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INVOLUTION DIFFEOMORPHIC

OF

MANIFOLDS TO REAL

WITH A SET PROJECTIVE

OF

FIXED

POINTS

SPACE
UDC 517.83

A. S. Mishchenko

It is proved that if, on a manifold with an involution, the subset of fixed points is diffeomorphic to an even-dimensional real projective space, then the manifold is bordant to the complex projective space in the class of nonoriented bordisms.

In the classification of smooth periodic transformations of manifolds, it is important to investigate the relation between various invariants of the manifold and the structure of the set of fixed points of the transformation. An effective method of doing this is to apply bordism theory for the description of the set structure of the fixed points. The present note gives an illustration of the application of bordism theory to a problem concerning fixed points of an involution. We start with a description of the situation we shall study. All manifolds are assumed to be unoriented. We say that a manifold M with an involution is given if we have a smooth diffeomorphism T : M ~ M, T 2 = id. Two closed manifolds M and M' with involution are bordant if there is a manifold with a border W with an involution such that the boundary OW is equivalently diffeomorphic to the unconnected sum M U M'. If M is a manifold with an involution T, then the set N of fixed points of T, i.e., those points x such that Tx = x, is a finite union of submanifolds N = U Ni. A tubular neighborhood Ui of a submanifold Ni is diffeomorphic to the space of the normal vector fibering ~?i of the imbedding of N i in M, and the involution T induces a Z2-fibering strUcture in this fibering. The bordism class with involution M determined the formal sum of bordisms (also nonoriented) of real fiberings ~[~?i]and is completely determined by this formal sum. We are interested in the following problem. Let the set N of fixed points of the involution T consist of a single connected component and be diffeomorphic to the real projective space RP 2n. What can be said concerning the manifold M itself? An example of such an involution is the complex-conjugate transformation of coordinates in a complex projective space CP 2n. Conner and Floyd ([1], p. 228) assume that this is a unique situation; i.e., if the Set N of fixed points of a manifold M with involution T is diffeomorphic to RP 2n, then M is bordant to the manifold CP 2n with the above involution. It is sufficient to prove that the normal fibering ~? of the imbedding of N in M is bordant to the normal fibering of the imbedding of RP 2n in CP 2n. Conner and Floyd [1] prove that dim M = 4n, and the fibering ~? is stably isomorphic to the fibering (2s + 1) ~ (s -> n), where ~ is the one-dimensional Hopf fibering over RP 2n. The group of (unoriented) bordisms of real fiberings ~. (BO(n)) has a ring structure if we take multiplication to be the cartesian product of bases and the direct sum of fiberings. This ring is isomorphic to the ring of polynomials ~.[Xn] , where the xn are one-dimensional Hopf fiberings over RP n. In particular x 0 is the one-dimensional (trivial) fibering over one-point space. THEOREM. If M projective space RP 2n, fibering of RP 2n, and a which the involution is m is a manifold with involution whose set of fixed points is diffeomorphic to the real then the normal fibering of the imbedding of RP 2n in M an is bordant to the tangent manifold with involution M 4n is bordant to the complex projective space CP 2n, on the taking of the complex conjugate of homogeneous coordinates.

Proof. Each vector fibering 77 over RP 2n is equal to k~ + N. On the other hand each fibering ~ can be expressed in the form ~?' + N, where dim ~?' = 2n. Hence a bordism y representable by ~?can be written M. V. Lomonosov Moscow State University. Translated from Matematicheskie Zametki, Vol. 9, No. 3, pp. 249-252, March, 1971. Original article submitted December 17, 1969.
9 1971 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15.00.

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y = y0(x0) N. I_~t Yk be the bordisms represented by the fiberings k~, Yk = Y0k(X0)k-2n. Let k 0 be the smallest number such that the complete Stiefel class is equal to unity: w(k0~ ) = 1. Then the bordisms {Y0k} (1 -< k < k0) run over the whole set of bordisms {Y0k}. We must prove that a(Y0k ) ~ 0 for k ~2n + 1, where : ~lk(BO(n)) ~ ~ln+k-1 (RP~176 This is equivalent to the fact that a(Yk) is a polynomial of degree higher than k -2n for k ~ 2n + 1. The element c~(yk) is represented by the projection Pk of the fibering k~ over RP 2n with the canonical fibering ~ imposed on it. We have to prove that, for k ~ 2n + 1, there is a number s > k -2n and a Stiefel class w of the manifold Pk such that (cSw, Pt0 ~ 0. The cohomologies of Pk form the ring H*(P~, Z~) = Z2 [u, c] / {(c § u)~, u~n+~}. The complete Stiefel class of tangent fibering is equal to w = (1 + u)2n+l(1 + c + u) k (cf. [1], See. 24). Letrbe the smallest number of then Y0k runs through the whole set of k -2n + 1 _ j ~< k -2, 2n + 1 <- j, such acteristic classes of tangent fibering. k-l-j>_ 2n +l, thenw(i ) = (u+c) i. which2 r > 2n. It is easily seen that, if2n +l_
thatj ~k-2n +l,j>_ 2n +1, j_
+1 +2 r > k > 2n +1) andC] n = 1. Suehanumberex-

2n= tccs_l...z~0, ai=0,t,

k=~s+l~s...~t. Then m = fls.-.fil I < 2n. j = fis .../~tat-1 ...1. = C~s fls-1 = (Vs-~ ..... +l,j-
Here/Js+ 1 = 1 or fis+l = 0, fis = 1. Consider the following two cases. 1) fls+~ = 1. Let fis = as ..... fit = at, fit-1 = 0, ~t-1 = 1 for some number t. Then we must take Plainlyj >_2n +1, j_> k-2n +1, j-_2n +t,j->k-2n = 1, and the theorem is proved.

Remark. For each manifold Mn, there is a manifold with involution N 2n such that Mn is its set of fixed points. For N 2n we must take M n x Mn, and the involution T is defined by T(x, y) = (y, x). Then the normal fibering of the imbedding of Mn in N m is isomorphic to the tangent fibering of Mn, The question arises of whether there are other manifolds with involution N' having a set of fixed points diffeomorphic to the fixed manifold Mn. There is a theorem which asserts that, ff Mn = RP 2k, then N' is bordant to N. However when Mn ~ RP 2k this is not in general true. For example if M 5 = (RP 4 + RP 2 · Rp2)Rp I, besides then the manifold with involution M 5 x M 5, there is another manifold with involution of dimension 8, bordant to the manifold (Rp2) 4 + (Rp4) 2 (and not bordant to M 5 x Ms), whose fixed points are bordant to M 5. Calculalations with examples of this type reduce to the methods used in [2] and [3]. LITERATURE
1.

CITED

2. 3.

P. E. Conner and E. E. Floyd, Smooth Periodic Mappings [Russian translation], Moscow (1969). A. S. Misehenko, "Manifolds with the action of the group Zp and fixed points," Matem. Zametki, 4, No. 4,381-386 (1968). A. S. Mishchenko, "Bordants with the action of the group Zp and fixed points," Matem. Sb., 8__~0,No. 3, 307-313 (19697.

149