Mat S b o r n ik
Tom 95 (137) (1974), N o. 1
M a t h . U SSR Sbo r n ik
Vol. 24 (1974), N o. 1
REMARK ON TOE P AP ER "ACTION S OF F IN ITE
CYCLIC G ROUPS ON QUASICOMPLEX M AN IF OLD S"
UDC 513.836
I. M. KRICEVER
In the author's paper [l] two theorems were formulated (Theorems 1.11 and 2.2),
assertin g that certain conditions on a collection of Ζ
bundles were necessary and
sufficient for this collection to be the collection of normal bundles to the fixed submani
folds of a unitary Ζ
manifold. U nfortunately, these conditions are in fact only suffi
cient. However, the methods of [l] allow one, at the cost of purely technical compli
cation s, to obtain n ecessary and sufficient conditions.
We consider the category whose objects are collections (Χ, \ ζ.\ ), where X is a C
manifold and \ ^ is a finite collection of G bundles
U
u
in this category (n = iim^X; µ = (µ
over X. The bordism groups
. . .) is a "m u lt i in d e x", µί
dim if.) re
place for us the bordism groups of (7 manifolds, which, incidentally, are the special
case of 1)
for µ. = 0.
f
η,µ
ι
For a collection of Ζ , bundles \ ζ.j over X (p prime), let £ .
of ζί to a fixed submanifold Fs
of X, and νs
be the restriction
the normal bundle of Fs
in X. As usu
al, t h e collection of Ζ , bundles ν , \ ζ• \ over the trivial Ζ , manifolds F
S
• H fZ
de
^
1~)
IS
fines a homomorphism
e> kuznpk + RZpk
^V
where the sum is taken over those collection s of nonnegative in tegers (/, \ n.\ , \ n. \ )
for which 2 ( Σ . «. + I) = η and S, «. •
7
7
7
We d en o t e by Ψ: R^^
(i, / ) * (pi + s . j') (0 < s < p
F or
U
y•
» R*%
the homomorphism induced by t h e c h a n ge of in d ic e s
1, / Ξ s (mod p)
ω — (i,, . . . . i ), let νω(ιι.,
tion of t h e se r i e s Σ * = 1 u/
µ ·.
l
1
/ ' = (/
s)/ p),
[CP{us)]~ l,
wh ere CP(u) = 1™ [CPm]um.
le c t io n ωί of len gth µί there is defined a hom om orphism Vω.,
t ive gen er a t o r [Μ] Χ 11^ ( C P / ·? )
an d a lso / ^ (s, / ' ) .
. . . , u ) be t h e se r ie s o bt a in ed by sym m etriza
of th e group ^ 2 iV^y= O~
F or each col
wh o se va lu e on t h e addi»
®U{n. .)) is equ al to [u]N ,_
χ [Μ] χ {the series o bt a in ed from ν ω { . . . , f ([u\ . , vs), . . . ) by r ep la c in g νs by
fell
We define homomorphisms D a. as follows:
AMS (MOS) subject classifications
if (/', p) = I, then
(1970). Primary 55C35, 57D85; Secondary 57D 90.
Copyright © 1976. American Mathematical Society
145
146
Ι. Μ. KRICEVER
s
[Μ] χ Π {CPi )
= [ < * , · [Μ] · Π If )
Bms ([„],),
where dim , [M] = m; but if p divides /', then
?) = [«0 , · [Μ]
Da, ({Μ) Χ Π (CP?))
\
s=i
We denote by [Da]
i£l)
/
ω
s= n \
l
"J /
Ψ'
/
(ω,, . . .), the tensor product of the homomorphisms D a.
ω
1
[ D a ] _ :R2^ ^
U* [[u]]/ 0p ([u] .
) = 0.
Ρ*"1
ω
We introduce an additional graduation in R
, by making each collection (in.!,
\ ni .\ ) correspond to the number d = X,· ^ =^n.. Now let pd be the "homogeneous com
pon en t" of ρ € R+ P^.
z k
Theorem. A bordism class ρ e R p
belongs to Im β if and only if ψ (ρ) ε
Im β
and for any ω the quantity
u
is divisible by u
n
in the ring U [[u]]/ d A[u] ,
7
V
ρ R> "* i
) = 0.
ra
To describe Im β
in the case when m is divisible by at least two primes, it is
n ecessary to insert an analogous correction into the hypothesis of Theorem 2.2 of [ l] .
Received 20/ MAR/ 74
BIBLIOGRAPHY
1. I. M. Kricever, Actions of finite cyclic groups on quasicomplex
manifolds,
(132) (1973), 306 319 = Math. USSR Sb. 19 (1973), 305 319
Translated by N. STEIN
Mat. Sb. 90