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Journal of Mathematical Sciences, Vol. 105, No. 2, 2001

ON THE COBORDISM CLASSIFICATION OF MANIFOLDS WITH Z/p-ACTION WHOSE FIXED-POINT SET HAS TRIVIAL NORMAL BUNDLE
T. E. Panov UDC 515.164.24

Definition 1. An action of the group Z/p on a stably complex manifold M 2n is said to be simple if the fixed-point set consists of finitely many connected submanifolds with trivial normal bundle. A simple action is said to be strictly simple if the weight sets of action (i.e., the sets of eigenvalues for the differential of the action of the generator Z/p at the fixed points) are identical for all fixed submanifolds of the same dimension. Here "a stably complex manifold" stands for "a manifold with complex structure in the stable tangent bundle." We obtain a complete classification of the complex cobordism classes U containing a manifold with simple action of Z/p. The description is given both in terms of co efficients of the universal formal group of "geometric cobordisms" (Theorem 1) and in terms of characteristic numbers (Theorem 2 and Corollary 2). The classification problem for strictly simple actions of Z/p was completely solved by Conner and Floyd in [1]. (In [1], the strictly simple action from Definition 1 was simply called the "action of Z/p with fixed-point set having a trivial normal bundle.") Note that even in the special case of action with a finite number of isolated fixed points the notions of simple and strictly simple action are distinct (see examples below). The Conner­Floyd results follow from the results of our paper. At the same time, we believe that the approach used in [1] do es not allow one to obtain our more general result. The applications of the formal group theory to problems connected with Z/p-actions were first discussed in the pioneering article [2]. The formal group theory itself comes to topology due to the so-called formal group of geometric cobordisms. The problem solved here was first stated in [3]. There one obtained a formula expressing the mo d p cobordism class of manifold M 2n with simple action of Z/p via certain action invariants (see (8)). Actually, the first results on the problem were obtained even earlier, in [4]. In particular, the statement mentioned in our article as Corollary 3 was proved. In [4], as well as in our paper, the set of cobordism classes of manifolds with simple Z/p-action is described as an U -mo dule spanned by certain co efficients of the power system defined by the formal group of geometric cobordisms. (Here U is the complex cobordism ring of points, which is isomorphic to the polynomial ring Z[a1 ,a1 ,... ], deg ai = -2i as was shown by Milnor and Novikov.) In this article, we propose a new choice of generators for the above U -mo dule. Moreover, this choice allows us to solve the classification problem in terms of the characteristic numbers. Therefore, we consider an operator g of prime perio d p > 2 (i.e., g p = id) acting on a stably complex manifold M 2n in such a way that the fixed-point set is a union of connected submanifolds with trivial normal bundle (e.g., is a finite number of fixed points). This means that we have a simple Z/p-action. Let the (j ) fixed submanifolds represent the cobordism classes j U and have weights (xk ) (Z/p) (these are the nonunit eigenvalues for the differential of g at the fixed points) in their trivial normal bundles. These data define the cobordism class of M 2n in U up to elements from pU (see [3]). This follows from the fact that the cobordism class of manifolds with free Z/p-action (i.e., without fixed points) belongs to pU , and vice versa, in any cobordism class from pU , one could obviously find a manifold with free action of Z/p (e.g., rotating p components). (j ) (j ) With each fixed submanifold of the cobordism class j U with weights (x1 ,... ,xmj ), 2mj +dim j = (j ) (j ) 2n, one could asso ciate the so-called "Conner­Floyd invariant" 2mj -1 (x1 ,... ,xmj ) U2mj -1 (B Z/p)(see [2]).
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 68, Pontryagin Conference­8. Top ology, 1999.

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1072­3374/01 /1052­1876 $ 25.00 c 2001 Plenum Publishing Corp oration


To define it, we mention that the bordism group U (B Z/p) is isomorphic to the group of principal Z/pequivariant bordisms (see [1]). This isomorphism takes the equivariant bordism class of the manifold N with free action of Z/p to the bordism class in U (B Z/p) given by the classifying map N/(Z/p) B Z/p. Then (j ) (j ) 2mj -1 (x1 ,... ,xmj ) is the equivariant bordism class of the unit sphere in the fiber of the (trivial) normal bundle to j . If we consider this unit sphere as (z1 ,... ,zmj ) Cmj |z1 |2 + ... + |zmj |2 = 1 , then the (free) action of Z/p on it is given by g : (z1 ,... ,zmj ) (exp(2ix1 /p)z1 ,... , exp(2ix(j ) /p)zmj ). mj The complex cobordism ring of B Z/p is U (B Z/p) = U [[u]]/[u]p = 0, where [u]p = pp (u) is the p-th power in the (universal) formal group of geometric cobordisms (see [3]). We have D(i (1,... , 1)) = un-i, where D is the Poincar´ tiyah duality operator from U (L2n-1 ) to U (L2n-1 ). From this we deduce that e­A p p the U -mo dule U (B Z/p) is generated by the elements 2i-1 (1,... , 1) with the following relations: [u]p 2i-1 (1,... , 1). u Here is the cobordism -pro duct: uk 2i-1 (1,... , 1) = 2(i-k)-1 (1,... , 1). It was shown in [5] that 0=
k (j )

(1)

2k-1 (x1 ,... ,xk ) =
j =1

u [u]x

2k-1 (1,... , 1).
j

(2)

Adding the elements 2k-1 (x1 ,... ,xk ), xj = 1 mo d p to the set of generators adding relations (2) to relations (1), we come to the U Z(p) -free resolution Z(p) is the ring of rational numbers whose denominators are relatively prime p-adics): 0 - F1 - F0 - U (B Z/p) - 0. Here F0 is the free U Z(p) -mo dule spanned by 2k-1 (x1 ,... ,xk ) and F1 spanned by the relations
k

for the mo dule U (B Z/p) and of the mo dule U (B Z/p) (here with p, i.e., the ring of integer

is the free U Z(p) -mo dule

a(x1 ,... ,xk ) = 2k-1 (x1 ,... ,xk ) - (
j =1

u ) 2k-1 (1,... , 1) [u]xj

[u]p 2k-1 (1,... , 1). u Therefore, each simple action Z/p on M 2n gives a certain relation between the elements 2k-1 (x1 ,... ,xk ) in U (B Z/p). Since any element from U (B Z/p) corresponds to the bordism class of a manifold with free (j ) (j ) Z/p-action, the converse is also true: any relation in U (B Z/p) of the form j j 2mj -1 (x1 ,... xmj ) = 0, j U , 2mj + dim j = 2n is realized on a certain manifold M 2n with a simple action of Z/p whose cobordism class in U is uniquely determined up to the elements from pU . This manifold M 2n is constructed as follows. The relation in U (B Z/p) gives us the manifold with free Z/p-action whose boundary is a union of manifolds of the form j â S2mj -1 . Then we glue "covers" of the form j â D2mj to the boundary to get the closed manifold M 2n , which realizes the above relation. Hence, we define the "realization homomorphism" : F1 U /pU = U Z/p. It assigns to the relation between the elements 2k-1 (x1 ,... ,xk ) U (B Z/p) a mo d p cobordism class of the manifold that realizes this relation as described above. The Conner­Floyd results (cf. [1]; see also [4]) give us the following values of on the basic relations from F1 : ak =
k

and

(a(x1 ,... ,xk )) =
i=1

u [u]x

i

k

mo d p U /pU , 1877


[u]p mo d p U /pU , uk k + where k stands for the co efficient of u . Therefore, Im = (1) Z/p, where (1) = (1) · U is + the U -mo dule spanned by the positive part (1) of the co efficient ring (1) of the power system [u]k (here [u]k is the k th power in the formal group of geometric cobordisms). The homomorphism lifts to the homomorphism : F1 (1) Z(p) or to the homomorphism : F1 p (1) Z(p) , where p (1) is the U -mo dule spanned by + (1) and p. Thus, the problem of description of the cobordism classes of manifolds with simple Z/p-action is equivalent to the problem of description of the U Z/p-mo dule (1) Z/p or U Z(p) -mo dules (1) Z(p) and p (1) Z(p) . These mo dules are ideals in U Z/p and U Z(p) , respectively. (k ) (k ) Let [u]k = ku + n1 n un+1 , and so, n -2n are the co efficients of the power system. U (ak ) = - Theorem 1. One could take the fol lowing coefficients n -2n as generators of the U Z(p) -module U (1) Z(p) : n 1 if n is not divisible by p - 1, (p) n if n is divisible by p - 1. Here p1 is any generator of the cyclic group (Z/p) . n = Remark. It follows from the Dirichlet theorem that one could cho ose a prime generator p1 of the cyclic group (Z/p) . Proof of Theorem 1. First, let us consider the co efficients n for nonprime r. Therefore, let r = p1 q with prime p1 . Since [x]r = [[x]p1 ]q , we have rx +
n (r ) (p )

n xn

(r )

+1

= q [x]p1 + n n ([x]p1 )n+1 (p ) = p1 qx + q n n 1 xn+1 (q ) (p ) + n n (p1 x + m m1 xm+1 )n+1 .
(p ) (q )

(q )

Taking the co efficients of xr

+1

in both sides of the above identity, we get
( ( ( nr) = P (1 1 ,... ,np1 ) ,1 ,... ,nq) ),

where P is a certain polynomial with integer co efficients (without constant term). Therefore, we can write (r ) (p ) (p ) (q ) (q ) (r ) n = 1 1 1 + ... + n n 1 + µ1 1 + ... + µn n , i ,µi U . Hence the co efficients n , r = p1 q could be excluded from the set of generators for the U Z(p) -mo dule (1) Z(p) . Now, if q is still not prime, we repeat the above pro cedure, and so on. Finally, we obtain the set of generators that consists of only the (p ) co efficients n 1 with prime p1 . Among all prime p1 , there is one particular p1 = p (the order of the group acting on the manifold). Now, we are going to show that the above set of generators could be restricted to the set described in the statement of the theorem. (p ) Note that for any (prime) generator p1 of the cyclic group (Z/p) , one could take the co efficient 1 1 as a generator of (1) Z(p) in dimension 1 (i.e., in -2 ). Indeed, let p2 be any prime. Then [[x]p2 ]p1 = [[x]p1 ]p2 . U Hence p1 p2 x + p1 = p2 p1 x + p2
n

n 2 xn
(p )

(p )

+1

+ +

n

n 1 (p2 x + n 2 (p1 x +
(p )

(p )

m

m2 xm+1 )n
(p ) (p )

(p )

+1

n

n 1 xn

+1

n

m

m1 xm+1 )n+1 .
(p ) (p )

(3)
(p )

Taking the co efficient of x2 in both sides of the above identity, we get p1 1 2 + p2 1 1 = p2 1 1 + p2 1 2 . 2 1 (p ) (p ) Hence, (p1 - p2 )1 2 = (p2 - p2 )1 1 . Since p1 is a generator of (Z/p) , p1 - p2 is invertible in Z(p) . Therefore, 1 2 1 (p2 ) (p1 ) (p ) 1 = 1 with Z(p) U Z(p) . Thus, for any prime p2 = p1 the co efficient 1 2 is a multiple of ( 1p1 ) , and that is why it could be excluded from the set of generators of (1) Z(p) . Now, consider the co efficient system 1 ,... ,k ,... intro duced in the statement of the theorem (that is, i is the co efficient by xi+1 in the series [x]p1 if i is not divisible by p - 1 and is the co efficient by xi+1 in 1878


the series [x]p if i is divisible by p - 1). Suppose that we have proved that this co efficient system is a set of generators for (1) Z(p) in all dimensions up to n - 1. Hence, for any q and k n - 1, one has
( kq) = (q) 1 + ... + (q) k , 1 k (q ) (q )

(4)

where i U Z(p) . We are going to prove that n could also be decomposed in such a way. It follows from the above argument that we can consider only prime q . (p ) First, suppose that n is not divisible by p - 1. Hence n = n 1 , where p1 is a generator of (Z/p) . Let p2 be any prime. Taking the co efficient of xn+1 in both sides of (3), we obtain
( ( p1 np2 ) + pn+1 np1 ) + µ1 1 + ... + µ 2 n-1 n-1

( ( = p2 np1 ) + pn+1 np2 ) + 1 1 + ... + n-1 n-1 . 1

Here we expressed the co efficients k 1 , k 2 , k < n, as linear combinations of generators 1 ,... ,n-1 , i.e., µi ,i U Z(p) . Therefore,
( ( p1 (1 - pn )np2 ) = (p2 - pn+1 )np1 ) +(1 - µ1 )1 + ... +(n 1 2 -1

(p )

(p )

-µ

n-1

)n-1 .

(5)

Since p1 is a generator of (Z/p) and n is not divisible by p - 1, we deduce that p1 (1 - pn ) is invertible in Z(p) . 1 (p2 ) (p ) Thus, from (5) we obtain that n is a linear combination of 1 ,... ,n-1 and n = n 1 with co efficients from U Z(p) . Now, suppose that n is divisible by p - 1. Before we pro ceed further, let us make some preliminary remarks. It is well known (Milnor, Novikov) that a complex cobordism co efficient ring U is a polynomial ring: U = Z[a1 ,a2 ,... ,an ,... ], an -2n . The ring U is also the co efficient ring of the (universal) U formal group of geometric cobordisms. The co efficient ring of the logarithm of this formal group is the ring CPn CPn n+1 U (Z) = Z[b1 ,b2 ,... ,bn ,... ], where bn = . (This logarithm is g (u) = u + u , cf. [2].) One n +1 n n +1 could find two sets {a } and {b } of multiplicative generators in the rings U and U (Z) such that the i i inclusion 0 : U U (Z) is as follows: 0 (a ) = i p · b if i = pk - 1 for some k > 0, i b otherwise. i

Let B + be the set of elements of degree >0 in the ring B = U (Z). Then (B + )2 consists of elements in U (Z) that can be decomposed into the pro duct of two nontrivial factors. The map 0 : U U (Z) sends the (p) co efficients n of the series [x]p to the element of the form (p - pn+1 )bn +((B + )2 ). Therefore, the co efficients (p) pk -1 could be taken as multiplicative generators of U Z(p) in dimensions pk - 1. In other dimensions l = pk - 1, we have l
(p)

pU , i.e., l

(p)

is divisible by p in U .

Now let us return to the pro of of Theorem 1. We rewrite identity (3), replacing p1 by p: pp2 x + p = p2 px + p2 Taking the co efficient of xn pn =
(p2 ) +1 m

m2 xm+1 +
m

(p )

m

m (p2 x + 1 2 x2 + 2 2 x3 + ... )m+1
m

(p)

(p )

(p )

m xm+1 +

(p)

m2 (px + 1 x2 + 2 x3 + ... )m+1 .

(p )

(p)

(p)

(6)

in both sides, we get
(p) m
+ pn+1 n + 2 +
(p ) pn+1 n 2

m (p2 x + 1 2 x2 + 2 2 x3 + ... )m+1 (px +
(p) 1 x2

(p)

(p )

(p )

n+1

(p) p2 n

+

(p2 ) m
+

(p) 2 x3

+ ... )

m+1 n+1 (p )

,

where · n+1 stands for the co efficient of xn+1 . Let us again write the co efficients m2 for m < n as linear (p) combinations of generators 1 ,... ,m . Since m pU for m = pk - 1, we could rewrite the last identity 1879


as
( p(1 - pn )np
2

)

( = p2 (1 - pn )np) + p(µ1 1 + ... µnn ) 2

-
k :pk -1
p
)

(p) k -1

p2 x + 1 2 x2 + 2 2 x3 + ...
(p) p 2 -1 x
2

(p )

(p )

p

k

(7)
n+1 m+1

+ The pk - The also from

2

p-1 xp + p

(p)

+ ... + p
(p)

(p) p k -1 x

k

+ ...
n+1

.
(p)

last two summands in the above formula can be rewritten as p-1 1 + p2 -1 2 + ... + pk -1 k , 1 < n, i U . The other terms in the above identity belong to pU Z(p) (i.e., are divisible (p) co efficients pi -1 are the multiplicative generators of U Z(p) in the dimensions pi - 1. Therefo have i pU Z(p) , i.e., i is divisible by p in U Z(p) . Let i = pi with i U Z(p) . (7), we get p(1 - pn )n
(p2 )

(p)

where by p). re, we Then

= p2 (1 - pn )n + p(µ1 1 + ... + µn n ) 2 ( ( ( +p(pp)1 1 + pp) 1 2 + ... + pp)-1 k ), 2- k -

(p)

where pk - 1 < n, µi ,i U Z(p) . Since n is divisible by p - 1, we obtain that 1 - pn is divisible by p (for 2 p2 = p). Hence the entire identity above is divisible by p. Dividing it by p and noting that 1 - pn is invertible in p (1-pn ) (p) p (1-pn ) (p ) 2 2 U Z(p) , we obtain that n 2 = p2(1-pn ) n + 1 1 + ... + n-1n-1 . Thus, setting n = p2(1-pn ) U Z(p) , we obtain the decomposition of type (4) for n
(p2 )

(since now n = n ), which completes the pro of of Theorem 1.

(p)

Corollary 1. Let p1 be a generator of the cyclic group (Z/p) . The fol lowing set could be taken as a set of generators of the U Z(p) -module p (1) Z(p) : 0 = p; (p ) n = n 1 if n is not divisible by p - 1; (p) pk -1 = pk -1 , k = 1, 2,... (and no generators in other dimensions). The fol lowing set could be taken as a set of generators of the U Z/p-module (1) Z/p: (p ) n = n 1 if n is not divisible by p - 1; (p) pk -1 = pk -1 , k = 1, 2,... (and no generators in other dimensions). Proof. p - 1 and to pU . mo dules. Let us consider the set of generators of (1) Z(p) constructed in Theorem 1. If n is divisible by n = pk - 1, then the elements n are divisible by p, i.e., belong to pU . All other n do not belong Therefore, one could take the sets described in the corollary as generators of the corresponding The corollary is proved.

Below we are going to use the description of the U Z(p) -mo dule p (1) Z(p) in order to prove a result similar to the well-known Stong­Hattori theorem [1]. Namely, we are going to describe the set of cobordism classes of manifolds with a simple Z/p-action in terms of characteristic numbers. As was mentioned in [3], the homomorphism : F1 p (1) Z(p) can be extended up to a homomorphism p : F0 U (Z) Z(p) . For this p one has
k

p (x1 ,x2 ,... ,xk ) =
j =1

u [u]x

j

pu [u]p

,
k

where p (x1 ,x2 ,... ,xk ) := p (2k-1 (x1 ,x2 ,... ,xk )), 2k-1 (x1 ,x2 ,... ,xk ) F0 . In particular, p (1,... , 1) = pu . [u]p k 1880


Thus, for any (simple) action of Z/p on M 2n , the mo d p cobordism class of M 2n is expressed in terms of (j ) cobordism classes j U of fixed submanifolds and weights (xk ) (Z/p) in the corresponding (trivial) normal bundles as follows: [M 2n ]
j

j p (x1 ,... ,x(j ) )mo d pU mj
(j ) (j )

(j )

(8)

Now, the following question arises: which elements of the form j j p (x1 ,... ,xmj ) U (Z)Z(p) represent cobordism classes of manifolds with simple Z/p-action? This question was first posed in [3] and is analogous to the Milnor­Hirzebruch problem of describing the set of elements in U (Z) representing the cobordism classes of (stably complex) manifolds. While the Milnor­Hirzebruch problem is solved in the Stong­Hattori theorem, the answer to the above question is given in our Theorem 2. We will need the following definition. Definition 2. Let = i ki · (i), i, ki Z, i > 0, ki 0, be a partitioning of n = = i ki · i (i.e., n is decomposed into a sum of positive integers, and the number i enters this sum ki times). We say that the partitioning is divisible by p - 1 if all i such that ki = 0 are divisible by p - 1 (i.e., all the summands are divisible by p - 1; obviously, such partitionings exist only for those n divisible by p - 1). We say that the partitioning is non-p-adic if, for any j > 0, one has kpj -1 = 0 (i.e., there are no summands of the form pj - 1). Theorem 2. The element U (Z)-2n Z(p) belongs to represents the cobordism class of the manifold with simple teristic numbers s ( ), = i ki · (i), = i ki · i n by p - 1 the cohomological characteristic numbers s ( ), the U Z(p) -module p (1) Z(p) and, therefore, Z/p-action if and only if al l its K -theory characbelong to Z(p) , and for al l partitionings divisible = n are zero mo d p.

Proof. (a) Necessity. Let p (1) Z(p) . Note that the set of generators for the U Z(p) -mo dule p (1) Z(p) described in Corollary 1 has the following property: each of its elements i p (1) Z(p) is at the same time a multiplicative generator of U Z(p) in dimension -2i. However, the above set of generators of p (1) Z(p) has no elements in dimensions -2i such that i is divisible by p - 1 and i = pk - 1. Therefore, we add the missing generators i in these dimensions to get the whole set of multiplicative generators for U Z(p) . Now, we have U Z(p) = Z(p) [1 ,2 ,... ]. Since p (1) Z(p) U Z(p) , we see that all K -characteristic numbers s ( ) are in Z(p) . If n is not divisible by p - 1, then there are no partitionings divisible by p - 1. Now, let n = m(p - 1). One could write as a homogeneous polynomial of degree -2m(p - 1) in i : =
=m(p-1) k k where = i ki · (i), = 11 · 22 · ... . Now, it follows from the description of p (1) Z(p) given in Corollary 1 that p (1) Z(p) if and only if for all non-p-adic and divisible by p - 1 partitionings the co efficients r in decomposition (9) are zero mo dulo p. Consider the Chern­Dold character chU in cobordisms [3]: chU (u) = t + i1 i ti+1 . Here u = cU ( ) 1 U 2 (CP ) is the first cobordism Chern class of the universal line bundle, t = cH ( ) H 2 (CP ) is that in 1 cohomologies, and the co efficients i are from U (Z). Then, for any -2n , = s ( ) holds and U =n

r = rm(p

-1) m(p-1)

+ ... ,

(9)

B = U (Z) = Z[1 ,2 ,... ] (i.e., the co efficient ring of the Chern­Dold character coincides with U (Z)). Moreover, i = ei · i + ((B + )2 ) if i = pk - 1 and i = pei · i + ((B + )2 ) if i = pk - 1 with invertible ei Z(p) . Now, let us write as a homogeneous polynomial in i . Since all i have integer homological characteristic numbers, to prove the necessity of the theorem it suffices to show that the co efficient of in the decomposition of is zero mo dulo p for all partitionings = i ki · (i) divisible by p - 1. This co efficient is the corresponding homological characteristic number s ( ), which could be decomposed as follows: s ( ) =
:

r s ( ),

(10)

1881


where means that refines . This co efficient is divisible by p. Indeed, if the partitioning = i ki · (i) is divisible by p - 1 and is non-p-adic, then r is zero mo dulo p, since p (1) Z(p) (see ab ove). If summands of the form pk - 1 enter the partitioning , then pU (Z) Z(p) , i.e., s ( ) is divisible by p. The necessity of the theorem is proved. (b) Sufficiency. Since all K -characteristic numbers of are in Z(p) , one could deduce from the Stong­Hattori theorem [6] that U Z(p) . Suppose, moreover, that the characteristic number s ( ) is zero mo dulo p for any divisible by p - 1 partitioning = i ki · (i), = n. Consider again the generator set 1 ,2 ,... for U Z(p) constructed above. In order to prove that p (1) Z(p) , one needs to show that for every divisible by p - 1 and non-p-adic partitioning = i ki · (i), the co efficient r in decomposition (9) is zero mo dulo p. Let be such a partitioning. We can rewrite identity (10) as follows: s ( ) = r s ( )+
, =

r s ( ).

(11)

One can assume by induction that if a partitioning such that , = , = m(p - 1) is non-p-adic, then the co efficient r is divisible by p. If the partitioning = i ki · (i) is not non-p-adic (i.e., there are some summands of the form pk - 1), then s ( ) is divisible by p. In any case, the second summand on the right-hand side of (11) is zero mo dulo p. The left-hand side of (11) is zero mo dulo p by assumption. Since is non-p-adic, we have = e · + ... with invertible e Z(p) . Therefore, s ( ) is not divisible by p. Thus, it follows from (11) that r is zero mo dulo p. The theorem is proved. Corollary 2. The element U represents the cobordism class of manifolds with Z/p-action whose fixedpoint set has trivial normal bund le if and only if for al l divisible by p - 1 partitionings , the cohomological characteristic numbers s ( ), = n, are zero modulo p. Corollary 3. Each cobordism class of dimension n 4p - 6 contains a manifold M n with simple action of Z/p. In dimension n = 4p - 4, there exist manifolds (e.g., CP2p-2 ) whose cobordism class do es not contain a manifold with a simple action of Z/p. In [1], it was shown by means of metho ds not involving the formal group theory that a cobordism class U contains a manifold with strictly simple action of Z/p if and only if al l the characteristic numbers ( ) are zero mo dulo p. More precisely, it was shown in [1] that the set of cobordism classes of manifolds with strictly simple Z/p-action coincides with the U -mo dule spanned by the set Y 0 = p, Y 1 ,Y 2 ,... , where i Y i p -1 are the so-called "Milnor manifolds." The manifold Y i is uniquely determined by the following U conditions: s(pi -1) (Y i ) = p and s (Y i ) is divisible by p for every . For the purposes of description of the set of cobordism classes in terms of characteristic numbers, one could consider U Z(p) -mo dules instead of U (p) mo dules. Hence one could take the elements pi -1 from Corollary 1 as the representatives of the cobordism classes of Y i . Now we see that the U Z(p) -mo dule U [p, Y 1 ,Y 2 ,... ] Z(p) studied by Conner and Floyd is included in our U Z(p) -mo dule p (1) Z(p) . Indeed, the set of generators for the Conner­Floyd mo dule is a subset of the generator set for p (1) Z(p) . Finally, we note that if a certain cobordism class U contains a representative M with strictly simple action of Z/p, then any simple action of Z/p on M need not be strictly simple. Indeed, let us consider M1 = CPp-1 on which the generator Z/p acts as follows: (z1 : ... : zp ) = (z1 : z2 : ... : p-1 zp ) (this simple action with p fixed points is strictly simple as well), and M2 = CP1 , (z1 : z2 ) = (z1 : z2 ) (this simple action with 2 fixed points is not strictly simple). Then one has two simple actions of Z/p on M = M1 â M2 : (a, b) = (a, b) and (a, b) = (a, b), a CPp-1, b CP1 . The first one is strictly simple, while the second one is not. The author is grateful to Prof. V. M. Buchstaber for useful recommendations, stimulating discussions, and attention to this research. 1882


REFERENCES 1. P. E. Conner and E. E. Floyd, Differentiable Periodic Maps, Springer-Verlag, Berlin (1964). 2. S. P. Novikov, "The metho ds of algebraic topology from the viewpoint of cobordism theory," Math. USSR, Izv., 1, 827­913 (1967). 3. V. M. Buchstaber and S. P. Novikov, "Formal groups, power systems, and Adams operators," Math. USSR, Sb., 13, 80­116 (1971). 4. G. G. Kasparov, "Invariants of classical lens manifolds in cobordism theory," Math. USSR, Izv., 3, 695­705 (1969). 5. S. P. Novikov, "Adams operators and fixed points," Math. USSR, Izv., 2, 1193­1211 (1968). 6. R. E. Stong, Notes on Cobordism Theory, Princeton Univ. Press (1968). 7. V. M. Buchstaber, "The Chern­Dold character in cobordisms. I," Math. USSR, Sb., 12, 573­594 (1970).

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