented manifold M . Let E be an el liptic complex on M and let i : (g Ei ) (Ei ) be a lifting of g to the components of the complex E such that the corresponding "geometric" endomorphisms Ti (g , ) : (Ei ) (Ei ) define an endomorphism T (g , ) of the complex E (that is, the Ti commute with the difn ferentials di ). Then the "equivariant index" ind(g , E ) := i=1 (-1)i tr Ti (where Ti : H i (E ) H i (E )) is given by the formula
q

ind(g , E ) =
j =1

(Pj ),

(5)

where (Pj ) C depends only on local properties of T and i at the point Pj . In particular, if the operator i induces an endomorphism i (Pj ) : Ei,Pj Ei, at each fixed point Pj , then
n

Pj

(Pj ) =
i=0

(-1)i

tr i (Pj ) . det 1 - JPj (g )

(6)

Example 3.1. If Ei = i (M ) is the de Rham complex, then i (Pj ) = i JPj (g ). Example 3.2. If Ei = p,i (M ) is the Dolbeault complex, then ¯ i (Pj ) = p JPj (g ) i J Pj (g ), where JPj (g ) is the holomorphic part of the Jacobi matrix JPj (g ).


CALCULATION OF HIRZEBRUCH GENERA FOR MANIFOLDS

9

Let us find (Pj ) for the Dolbeault complex p, . Since det 1 - JPj (g ) ¯ det 1 - JPj (g ) det 1 - J Pj (g ) , we deduce from (6) that (Pj ) = = tr p JPj (g )
i

=

¯ (-1)i i J Pj (g )

¯ det 1 - JPj (g ) det 1 - J Pj (g ) ¯ tr p JPj (g ) tr p JPj (g ) det 1 - J Pj (g ) = . ¯ det 1 - JPj (g ) det 1 - JPj (g ) det 1 - J Pj (g ) (-1)i tr i A
i

Here we have used the following well-known identity from linear algebra: det(1 - A) = for any linear operator A. Thus, p (Pj ) = tr p JPj (g ) det 1 - JPj (g ) . (8) (7)

Theorem 3.2 and formula (8) enable us to find the fixed point contribution functions (Pj ) for the complexes 0, , E , L, Xy . These complexes calculate respectively the Todd genus, the Euler number, the L-genus and the y -genus of the manifold M . § 4. Calculations for the Todd genus, Euler number and the L-genus 4.1. Calculations for the Euler numb er. Let us consider the complex Ei = n p p,i p=0 (-1) , for which ind(E ) = e(M ). In this case
n

i (Pj ) =
p=0

¯ (-1)p p JPj (g ) i J Pj (g ),

and the fixed point contribution functions are as follows (see (8)):
n

e (Pj ) =
p=0

(-1) p (Pj ) =

p

n p=0

(-1)p tr p JPj (g )

det 1 - JPj (g )
q

= 1.

(9)

Here we have used formula (7). From this and theorem 3.2 we deduce that ind(g , E ) =
j =1 1 Since p lZ/p ind(g l , E ) = s is the alternating sum of dimensions of the invariant subspaces for the action of g on the cohomology of the complex E , and since ind(1, E ) = ind(E ) = e(M ), we have p-1

e (Pj ) = q .

ind(1, E ) = e(M ) = -
l=1

ind(g l , E ) + ps = ps - q (p - 1) = q + p(s - q ).

Therefore, we obtain the following formula for the Euler number: e(M ) q (mod p). (10)


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TARAS E. PANOV

4.2. Calculations for the To dd genus. The calculation of the Todd genus of a stably complex manifold in terms of the Z/p-action was carried out by Buchstaber and Novikov in [5]. To make our exposition complete, we give their results here. Definition 4.1. The Atiyah­Bott function ABtd (x1 , . . . , xn ) of a given fixed point is the following function of the set of weights x1 , . . . , xn , xi Z/p:
n

ABtd (x1 , . . . , xn ) = - Tr
k=1

1 1-e
2 ixk /p

, .

(11)

where Tr : Q( ) Q is the number-theoretical trace, := e It was shown in [5] that
q

2 i/p

ABtd (x
j =1

(j ) 1

, . . . , x(j ) ) td(M ) mod p, n

(12)

The number-theoretical trace in the definition of Atiyah­Bott functions for the Todd genus was also calculated in [5]: ABtd (x1 , . . . , xn ) -
n

p[u]td 1 p- [u]td p pu [u]td p

n k=1 n k=1

u [u]td xk u [u]td xk

mod p,
n

(13) mod p.

ABtd (x1 , . . . , xn )
m=0

m

Here [u]td is the q th power in the formal group law corresponding to the Todd genus q (see formula (2T )), and h(u) k is the coefficient of uk in the power series h(u). 4.3. Calculations for the L-genus. Now let us consider the elliptic complex n L = p=0 p, . Its index is the L-genus: ind(L) = L(M ). In this case, the lifting of g to the components of the complex L has the following form:
n

i (Pj ) =
p=0

¯ p JPj (g ) i J Pj (g ),

and the fixed point contribution functions are (see formula (8))
n

L (Pj ) =
p=0

p (Pj ) =

n p=0

tr p JPj (g )
n

det 1 - JPj (g ) =
k=1

=

det 1 + JPj (g ) det 1 - JPj (g )

1 + e2 1 - e2

ix ix

(j ) k (j ) k

/p /p

.

(14)

Here we have again used formula (7). Hnece it follows from theorem 3.2 that the equivariant index is
q q n

ind(g , L) =
j =1

L (Pj ) =
j =1 k=1

1+e 1-e

2 ix 2 ix

(j ) k (j ) k

/p /p

.


CALCULATION OF HIRZEBRUCH GENERA FOR MANIFOLDS

11

1 As before, p lZ/p ind(g l , L) = s is the alternating sum of the dimensions of the invariant subspaces for the action of g on the cohomology of the complex L, and we have ind(1, L) = ind(L) = L(M ). Hence p-1 q p-1 n

L(M ) = ind(1, L) = -
l=1

ind(g , L) + ps = -
j =1 l=1 k=1

l

1+e 1-e

2 ix 2 ix

(j ) k (j ) k

l/p l/p

+ ps.

Again, we consider the number-theoretical trace Tr : Q( ) Q and introduce the Atiyah­Bott functions ABL (x1 , . . . , xn ) as
n

ABL (x1 , . . . , xn ) = - Tr
k=1

1 + e2 1 - e2

ixk /p ixk /p

.

(15)

Then we have

q

AB
j =1

L

x

(j ) 1

, . . . , x(j n

)

L(M ) mod p.

(16)

Relations (15) and (16) are analogous to relations (11), (12) for the Todd genus. It remains to calculate the number-theoretical trace in the definition of the Atiyah­ Bott functions ABL (x1 , . . . , xn ). n 1+ xk We set = 1- . Then = e2i/p = 1- . We must calculate Tr k=1 1- xk . 1+ 1+ We shall carry out our calculations in the p-adic extension Qp ( ) of the field Q( ). Let us write A B if A and B are equal modulo pZp Qp . First of all, we prove the following statement. Lemma 4.2. Tr(k ) 0 for any k 1.
p-1

Proof. Since 2 = p + 1 = ( + 1)( = (1 - )(
p-1

-

p-2

+

p-3

- · · · - + 1), we have -
p-2

-

p-2

+ 2

p-3

- · · · - + 1)

=

p-1

+

p-3

- · · · - .

Therefore,
p-1

Tr =
m=1

k

( m )

p-1

- ( m )p

-2

+ ( m )p

-3

- ··· -

mk

0

because
p-1

( )
m=1

m p-1

- ( )
p-1 m=0

m p-2

+ ··· -

mk

p-1

=
m=0

( m )

p-1

- ( m )p

-2

+ ··· -

mk

,

and because

( m )

r

0 for any r. The lemma is proved.
x x x

We further deduce that
n k=1

1+ 1-

x x

k k

n

=
k=1

1+ 1-

1- 1+ 1- 1+

k

k

1 =n

n k=1

1+ 1-

1- 1+ 1- 1+

k

xk

1 =: n



Ak k ,
k=0


12

TARAS E. PANOV

where the Ak Zp are p-adic integers (since 1/xk Zp ). Therefore, we can write Tr 1 n


Ak
k=0

k

Tr

1 n

n

n

Ak
k=0

k

=
k=0

Ak Tr(

k -n

).

Set Tr -s = Bs and introduce two formal power series A(u) = k=0 Ak uk , B (u) = n u k k=0 Bk u . It follows from (2L ) that A(u) = k=1 [u]L . Therefore, we must calculate the coefficient of un in the series A(u)B (u). We have

xk



B (u) = Tr 1 +
s=1



-s s

u

= Tr

1 1-

-1

u 1 -u .

= Tr

-u

= Tr 1 +

u -u

= (p - 1) + u Tr

Observe that if (u) is the minimal polynomial for an element with respect (u) 1 to the extension Qp ( ) | Qp , then Tr -u = - (u) . Since 0= -1 = -1
p 1- 1+ 1- 1+ p

-1 -1
-1

=

(1 - )p - (1 + )p (1 + ) 1 = p (-2 ) (1 + ) (1 + )p
(1+u)p -(1-u)p 2u

(1 + )p - (1 - )p , 2
1- 1+

we deduce that (u) = Hence, - Tr

is the minimal polynomial for = u)p-1 - u)p - (1 - u - (1 - u 1 , u )p-1 , )p
n k=1

.

1 (u) (1 + u)p-1 + (1 - = =p -u (u) (1 + u)p - (1 - 1 (1 + u)p-1 B (u) = (p - 1) + u Tr = -u (1 + u)p
n

Tr
k=1

1+ 1-

xk xk

A(u)B (u)

n

=

(1 + u)p-1 - (1 - u)p p (1 + u)p - (1 - u)p
n k=1

-1

u [u]Lk x

,
n

We deduce that (1 + u)p-1 - (1 - u)p ABL (x1 , . . . , xn ) - p (1 + u)p - (1 - u)p
-1

u [u]Lk x

mod p.
n

This formula is analogous to the first formula in (13). Further, p (1 + u)p-1 - (1 - u)p (1 + u)p - (1 - u)p =p
-1

=p

(1 - u)(1 + u)p - (1 + u)(1 - u)p (1 - u)(1 + u) (1 + u)p - (1 - u)p

(1 + u)p - (1 - u)p - u (1 + u)p + (1 - u)p (1 - u2 ) (1 + u)p - (1 - u)p p pu p pu = - = - 2 2 (1+u)p -(1-u)p 1-u 1-u (1 - u2 )[u] (1 - u2 ) (1+u)p +(1-u)p pu - L (1 + u2 + u4 + · · · ). [u]p

L p


CALCULATION OF HIRZEBRUCH GENERA FOR MANIFOLDS

13

From this we obtain (1 + u)p-1 - (1 - u)p ABL (x1 , . . . , xn ) - p (1 + u)p - (1 - u)p pu [u]L p
n k=1 -1 n k=1

u [u]Lk x

n

u (1 + u2 + u4 + · · · ) [u]Lk x

mod p,
n

whence
[n/2]

ABL (x1 , . . . , xn )
i=0
k

pu [u]L p
k

n k=1

u [u]Lk x

.
n-2i

- u) At the same time, [u]L = (1+u)k -(1-u)k = tanh(k arctanh u), and the sek (1+u) +(1 ries arctanh u (as well as tanh u) contains only odd powers of u. Therefore, the n pu series [u]L k=1 [uu contains only even powers of u. Thus we can finally write ]L
p

the formulae for the Atiyah­Bott functions for the L-genus which are analogous to formulae (13) for the Todd genus: (1 + u)p-1 - (1 - u) ABL (x1 , . . . , xn ) - p (1 + u)p - (1 - u)
n p-1 p k=1 n

xk

u [u]Lk x

mod p,
n

ABL (x1 , . . . , xn )
m=0

pu [u]L p

n k=1

u [u]Lk x

(17)

mod p.
m

By (16), we see that
q n

L(M )
j =1 m=0 (1+u)p -(1-u)p (1+u)p +(1-u)p

pu [u]L p

n k=1

u [u]L(

mod p,
j)

(18)

xk

m

where [u]L = p

.

Remark. Relations similar to (12), (16) could also be obtained for the Euler number. In this case, the Atiyah­Bott functions are
n

ABe (x1 , . . . , xn ) = - Tr
k=1

1- 1-

= - Tr(1) = -(p - 1) 1 mod p.

Therefore,
q

q=
j =1

ABe (x

(j ) 1

, . . . , x(j ) ) e(M ) mod p. n

(19)

This relation is analogous to relations (12) and (16) for the Todd genus and the Lgenus.


14

TARAS E. PANOV

§ 5. General results on the calculation of Hirzebruch genera via invariants of the Z/p-action Here we consider another approach to calculating the equivariant index m ind(g , E ) = i=0 (-1)i tr(g , H i ) of an elliptic complex E . This approach is taken from [3] (see also [7]). In what follows we adopt somewhat weaker assumptions about the action of an operator g , g p = 1, on a stably complex manifold M 2n . Namely, we remove the transversality condition. Let M g = x M | g x = x be the fixed point set, and let g M g = M be its decomposition into connected components. Then the equivariant g index can be computed as the sum of the contributions (M ) corresponding to the g fixed point components M (see [7]). These contributions are calculated as follows. g Let Y = M be one of the fixed point components of M 2n . For each point p Y , g acts linearly on the tangent space Tp M . This tangent space decomposes into the direct sum of the eigenspaces Np, for eigenvalues , || = 1. In this way we obtain the eigenbundle N over Y . The N1 is just the tangent bundle T Y to Y . With d = rk N , we therefore have:
d


T M |Y =


N ,

c(N ) =
i=1

(1 + x ), i

(20)

that is,
d


n

c(T MY ) =
i=1

(1 +

x i

)=

(1 + xi ).
i=1

The recipe for calculating (Y ) is as follows. Consider the index formula (3) in Theorem 2.1:
m n

ind(E ) =
i=0

(-1) ch(Ei ) cn (M )
j =1

i

1 1-e

-x

j

1 1-e

xj

[M ]

(21)

and replace M by Y and exi by -1 exi (where xi belongs to the eigenvalue ). Apply the same process to the terms ch(Ei ). This can obviously be done if the Ei are associated to the tangent bundle of M . This "recipe" is taken from [7]. In the case of finite number of fixed points we have Y = pt, cn (Y ) = 1, and so we must replace x1 x2 . . . xn by 1, and exj by -1 . Therefore, introducing j the "weight" xi by the formula j = exp
2 ix p
j

, we just have to replace x by j

- 2p i xj . (Here in the first case x stands for the first Chern class of "virtual" line j subbundle in T M corresponding to the eigenvalue j , while in the second case xj is the "weight" of the fixed point and is defined only modulo p.)

Example 5.1. Let us consider the y -genus of a manifold M . Applying our recipe to the formula from Theorem 2.2, we obtain the following formula for the contribution of each fixed point P :
n

(P ) =
k=1

1 + y e2ixk /p . 1 - e2ixk /p

(22)


CALCULATION OF HIRZEBRUCH GENERA FOR MANIFOLDS

15

Putting y = -1, 0, 1, we obtain the formulae for the contribution function for the Euler number, the Todd genus and the L-genus:
n

e (P ) = 1,

td (P ) =
k=1

1 1 - e2
ixk /p

n

,

L (P ) =
k=1

1 + e2 1 - e2

ixk /p ixk /p

.

These formulae coincide with (9), (14) and the formula in [5], which were deduced from the Atiyah­Bott theorem 3.2. Now consider the general case of an arbitrary Hirzebruch genus :
n

(M ) =
i=1 - f 1

xi [M ], f (xi )

where g (u) = (u) is the logarithm of the corresponding formal group law. Suppose that there is an elliptic complex E associated to T M whose index equals (M ). Applying the above recipe, we see that the contribution functions of the fixed points for the action of g on M are given by the formula
n

(P ) =
k=1

1 . f (-2 ixk /p)

(23)

Here the xk are the "weights" of the fixed point P . They are determined by the ix formula k = exp 2p k , xk = 0 mod p, where the k are the eigenvalues of the Jacobi matrix JP (g ) of the map g at the point P . The equivariant index of E is then determined by the formula
q n

ind(g , E ) =
j =1 k=1

1 f


-2 ix

(j ) k

/p

.

1 Further, p lZ/p ind(g l , E ) = s is the alternating sum of the dimensions of the equivariant subspaces for the action of g on the cohomology of E , and ind(1, E ) = ind(E ) = (M ). Therefore, p-1 q p-1 n

(M ) = ind(1, E ) = -
l=1

ind(g , E ) + ps = -
j =1 l=1 k=1

l

1 f


-2 ix

(j ) k

l/p

+ ps.

Consider the number-theoretical trace Tr : Qp ( ) Qp , where := exp
p-1 n l=1 k=1

2 i p

. Then

1 f -2 ix
(j ) k

n

l/p

= Tr
k=1

1 f -2 ix
(j ) k

/p

.

Thus, we obtain the following result. Theorem 5.1. Suppose that there is an el liptic complex of bund les associated to T M whose index is equal to the Hirzebruch genus (M ) of the manifold M . Let g be a holomorphic transversal endomorphism acting on M such that g p = 1. Then we have the fol lowing formula for (M ):
q n

(M ) -
j =1

Tr
k=1

1 f


-2 ix

(j ) k

/p

mod p.

It is now convenient to make the following definition.


16

TARAS E. PANOV

Definition 5.2. The Atiyah­Bott fixed point function AB (x1 , . . . , xn ) corresponding to the genus is defined to be the following function of the set of weights x1 , . . . , xn , xi Z/p:
n

AB (x1 , . . . , xn ) = - Tr
k=1

f



1 . -2 ixk /p

Then we immediately get
q

AB x
j =1

(j ) 1

,...,x

(j ) n

(M ) mod p.

(24)

Now we set = f (-2 i/p). Then -2 i/p = g (), and f (-2 i/pxk ) = f xk g () = []k . Hence the following statement holds: x Prop osition 5.3. The Atiyah­Bott fixed point function AB (x1 , . . . , xn ) corresponding to the genus can be computed as
n

AB (x1 , . . . , xn ) = - Tr
k=1

1 , []k x
k

= f

-2 i p

.

Lemma 5.4. For any k > 0 we have Tr Proof. Let = f


0 (that is, Tr k pZp ).
p-1

-

2 i p

= C0 + C1 + · · · + C
p-1



p-1

Qp ( ), Ci Zp . m . In particular,

Then C0 + C1 m + C2 2m + · · · + C C0 + C1 + · · · + Cp-1 = 0. Therefore,
p-1



(p-1)m

= f -

2 i p

Tr k =
m=1 p-1

f



-

2 i m p
m

p-1

=
m=1

C0 + C1
2m

m

+ C2

2m

+ ··· + C 0,

p-1



(p-1)m

=
m=0

C0 + C1
rm

+ C2

+ ··· + C

p-1

(p-1)m

since

p-1 m=0

0 for any r. The lemma is proved.
td

= 1 - e2 formula (2td )). Hence, C0 = 1, C1 = -1, Ci = 0 for i > 1. Example 5.2. For the Todd genus, = f - ^ Example 5.3. For the L-genus, = fL -
p-3 2 i p

2 i p

i/p

= 1 - (see
p-1 p-2

=

1-e2 1+e2 2i+1

i/p i/p

=

1- 1+ 2i

=

-

+

- · · · - (see Lemma 4.2). Hence, C0 = 0, C

= -1, C

= 1 for i > 1.

^ § 6. Calculations for the A-genus and the y -genus ^ ^ 6.1. Calculations for the A-genus. We consider the A-genus
n

^ A(M ) =
j =1

xj e-xj /2 [M ] 1 - e-xj


CALCULATION OF HIRZEBRUCH GENERA FOR MANIFOLDS

17

for a stably complex manifold M such that c1 (M ) 0 mod 2. In Theorem 2.4, we ^ gave an elliptic complex whose index is A(M ). The Atiyah­Bott functions for the ^ A-genus are as follows (see Definition 5.2): ABA (x1 , . . . , xn ) = - Tr ^
k=1 q

n

exp 1 - exp
(j ) n

2 ix 2p


k

n k=1

2 ix p

= - Tr
k

x k /2 1 - x

k

, (25)

AB
j =1

^ A

x1 , . . . , x

(j )

^ A(M ) mod p.

(26)

Now we have to calculate the number-theoretical trace in the/2 definition of the xk Atiyah­Bott functions ABA (x1 , . . . , xn ). By Proposition 5.3, 1- xk = []1 , where ^ A [ = 2 sinh m arcsinh is the mth power in the formal group law correspond^ ing to the A-genus (see formula (2A )), and = 2 sinh - 2i = -2 sinh i = 2p p e
- i/p

u]A m

u 2

xk

- e

i/p

=

-1/2

-

1 /2

=

(p+1)/2

-

(p-1)/2

.
i p

We claim that the minimal polynomial for the element = -2 sinh

is

2 sinh p arcsinh (u) = u Indeed,

u 2

.

() =

2 sinh p arcsinh

-2 sinh

i p

2

=

-2 sinh p ·

i p

=0

and (u) is a polynomial of degree p - 1 with leading term up-1 . For instance, for p = 3 we get (u) = u2 + 3, for p = 5 we get (u) = u4 + 5u2 + 5, and so on. Further, we have
n k=1 n k=1 n k=1

x k /2 = 1 - xk =

f

1 n

^ A n

1 = xk gA () ^ [
^ ]Ak x

1 [ ]
^ A xk

=:

k=1

1 n

Ai i =
i=0

1 A(), n


18

TARAS E. PANOV
n u ^ k=1 [u]A x

where A(u) :=

=

from lemma 5.4 that Tr(k ) the required trace is
n

k

i=0

Ai ui , Ai Zp are p-adic integers. It follows
(p+1)/2

0 for any k > 0. (Here =

-

(p-1)/2

.) So

Tr
k=1

xk /2 1 - x

k

= Tr
n

1 n



Ai
i=0

i

Tr

1 n

n

Ai
i=0

i

=
i=0

Ai Tr(

i- n

) = A(u)B (u)

n

,

where Bs := Tr Tr
1 -u

-s

, B (u) := . Therefore,

i=0

Bi ui . As before, B (u) = (p - 1) + u Tr

1 -u

,

=-

(u) (u)

2 sinh p arcsinh (u) = u p cosh p arcsinh (u) = 2 u 1+ u 4 (u) p = (u) 2 1+
u2 4

u 2 u 2

, 2 sinh p arcsinh - u2
u 2 u 2 u 2

,

cosh p arcsinh sinh p arcsinh

-

1 , u

B (u) = p - 1 - u

(u) =p- (u)

p

u 2 u2 4

1+

cosh p arcsinh sinh p arcsinh =

u 2 u 2 u 2 u 2

=

p sinh (p - 1) arcsinh u 2 u cosh arcsinh 2 sinh p arcsinh

p sinh (p - 1) arcsinh 1+
u2 4

u 2

.

sinh p arcsinh

Here we have used the following formulae: cosh arcsinh(u/2) = sinh(x - y ) = sinh x cosh y - sinh y cosh x. We further deduce that p sinh (p - 1) arcsinh u 2 u cosh arcsinh 2 sinh p arcsinh

1 + u2 /4 and

B (u) =

u 2 u 2

2p sinh (p - 1) arcsinh u 2 = u sinh (p - 1) arcsinh 2 + sinh (p + 1) arcsinh 2[u]A-1 p =p A [u]p-1 + [u]A+1 p [u] p [u]
A p-1 A p-1

- [u] + [u]

A p+1 A p+1

.

Here [u]A = 2 sinh m arcsinh(u/2) , and we have used the formula 2 cosh x sinh y = m sinh(y + x) + sinh(y - x). Thus we obtain the following formulae for the Atiyah­Bott


CALCULATION OF HIRZEBRUCH GENERA FOR MANIFOLDS

19

functions ABA (x1 , . . . , xn ) = - Tr ^ ABA (x1 , . . . , xn ) = - p ^

n xk k=1 1-

/2 xk

^ for the A-genus:
u 2 u 2 n k=1

sinh (p - 1) arcsinh 1+
u 4
2

u
^ [u]Ak x n

,

sinh p arcsinh
n

2[u]A-1 p ABA (x1 , . . . , xn ) = - p A ^ [u]p-1 + [u]A+1 p ABA (x1 , . . . , xn ) ^ p [u] [u]
A p+1 A p-1

u
^ [u]Ak x n

, .
n (j ) (j ) n

(27)

- +

k=1 n A [u]p-1 u A ^ [u]p+1 k=1 [u]Ak x

^ ^ Then the A-genus itself is calculated as A(M )

q j =1

ABA x1 , . . . , x ^

.

6.2. Calculations for the y -genus. Let us consider the y -genus
n

y (M ) =
j =1

xj (1 + y e-xj ) [M ] 1 - e-xj

of a manifold M . An elliptic complex whose index is y (M ) is given by Theorem 2.2. The Atiyah­Bott functions for the y (M )-genus are as follows (see Definition 5.2): ABy (x1 , . . . , xn ) = - Tr
k=1 n n

1 + y exp 1 - exp

2 ix p 2 ix p


k



k

= - Tr
k=1 q

1 + y xk , 1 - xk

(28) (29)

ABy x1 , . . . , x
j =1

(j )

(j ) n

y (M ) mod p.

These formulae could be also deduced from the Atiyah­Bott theorem 3.2 as was done for the L-genus in § 4.3. We shall now calculate the number-theoretical trace in the definition of the xk Atiyah­Bott functions ABy (x1 , . . . , xn ). By Proposition 5.3, 1+yxk = []1 y , 1- where [u = (for y = -1) is the mth power in the formal group law corresponding to the y -genus (see formula (2y )) and ^ = f - 2 i p = 1 - e2i/p 1- . = 2 i/p 1 + y 1 + ye
y ]m (1+y u)m -(1-u)m (1+y u)m +y (1-u)m
xk

y

1- Then = 1+y . In what follows we assume that y Zp and y = -1 mod p. The case y = -1 mod p corresponds to the Euler number which has already been considered above. Note that for y = 0, 1 we get = 1 - and = 1- respectively. 1+ This coincides with the corresponding values for the Todd genus and the L-genus obtained above.


20

TARAS E. PANOV

It is easy to see that the minimal polynomial for the element = (1 + y u)p - (1 - u)p . (1 + y )u

1- 1+y

is

(u) = Indeed, p - 1 = 0= -1

1- 1+y 1- 1+y

p

-1

-1

=

1 (). (1 + y )p-1
-1

Hence () = 0, and (u) is a polynomial of degree p - 1 with leading term up Further, we have
n k=1

.

1 + y xk = 1 - xk =

n k=1

1 ^ fy xk gy () ^
n k=1 i=0 k

n

=
k=1 i=0

1 y []xk 1 A(), n

1 n

1 y =: n []xk

Ai i =

where A(u) :=

follows from Lemma 5.4 that Tr( )

n u k=1 [u]y xk

=

Ai ui with Ai Zp being p-adic integers. It 0 for any k > 0.

Remark. The presentation of as = C0 + C1 + · · · + Cp-1 p-1 , Ci Zp , which is used in the proof of Lemma 5.4, is obtained as follows. Since 1 + y p = 1 + (y )p = (1 + y ) (y )p we have = (1 - ) (y )p 1- = 1 + y â y
p-1 -1 -1

- (y )p

-2

+ (y )p

-3

- · · · - y + 1 ,

- (y )p-2 + (y ) 1 + yp
p-2

p-3

- · · · - y + 1

=

1 1+y

p

+y

p-2



p-1

- (y

+y

p-3

)

p-2

+ · · · - (y + 1) + 1 - y
y i +y i-1 1+y p

p-1

.

Since y p + 1 y + 1 = 0 mod p, we obtain Ci = (-1)i C0 =
1-y p-1 1+y p

Zp (i > 0),

Zp .

So, the number-theoretical trace we are interested in is
n

Tr
k=1

1 + y xk 1 - xk

Tr

1 n

n

n

Ai
i=0

i

=
i=0

Ai Tr(

i-n

) = A(u)B (u)

n

,


CALCULATION OF HIRZEBRUCH GENERA FOR MANIFOLDS

21
1 -u

where Bs := Tr Tr
1 -u

-s

, B (u) :=

i=0

Bi ui . As before, B (u) = (p - 1) + u Tr

,

=-

(u) (u)

. Hence, the following formulae hold:

(1 + y u)p - (1 - u)p , (1 + y )u py (1 + y u)p-1 + p(1 - u)p-1 (1 + y u)p - (1 - u)p (u) = - , (1 + y )u (1 + y )u2 (u) y (1 + y u)p-1 + (1 - u)p-1 1 =p -, p - (1 - u)p (u) (1 + y u) u (u) B (u) = (p - 1) - u (u) y (1 + y u)p-1 + (1 - u)p-1 (1 + y u)p-1 - (1 - u)p = p - up =p (1 + y u)p - (1 - u)p (1 + y u)p - (1 - u)p (u) = Therefore,
n

-1

.

ABy (x1 , . . . , xn ) = - Tr
k=1

1 + y xk 1 - xk
-1 n k=1

(1 + y u)p-1 - (1 - u)p =- p (1 + y u)p - (1 - u)p We deduce that B (u) = p

u y [u]xk

.
n

(1 - u)(1 + y u)p - (1 + y u)(1 - u)p (1 + y u)p-1 - (1 - u)p-1 =p (1 + y u)p - (1 - u)p (1 - u)(1 + y u) (1 + y u)p - (1 - u)p p pu = - (1 - u)(1 + y u) (1 - u)(1 + y u) (1+yu)p -(1-u)pp (1+y u)p +y (1-u) - pu 1 , y [u]p (1 - u)(1 + y u) 1 = (1 - u)(1 + y u) ABy (x1 , . . . , xn ) =
m=0

1 + (-1)m y 1+y

m+1

um ,

pu 1 y [u]p (1 - u)(1 + y u) pu [u]p
n
y

n k=1

u y [u]xk

n m+1

k=1

u y [u]xk

m=0

1 + (-1)m y 1+y

um
n

.

We finally obtain the following formulae for the Atiyah­Bott functions ABy (x1 , . . . . . . , xn ) = - Tr
n 1+y xk k=1 1- xk

of the y -genus (y = -1 mod p):
1 n k=1 n k=1

(1 + y u)p-1 - (1 - u)p- ABy (x1 , . . . , xn ) = - p (1 + y u)p - (1 - u)p
n

u y [u]xk

,
n

ABy (x1 , . . . , xn )
m=0

1 + (-1) y 1+y

m m+1

pu [u]p y

u y [u]xk

(30) .

n-m


22

m m

TARAS E. PANOV

(1+ u - y Here [u]m = (1+yyu)) +y(1-u))m is the mth power in the formal group law correm (1-u sponding to the y -genus. The y -genus itself is the calculated as follows: q

y (M )
j =1 q

ABy x1 , . . . , x(j n
n

(j )

)


j =1 m=0

1 + (-1)m y 1+y

m+1

pu [u]p

n
y

k=1

u y [u] (j
x
k

mod p.
)

n-m

Formulae (13), (17) for the Todd genus and the L-genus are obtained from this formula by substituting y = 0 and y = 1 respectively. § 7. The Conner­Floyd equations and the calculation of Hirzebruch genera in terms of invariants of the action Here we consider the connection of the results in § 5 with the so-called Conner­ Floyd equations, which were introduced by Novikov in [12], [13]. (Similar relations were also obtained in [8], [11].) Namely, it was shown there that the sets (j ) (j ) (j ) x1 , . . . , xn , xk Z/p are the sets of weights for some action of Z/p on a manifold M 2n if and only if they satisfy the following Conner­Floyd equations:
q j =1

pu [u]p

n k=1

u [u]x(

0,
m

m = 0 , . . . , n - 1.

(31)

j) k

Here [u]m is the mth power in the universal formal group law of geometric cobordisms (cf. [4], [12]). Applying a Hirzebruch genus : U , we obtain the Conner­Floyd equations corresponding to :
q j =1

pu [u] p

n k=1

u [u](
x

0,
m

m = 0 , . . . , n - 1,

(32)

j) k

where [u] is the mth power in the formal group law corresponding to the genus . m The following formula for the Todd genus was deduced from cobordism theory in [5]: q n pu u td(M ) . (33) td [u]p [u]tdj ) n ( j =1 k=1
x
k

Furthermore, it was shown there that formula (33) is exactly the difference between the formula
q n

td(M )

-
j =1

Tr
k=1

1 1-
x
(j ) k

q

n

j =1 m=0

pu [u]td p

n k=1

u [u]tdj (
x
k

,
)

m

deduced from the Atiyah­Bott theorem (see § 4.2) and the sum of the Conner­Floyd equations (32) for the Todd genus. (Here = e2i/p , [u]td = 1 - (1 - u)m .) m Below we generalize this result considering the case of an arbitrary genus that satisfies the hypotheses of Theorem 5.1.


CALCULATION OF HIRZEBRUCH GENERA FOR MANIFOLDS

23

heorem 7.1. The difference between the formula of theorem 5.1,
q n

(M )

-
j =1

Tr
k=1

1 [](
x

,

=f



-

j) k

2 i p

and the sum of the Conner-Floyd equations (32) for the genus with some p-adic integer coefficients gives the fol lowing formula for the genus :
q

(M )
j =1

pu [u] p

n k=1

u [u](j
x
k

.
)

n

Proof. We have
n k=1

1 1 =n []xk =

n k=1 i=0

1 =: n []xk



Ai i =
i=0

1 A(), n

where A(u) :=

n u k=1 [u]k x n

Ai ui , Ai Zp are p-adic integers. Therefore,
i

Tr
k=1

1 []k x

1 = Tr n
n

Ai
i=0 i- n

1 Tr n
n

n

Ai
i=0

i

=
i=0

Ai Tr

=
k=1

u B (u) [u]k x

,
n

where Bs := Tr -s , B (u) := i=0 Bi ui . Now we introduce a power series h(u) as follows: h(u) = p [u] - u p . B (u)[u] p (34)

Then h(u) is a series with p-adic integer coefficients beginning with 1. Indeed, [u] - u p h(0) = p B (u)[u] p 1-
u [u] p u=0

=p
u=0

B (u)

=

p 1-

1 p

p-1

= 1,

since B (0) = B0 = Tr 0 = p - 1, [u] = pu + · · · . It follows from (34) that p B (u) = Hence, B (u) pu 1 pu - =- 1+ [u]p h(u) [u]p


pu 1 p - . h(u) [u]p h(u)

Hi ui ,
i=1


24

TARAS E. PANOV
1 h(u) n

where the Hi are the coefficients of the series
n

. Thus, u [u]k x H
n-m

Tr
k=1

1 []k x

pu - 1+ [u] p pu =- [u] p
n k=1



Hi u
i=1

i k=1 n-1

n

u [u]k x

-
n m=0

pu [u] p

n k=1

u [u]k x

,
m

q

n

(M )

-
j =1 q j =1

Tr
k=1 n k=1

1 []( u [u](

xk

j)

pu [u] p

n-1

q

+
j)

H
m=0

n-m j =1

xk

n

pu [u] p

n k=1

u [u](
x

.
m

j) k

This proves the theorem. Example 7.1. Consider the Todd genus td(M ). Then [u]td = 1 - (1 - u)p , B (u) = p p
1-(1-u)p- 1-(1-u)p
1

(see § 4.2). Therefore, h(u) = 1 - (1 - u)p - u =1-u 1 - (1 - u)p-1

(see formula (34)). Example 7.2. Consider the L-genus L(M ). Then (1 + u)p - (1 - u)p , (1 + u)p + (1 - u)p (see § 4.3). Therefore, [u]L = p h(u) = =
(1+u)p -(1-u)p (1+u)p +(1-u)p - u (1+u)p-1 -(1-u)p-1 (1+u)p +(1-u)p p p

B (u) = p

(1 + u)p-1 - (1 - u)p (1 + u)p - (1 - u)p

-1

(1 + u) - (1 - u) - u(1 + u)p - u(1 - u)p = (1 + u)(1 - u), (1 + u)p-1 - (1 - u)p-1 which is in accordance with the calculations from § 4.2, 4.3. ^ ^ Example 7.3. Consider the A-genus A(M ). Then [u]A = 2 sinh p arcsinh p (see § 6.1). Therefore, h(u) = = = = 2 sinh arcsinh u - u cosh arcsinh 2 2 sinh (p - 1) arcsinh u 2 sinh arcsinh 2 sinh cosh
p-1 2 u 2 u 2 ^

u , 2

B (u) = p

sinh (p - 1) arcsinh u 2 cosh arcsinh u sinh p arcsinh 2

u 2

- sinh arcsinh u 2 sinh (p - 1) arcsinh
u 2

cosh arcsinh
u 2

u 2 u 2

arcsinh

cosh

p-1 2 arcsinh p+1 u cosh 2 arcsinh 2 p-1 cosh 2 arcsinh

2 sinh

p+1 u cosh arcsinh 2 arcsinh 2 u cosh p-1 arcsinh u 2 2 2 u arcsinh 2 . u 2


CALCULATION OF HIRZEBRUCH GENERA FOR MANIFOLDS

25

§ 8. The elliptic genus for manifolds with Z/p-action Let M 2n be a 2n-dimensional real orientable manifold with a complex structure in its stable tangent bundle.
xi 2n Definition 8.1 (see [7]). A Hirzebruch genus (M 2n ) = ] is i=1 f (xi ) [M called the el liptic genus if f satisfies one of the following equivalent conditions: n

1) f

= 1 - 2 f 2 + f 4 , f (0) = 0; f (u)f (v ) + f (u)f (v ) 2) f (u + v ) = . 1 - f (u)2 f (v )2

2

Let us consider the lattice L = 2 i(Z + Z) in C, with Im > 0, and put 1 1 L = L\{0}. The Weierstraú -function is (z ) = z12 + L (z-)2 - 2 . It satisfies the following differential equation: (z )2 = 4(z )3 - g2 (z ) - g3 = 4 (z ) - e
1

(z ) - e2 (z ) - e3 ,

where e1 = ( i), e2 = ( i ), e3 = i( + 1) are the zeros of the derivative . The function f (z ) = 1 (z ) - e1 (that is, f (z ) = sn(z ), the elliptic sine) satisfies the conditions of Definition 8.1 for = - 3 e1 , = (e1 - e2 )(e1 - e3 ) (see [7]). 2 Thus it gives rise to the elliptic genus. However, = (e1 - e2 )(e1 - e3 ) = 0 and 2 - = 1 (e2 - e3 )2 = 0, since e1 , e2 , e3 are all different. The degenerate cases 4 ^ e1 = e2 and e1 = e3 ( = 0) correspond to the A-genus. (Putting = -1/8, we 2 12 obtain f = 1 + 4 f , that is, f (x) = 2 sinh(x/2).) The degenerate case e2 = e3 2 ( 2 - = 0) corresponds to the L-genus. (Putting = = 1, we obtain f = (1 - f 2 )2 , that is, f (x) = tanh x.) The differential equation defining the function f (x) for the elliptic genus implies that if we put deg = 4 and deg = 2, then the coefficient of x2k in the series f (x) becomes a weighted homogeneous polynomial of degree 2k in and with coeffin xi 2n cients in the ring Z 1 . Therefore, the elliptic genus (M 2n ) = ] i=1 f (xi ) [M 2 is a homogeneous polynomial of degree 2n in and . So can be regarded as a 1 homomorphism from the complex cobordism ring U to the ring Z 2 [, ] (or to the ring Zp [, ], p > 2). ~ The function f ( , x) is an elliptic function for the sublattice L = 2 i(Z · 2 + Z) L of index 2. The divisor of this function is (0) + ( i · 2 ) - ( i) - i(1 + 2 ) . We have the following decomposition of f ( , x) into an infinite product (see [7]): f ( , x) = 2 1-e 1+e
-x -x k=1

(1 - q k ex )(1 - q k e (1 + q k ex )(1 + q k e

-x -x

)(1 + q k )2 , )(1 - q k )2

q=e

2 i

.

(35)

Now we consider the following ob ject: L
(q ) i n

=
p=0



p,i


k=1

Sqk TC M
k=1

qk TC M

,

(36)

where TC M is the complexification of the tangent bundle to M , t E := k k k k (q ) is a power series in q whose k=0 ( E )t , St E := k=0 (S E )t . Then L


26

TARAS E. PANOV

coefficients are elliptic complexes associated to T M . Its index is a well-known power series in q , namely, the so-called twisted signature (see [7]): ind(L(q) ) = sign M , k=1 Sqk TC M k=1 qk TC M . It follows from the Atiyah­Singer theorem 2.1 and Lemma 2.3 that this index is

n

ind(L

(q )

)=
j =1

1+e xj 1-e

-x -x

j j

k=1

(1 + q k exj )(1 + q k e (1 - q k exj )(1 - q k e

-xj -xj

) )

[M

2n

].

(37)

It is clear from this formula that ind(L(q) ) is a power series in q with integer coefficients and with constant term (the coefficient of q 0 ) L(M ) = sign M . Comparing this expression with (35) and taking into account that

1 2

k=1

(1 - q k )2 = 1 k )2 (1 + q

/4

(38)

(see [7]), we obtain





(M ) = sign M ,
k=1

Sqk TC M
k=1

qk TC M

n/4

.

(39)

Suppose that an operator g , g p = 1, acts on a manifold M 2n with finitely many fixed points P1 , . . . , Pr . According to the recipe from § 5, the equivariant index ind g , L(q) of the complex L(q) equals the sum of the contribution functions L(q) (Pj ) of the fixed points. These contributions are obtained from formula (37) (j ) by replacing M 2n by Pj , x1 x2 . . . xn by 1 and xk by - 2p i xk :

n

1 + 1-
x x

L(q) (Pj ) =
k=1

(j ) k (j ) k

i=1

1 + qi 1 - qi

-x -x

(j ) k (j ) k

1 + qi 1 - qi

x x

(j ) k (j ) k

, =e
2 i p

.


CALCULATION OF HIRZEBRUCH GENERA FOR MANIFOLDS

27

1 We note that p lZ/p ind g l , L(q) = s(q ) is a power series in q whose coefficient at q k is the alternating sum of the dimensions of the invariant subspaces for the action of g on the cohomology of a certain complex. Namely, this complex is the coefficient of q k in the series L(q) :

L

(q )

= L + 2L TC M q + L (2TC M + TC M TC M + S 2 TC M + 2 TC M )q 2 + · · · ,
n

where L = p=0 p,i . Therefore, s(q ) is a series with integer coefficients. Further more, ind 1, L(q) = sign M , k=1 Sqk TC M k=1 qk TC M . Hence,


sign M ,
k=1

Sq k T C M
k=1 p-1 (q )

qk TC M ind(g l , L
l=1 lx lx
(j ) k (j ) k

= ind(1, L
r

)=-

(q )

) + ps(q )
-lx
(j ) k (j ) k

p-1 n

=-
j =1 l=1 k=1

1 + 1-

i=1

1 + qi 1-q

1 + qi 1-q

lx

(j ) k (j ) k

+ ps(q ).

i -lx

i lx

The left-hand side of this relation belongs to the ring Z [q ] of power series with integer coefficients, while its right-hand side a priori belongs to an extension of Z [q ] , namely, to the ring Z [q ] ( ), p = 1. We embed both rings in the p-adic extensions of the corresponding fields and consider the number-theoretical trace Tr : Qp {q }( ) Qp {q }, where Qp {q } = Qp [[q ]][q -1 ] is the field of Laurent series in q with rational p-adic coefficients and Qp {q }( ) is the algebraic extension of Qp {q } by the element , p = 1. Then
p-1 n l=1 k=1


lx

1 + (j 1 - l xk
n k=1

(j ) k )

i=1

1 + qi 1-q
xk
(j ) (j )

-lx

(j ) k (j ) k

1 + qi 1-q
-xk
(j )

lx

(j ) k (j ) k


x
(j ) k (j ) k

i -lx

i lx

= Tr

1 + 1-

i=1

1 + qi 1-q

1 + qi 1-q

.

xk

i -xk

(j )

i x

Thus the following proposition holds: Prop osition 8.2. We have the fol lowing formula for the index of the complex L in (36):
(q )

ind(L

(q )

) = sign M , Tr
n

Sqk TC M
k=1 xk
(j ) (j )

qk TC M
k=1 -xk
(j )

r

-
j =1

1 + 1-

i=1

1 + qi 1-q

1 + qi 1-q

x

(j ) k (j ) k

mod pZ [q ] . (40)

k=1

xk

i -xk

(j )

i x


28

TARAS E. PANOV

Using relations (38) and (39), we thus obtain the following formula for the elliptic genus (M ):
r

Tr

n k=1

1 · 1 + 2 1-
-x
(j ) k (j ) k

x x

(M ) -
j =1

(j ) k (j ) k (j ) k (j )



â
i=1

1 + qi 1-q

1 + qi 1-q

x

(1 - q i )2 (1 + q
i )2

mod pZp [q ] .

i -x

i xk

Taking into account the decomposition (35) of f ( , x) = sn x, we rewrite this formula as
r

(M ) -
j =1

Tr

n k=1

1 f , -
2 ix p
(j ) k



mod pZp [q ] ,

q=e

2 i

.

However, (M ) is a homogeneous polynomial in , with coefficients in the ring Z 1 Zp . So we want to obtain the expression for (M ) in the ring Zp [, ], 2 not in the ring Zp [q ] . (Zp [, ] is the subring of Zp [q ] .) As before, we put
ix = f , - 2p i . Then f , - 2p k = []xk , where []m = f mf -1 () is the mth power in the formal group law corresponding to the elliptic genus. We note that is an algebraic element over the field Q(, ). Indeed,

[]p = f

, -

2 ip p

= f ( , -2 i) = 0,

and [u]p is a rational function of u when p is odd. (For example, [u]3 = 2 +6u4 -2 u8 u 13-8u+8u6 -32 u8 . This follows from the differential equation and the addition -6u4 theorem for f ( , x) = sn x.) The numerator of [u]p is a polynomial in u with coefficients , , and is a zero of this polynomial. Obviously, this polynomial is of degree p2 in u, and its zeros are the values of f at all p-division points of the lattice 2 i(Z · 2 + Z), that is, they equal f , 2 i 2k +l , k , l Z. In particular, p , []2 , . . . , []p-1 , []p = 0 are among zeros. Finally, we have
r n

(M ) -
j =1

Tr
k=1

1 []x

mod pZp [q ] ,
k

=f

, -

2 i p

,

(41)

where both right- and left-hand sides are polynomials in , . Lemma 8.3. Let P (, ) be a homogeneous polynomial in , such that P (, ) 0 mod pZp [q ] . Then P (, ) 0 mod pZp [, ], that is, al l coefficients of P (, ) belong to pZp . Proof. The functions , are modular forms of weights 2 and 4 respectively on the subgroup 0 (2) SL2 (Z), 0 (2) := A SL2 (Z) A (mod 2) . 0


CALCULATION OF HIRZEBRUCH GENERA FOR MANIFOLDS

29

They have the following Fourier expansions near infinity (that is, for small q = e2i ): 3 1 = - e1 = + 2 4 =


dq
n=1 d|n d1 mo d 2

n

1 + 6(q + q 2 + 4q 3 + q 4 + 6q 5 + 4q 6 + · · · ), 4 = (e1 - e2 )(e1 - e3 ) = 1 + 16


(-1)d d3 q n =
n=1 d|n

1 - q + 7q 2 - 28q 3 + 71q 4 - · · · 16

1 Furthermore, (0) = - 8 , (0) = 0, 1+i = 0, 1+i = 0 (see [7]). 2 2 The proof is by induction on the degree deg P = 2n of the polynomial P . If deg P (, ) = 2, then P (, ) = b . Hence, b pZp , because pZp . Now assume / that the lemma holds for all polynomials of degree < 2k and let deg P (, ) = 2k . Write P (, ) = ak/2 + Q(, ) + b k , where deg Q(, ) < 2k , and therefore Q(, ) 0 mod pZp [, ]. Evaluation of P (, ) at the point = 1+i shows that 2

a

1+i k/2 2

is divisible by p, whence a is divisible by p. Evaluation of P (, ) at
k

= 0 shows that - 1 b is divisible by p, whence b is divisible by p. Thus all the 8 coefficients of P (, ) belong to pZp . The lemma is proved. From this lemma and (41) we obtain the following theorem. Theorem 8.4. Let g be a transversal endomorphism acting on M 2n such that g p = 1. Then we have the fol lowing formula for the el liptic genus (M 2n ):
r

(M ) -
j =1

Tr

n k=1

1 f , -
2 ix p
(j ) k



mod pZp [, ],

(42)

where Tr in the right-hand side denotes the number-theoretical trace Qp (, )( ) Qp (, ). All constructions in § 5 and § 7 (in particular, Lemma 5.4 and Theorem 7.1) can be repeated for the elliptic genus (we just replace the ring Zp by Zp [, ] in all formulae). Thus, the difference between formula (42) and a certain weighted sum of the following Conner­Floyd equations for the elliptic genus:
r j =1

pu [u]p

n k=1

u [u]x(

0 mod pZp [, ],
m

m = 0 , . . . , n - 1,

j) k

gives the following formula for the elliptic genus:
r

(M

2n

)
j =1

pu [u]p

n k=1

u [u]x

mod pZp [, ].
(j ) k

(43)

n

In the last two formulae, [u]m denotes the mth power in the formal group law corresponding to the elliptic genus.


30

TARAS E. PANOV

To give an example, we action of the group Z/p. that the generator g acts (0 z0 : 1 z1 : · · · : n zn ), which implies that n < p,
z0 zj

apply the formulae for the elliptic genus to a particular Namely, we consider the action of Z/p on CP n such in homogeneous coordinates as (z0 : z1 : · · · : zn ) 2 i i = e p yi . If y0 , . . . , yn are distinct as residues mod p, then this action has only finitely many fixed points on 0, . . . , 1, . . . , 0 ,
j

CP n . Namely, the fixed points are Pj = the local coordinates
zi zj

j = 0, . . . , n. In

,

z1 zj

,...,

z z

j j

,...,

zn zj

near Pj , the operator g acts linearly:



i z j z

i j

= exp

2 i p

(yi - yj )
2 i p

zi zj

. Therefore, the eigenvalues of the map g at

the point Pj are exp xi
(j )

(yi - yj ) , i = j , and the corresponding weights are

= yi - yj . Now we consider the elliptic genus . It is well known that (CP n ) = 1 1 - 2 u2 + u
4

.
n

(44)

Indeed, it follows from formula (1), that (CP n ) = g (u) n , where g (u) = - f 1 (u) is the logarithm of the corresponding formal group law. But g (u) = 1 , since (f )2 = 1 - 2 f 2 + f 4 . 1-2 u2 +u4 At the same time, it follows from (43) that
n

(CP )
j =0

n

pu [u]p

u
i=j

[u]

yi -y

mod pZp [, ].
j

n

Hence for any set y0 , y1 , . . . , yn of distinct residues modulo p, where n < p, we have
n j =0

pu [u]p

u
i=j

[u]y


j



i

-y

n

1 1 - 2 u2 + u

4

mod pZp [, ]
n

The function (1 - 2 u + u2 ) nomials Pn ( ), that is, 1-21

-1/2 u+ u

is the generating function for the Legendre poly= 1 + n>0 Pn ( )un . Hence, 2 Pn
n>0

1 =1+ 1 - 2 u2 + u4

k

(k u2 )n ,

with k 2 = . Therefore,
2m j =0

1 1 - 2 u2 + u u
i=j

4

= km P
2m

m

k k

, (45) mod pZp [, ].

pu [u]p

[u]

k P
j

m

yi - y

m

2m

In the simplest case 2m = n = p - 1, the set y0 - yj , y1 - yj , . . . , yj - yj , . . . , yp-1 - yj forms the complete set 1, 2, . . . , p - 1 of non-zero residues modulo p (since the yi are distinct). Hence, all the summands in the left-hand side of the last formula


CALCULATION OF HIRZEBRUCH GENERA FOR MANIFOLDS

31

are equal, and we obtain p2 up u[u]2 [u]3 · · · [u]
p-1

p-1

[u]p



1 1 - 2 u2 + u P
p-1 2

4

p-1

=k

p-1 2

k

mod pZp [, ].

(46)

This formula could be also deduced from the well-known relation [u]p P(p
-1)/2

( )up + · · ·

mod pZp [ ]

(see [10]). Here we put k = 1 for simplicity, and the dots stand for the higher order terms. Indeed, we could rewrite the above relation as [u]p = pu(1 + b1 u + · · · + bp with b1 , . . . , b
p-1 -1

up

-1

) + P(

p-1)/2

( )up + · · · ,

Z[ ]. Hence,

p2 up u[u]2 · · · [u]p-1 [u] =

p

p u(2u + · · · ) . . . (p - 1)u + · · · â pu(1 + b1 u + · · · + b
p-1

p-1 2

up up-1 ) + P
2

(p-1)/2

( )up + · · · ( )up
-1

p-1

= =

1 p (p - 1)! p(1 + c1 u + · · · + cp-1 up-1 ) + P 1 (p - 1)!
p-1

(p-1)/2

+ ···

p-1

1 + c1 u + · · · + cp-2 u

p-2

p + cp

-1

+ P(p

-1)/2

( )/p up

-1 p-1

,

where c1 , . . . , c get

belong to Z[ ]. Calculating the latter expression modulo p, we 1 P(p (p - 1)!

p2 up u[u]2 [u]3 · · · [u]

p-1

[u]p

-
p-1

-1)/2

( ) P(p

-1)/2

( ) mod pZp [ ],

since (p - 1)! -1 mod p, as required. § 9. Generalization to Z/p-actions having fixed submanifolds with trivial normal bundle Suppose that an operator g , g p = 1, acts on a stably complex manifold M 2n . Let the fixed point set M g be written as the union of connected fixed submanifolds: g g M g = M . Also suppose that all the M have trivial normal bundle in M . Let be a Hirzebruch genus that can be calculated as the index of an elliptic complex E of bundles associated to the tangent bundle T M . To avoid misunderstanding, we shall denote the first Chern classes of the "virtual" line subbundles of T M by zi . Then c(T M ) = (1 + z1 ) . . . (1 + zn ).


32

TARAS E. PANOV

g Let Y = Mk be one of the fixed submanifolds for the action of g . According to formulae (20) from § 5, we have: d

j

T M |Y =
j

Nj ,
d
j

c(Nj ) =
i=1 (j ) n

1 + zi

(j )

,

c(T MY ) =
j i=1

1 + zi

=
i=1

(1 + zi ),

where Nj is the subbundle of T M |Y corresponding to the eigenvalue j of the differential of g . Here p = 1 (hence, j = e2ixj /p ) and dj = dim Nj . Obviously, j N1 = T Y is the tangent bundle to Y . According to the results of Atiyah and Singer [3] described in § 5, the equivariant index ind(g , E ) of the complex E g can be computed as ind(g , E ) = (M ). We also have the following "recipe" for calculating the fixed point contribution functions (Y ). In the formula (21) for the index of E , replace M by Y and ez j = e2ixj /p , one should replace z Since ind(E ) = (M ), we have
n (j ) i
( j ) i

by

-1 zi je

(j )

. Equivalently, since

by z

(j ) i

- 2 ixj /p.

ind(E ) =
i=1

zi [M ] = f (zi )

d

j

z f

j i=1

(j ) i ( ) zi j

[M ].

Hence, our "recipe" shows that
d
j

(Y ) =
j =1 i=1 d


1 f zi
(j )

d

1

z f

- 2 ixj /p

i=1 d

(1) i (1) zi (1)

[Y ],

j 1 because cn (M ) = j i=1 zi j reduces to cn (Y ) = i=1 zi and all the weights xk corresponding to the eigenvalue = 1 are zero. Furthermore, since Y has triv( ) ial normal bundle in M , we have zi j = 0 for j = 1. Therefore, (Y ) = 1 j f (-2 ixj /p) (Y ), where the xj are the weights corresp onding to the eigenvalues

( )

j = e2ixj /p = 1, that is, the xj are non-zero modulo p. Finally, we have the following formula for the equivariant index: ind(g , E ) =
j

1 f -2 ix
( ) j

g (M ) .

Theorem 5.1 and Proposition 5.3 can naturally be extended to the case of actions having fixed submanifolds with trivial normal bundle. We get the following statement. Theorem 9.1. Let be a Hirzebruch genussuch that there is an el liptic complex of bund les associated to T M whose index is equal to (M ). Then we have the


CALCULATION OF HIRZEBRUCH GENERA FOR MANIFOLDS

33

fol lowing formula for (M ): (M ) -


Tr
j

1 f -2 ix
( ) j

/p

g (M )

=-


Tr
j

1 g (M ) mod p, [] xj

g where M g = M is the fixed point set, = f (-2 i/p), []k is the k -th power in the corresponding formal group, and Tr : Q(e2i/p ) Q is the number-theoretical trace. (If the genus takes its values in some ring instead of Z, replace Q by the corresponding quotient field ).

The author is grateful to Prof. V. M. Buchstaber and Prof. S. P. Novikov for suggesting the problem and wishes to express special thanks to V. M. Buchstaber for useful recommendations and stimulating discussions. References
1. M. F. Atiyah, R. Bott, A Lefschetz fixed point formula for el liptic complexes. I, Ann. of Math. 86 (1967), no. 2, 374­407. 2. R. Bott, C. Taubes, On the rigidity theorems of Witten, J. Amer. Math. Soc. 2 (1989), 137­186. 3. M. F. Atiyah, I. M. Singer, The index of el liptic operators. III, Ann. of Math. 87 (1968), 546­604. 4. V. M. Buchstaber, The Chern­Dold character in cobordisms. I, Mat. Sb. 83 (1970), no. 4, 575­595; English transl. in: Math. USSR-Sb. 12 (1970), 573­594. 5. V. M. Buchstaber, S. P. Novikov, Formal groups, power systems and Adams operators, Mat. Sb. 84 (1971), no. 1, 81­118; English transl. in: Math. USSR-Sb. 13 (1971), 80­116. 6. F. Hirzebruch, Topological Methods in Algebraic Geometry, 3rd edition, Grundlehren der mathematischen Wissenschaften, no. 131, Springer, Berlin­Heidelb erg 1966. 7. F. Hirzebruch, T. Berger, R. Jung, Manifolds and Modular Forms. Second Edition, MaxPlanc-Institut fur Mathematik, Bonn 1994. ¨ 8. G. G. Kasparov, Invariants of classical lens manifolds in cobordism theory, Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), no. 4, 735­747; English transl. in: Math. USSR-Izv. 3 (1969), 695­705. 9. I. M. Krichever, Generalized el liptic genera and Baker­Akhiezer functions, Mat. Zametki 47 (1990), no. 2, 34­45; English transl. in: Math. Notes 47 (1990), no. 2, 132­142. 10. P. S. Landweber (Ed.), El liptic Curves and Modular Forms in Algebraic Topology, Lect. Notes in Math., no. 1326, Springer, Berlin­Heidelb erg 1988. 11. A. S. Michshenko, Manifolds with the action of the group Zp and fixed points, Mat. Zametki 4 (1968), no. 4, 381­386; English transl. in: Math. Notes 4 (1968), 721­724. 12. S. P. Novikov, The methods of algebraic topology from the viewpoint of cobordism theory, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), no. 4, 855­951; English transl. in: Math. USSR-Izv. 1 (1967), 827­913. 13. S. P. Novikov, Adams operators and fixed points, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), no. 6, 1245­1263; English transl. in: Math. USSR-Izv. 2 (1968), 1193­1211. 14. S. Ochanine, Sur les genres multiplicatifs d´ efinis par des int´ ales el liptiques, Topology 26 egr (1987), 143­151. 15. T. E. Panov, El liptic genus for manifolds with the action of the group Z/p, Uspekhi Mat. Nauk 52 (1997), no. 2, 181­182; English transl. in: Russian Math. Surveys 52 (1997), no. 2, 418­419. Department of Mathematics and Mechanics, Moscow State University, 119899 Moscow, Russia E-mail address : tpanov@mech.math.msu.su