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Fixed Point Free Involutions on Cohomology Projective Spaces

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Fixed Point Free Involutions on Cohomology Projective Spaces Indian J. pure appl. Math., 39(3): 285-291, June 2008 °c Printed in India. FIXED POINT FREE INVOLUTIONS ON COHOMOLOGY PROJECTIVE SPACES HEMANT KUMAR SINGH1 AND TEJ BAHADUR SINGH Department of Mathematics, University of Delhi, Delhi 110 007, India e-mail: tej b [email protected] (Received 22 September 2006; after final revision 24 January 2008; accepted 14 February 2008) Let X be a finitistic space with the mod 2 cohomology ring isomorphic to that of CP n, n odd. In this paper, we determine the mod 2 cohomology ring of the orbit space of a fixed point free involution on X. This gives a classification of cohomology type of spaces with the fundamental group Z2 and the covering space a complex projective space. Moreover, we show that there exists no equivariant map Sm ! X for m > 2 relative to the antipodal action on Sm. An analogous result is obtained for a fixed point free involution on a mod 2 cohomology real projective space. Key Words: Projective space; free action; cohomology algebra; spectral sequence 1. INTRODUCTION Suppose that a topological group G acts (continuously) on a topological space X. Associated with the transformation group (G; X) are two new spaces: The fixed point set XG = fx²Xjgx = x; for all g²Gg and the orbit space X=G whose elements are the orbits G(x) = fgxjg²Gg and the topology is induced by the natural projection ¼ : X ! X=G; x ! G(x). If X and Y are G-spaces, then an equivariant map from X to Y is a continuous map Á : X ! Y such that gÁ(x) = Ág(x) for all g²G; x 2 X. Two actions of G on X are called equivalent if there exists an equivariant 1The author was supported by JRF of the Council of Scientific and Industrial Research. 286 HEMANT KUMAR SINGH AND TEJ BAHADUR SINGH homeomorphism of X onto itself. In 1919, Ker´ ekj´ art´ o´ and Brower independently showed that any periodic homeomorphism on a 2-dimensional sphere S2 is equivalent to an orthogonal action. An interesting problem in topology was to decide whether or not an action of a finite cyclic group G on an euclidean space Rn (or a sphere Sn) is equivalent to an orthogonal action. In 1925-26, Hopf raised the question of classifying the orbit spaces of free actions of finite cyclic groups on spheres. As unravelling the problem seemed impossible for some time, in the late 1930s, Smith considered a weaker problem: To study the homological relationship between the total space and the fixed point set of a homeomorphism of prime order. Following this thread of research, Smith [8] and Su [9] have obtained the cohomological structure of the fixed point set of involutions on a cohomology real projective space, and Bredon [2] has studied in detail the relationship between a mod p (p a ¤ prime) cohomology projective space X (that is, H (X; Zp) is a truncated polynomial ring over the field Zp generated by an element) and the fixed point set of G = Zp acting on X. In the early 1950s, Floyd [5] and Liao [6] began the study of the homological relationship between the total space and orbit space of a periodic map of prime order. In this note, we determine the mod 2 cohomology ring of the orbit space of a fixed point free involution on a cohomology complex (or real) projective space. For an odd integer n, the mapping [z0 : z1 : ¢ ¢ ¢ : zn¡1 : zn] ! [¡z1 : z0 : ¢ ¢ ¢ : ¡zn : zn¡1] defines a fixed point free involution on CP n. Cusick [4] has shown that only finite group which can act freely on a complex projective space is Z2. As above, one obtains a fixed point free involution on the real projective space RP n. Throughout this paper, H¤(X) will denote the Cechˇ cohomology of the space X with coeffi- n n ¤ n+1 cients in Z2. It is known that if X = RP or CP , then H (X) = Z2[a]=ha i, where deg a = 1 or 2, respectively. We recall that a paracompact Hausdorff space is finitistic if every open covering ¤ of X has a finite-dimensional open refinement (see [1]). By X »2 Y , we shall mean that H (X) and H¤(Y ) are isomorphic. We establish the following results. n Theorem 1 — Let G = Z2 act freely on a finitistic space X »2 CP . Then n must be odd and ¤ 3 (n+1)=2 H (X=G; Z2) = Z2[x; y]=hx ; y i; where deg x = 1 and deg y = 4. In particular, this gives a classification of the cohomology type of spaces with the fundamental group Z2 and the covering space a complex projective space. A classification of fixed point free involutions on complex projective space may be possible analogous to that of spheres. Consider the (2m + 1)-dimensional unit sphere S2m+1 ½ Cm+1. Given a positive integer p and integers q0; : : : ; qm relatively prime to p, there is a fixed point free homeomorphism h : 2m+1 2m+1 2i¼q =p 2i¼qm=p S ! S , (z0; : : : ; zm) 7! (e 0 z0; : : : ; e zm), of period p. This gives a free FIXED POINT FREE INVOLUTIONS ON COHOMOLOGY PROJECTIVE SPACES 287 2m+1 2m+1 action of G = Zp on S . The orbit space S =G is called a lens space, and denoted by 2m+1 i¼q =2 L (p; q0; : : : ; qm). If q0; : : : ; qm are odd integers, then [z0 : ¢ ¢ ¢ : zm] 7! [e 0 z0 : ¢ ¢ ¢ : i¼qm=2 2m+1 e zm] is a fixed point free involution on RP , and its orbit space is clearly the lens space 2m+1 L (4; q0; : : : ; qm). It is not difficult to see that the orbit space of any fixed point free involution on RP 2m+1 has the homotopy type of a lens space. The significance of the following result lies in the fact that the orbit spaces of different free actions of a group on a space may have different cohomology algebras. n Theorem 2 — Let G = Z2 act freely on a finitistic space X »2 RP . Then n must be odd and ¤ 2 (n+1)=2 H (X=G; Z2) = Z2[x; y]=hx ; y i; where deg x = 1 and deg y = 2. It follows that the mod 2 cohomology algebra of the orbit space of a fixed point free involution on 2m+1 a cohomology real projective space is that of a lens space L (4; q0; : : : ; qm), where q0; : : : ; qm are odd integers. As a consequence of the above theorems, we obtain the following: m Corollary — Suppose that G = Z2 acts on the unit m-sphere S by the antipodal map. Let X n n be a finitistic space with X »2 CP (resp. RP ), n odd. If G = Z2 acts freely on X, then there exists no equivariant map Sm ! X for m > 2 (resp. 1). 2. PRELIMINARIES In this section, we recall some facts which will be used in proofs of our theorems. Given a G- i ¼ space X, there is an associated fibration X ! XG ! BG, and a map ´ : XG ! X=G, where ¤ XG = (EG £X)=G and ¼ : EG ! BG is a universal G-bundle. If G = Z2 acts trivially on H (X) i ¼ and the Leray-Serre spectral sequence of X ! XG ! BG degenerates, then there must be fixed points of G on X (see [1]). When G acts freely on X, ´ : XG ! X=G is a homotopy equivalance. ¤ ¤ ¤ So the cohomology rings H (XG) and H (X=G) are isomorphic. To compute H (XG), we exploit the Leray-Serre spectral sequence of the map ¼ : XG ! BG with coefficients in Z2. If the system of local coefficient is simple, then the restriction of the product structure in the spectral sequence ¤;0 0;¤ ¤ ¤ to the subalgebras E2 and E2 gives the cup products on H (BG) and H (X), respectively. The edge homomorphisms, p p;0 p;0 p;0 p;0 p H (BG) = E2 ! E3 ! ::: ! Ep+1 = E1 ⊆ H (XG); and q 0;q 0;q 0;q q H (XG) ! E1 = Eq+1 ½ ::: ½ E2 = H (X) are, respectively, the homomorphisms ¤ p p ¼ : H (BG) ! H (XG); and 288 HEMANT KUMAR SINGH AND TEJ BAHADUR SINGH ¤ q q { : H (XG) ! H (X): ¤ We also recall that H (BG) = Z2[t], where deg t = 1, and the fact that the cup product in Er+1 » ¤ is induced from that in Er via the isomorphism Er+1 = H (Er). For the results about spectral sequences, a good reference is [7]. 3. PROOFS PROOF OF THEOREM 1 : If n is even, then G has fixed points, by the Floyd’s euler characteristic formula. Since G acts freely on X, n must be odd. If n = 1, then X is a mod 2 cohomology 2-sphere and X=G is a mod 2 cohomology real projective plane. Thus the theorem holds in this ¤ case. Assume that n > 2. As ¼1(BG) = G acts trivially on H (X), the spectral sequence of the i ¼ fibration X ! XG ! BG must not collapse at the E2-term. Since the system of local coefficients on BG is simple, we have p;q » p q E2 = H (BG) ­ H (X): p;q 2 It is obvious that E2 = 0 for q odd; so E2 = E3. Let a 2 H (X) be the generator of the ¤ cohomology ring H (X). Because the spectral sequence does not degenerate, we have d3(1 ­ a) = t3 ­ 1, and therefore p;2q p+3;2q¡2 d3 : E3 ! E3 p;q is the trivial homomorphism for q even, and an isomorphism for q odd.
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