Ó DOI: 10.2478/s12175-012-0057-2 Math. Slovaca 62 (2012), No. 5, 949–966 REALIZING COHOMOLOGY CLASSES AS EULER CLASSES Aniruddha C Naolekar (Communicated by J´ulius Korbaˇs ) s ◦ ABSTRACT. For a space X,letEk(X), Ek(X)andEk (X) denote respectively the set of Euler classes of oriented k-plane bundles over X, the set of Euler classes of stably trivial k-plane bundles over X and the spherical classes in Hk(X; Z). s ◦ We prove some general facts about the sets Ek(X), Ek(X)andEk (X). We also compute these sets in the cases where X is a projective space, the Dold manifold P (m, 1) and obtain partial computations in the case that X is a product of spheres. c 2012 Mathematical Institute Slovak Academy of Sciences 1. Introduction Given a topological space X, we address the question of which classes x ∈ Hk(X; Z) can be realized as the Euler class e(ξ)ofanorientedk-plane bundle ξ over X. We also look at the question of which integral cohomology classes are spherical. It is convenient to make the following definition. s º ≥ ÒØÓÒ 1.1 For a space X and an integer k 1, the sets Ek(X), Ek(X) ◦ and Ek (X) are defined to be k Ek(X)= e(ξ) ∈ H (X; Z) | ξ is an oriented k-plane bundle over X s k Ek(X)= e(ξ) ∈ H (X; Z) | ξ is a stably trivial k-plane bundle over X ◦ k k ∗ Ek (X)= x ∈ H (X; Z) | there exists f : S −→ X with f (x) =0 . s Note that we always have an inclusion Ek(X) ⊆ Ek(X) and the inclusion can ◦ be strict. The classes in Ek (X) are called spherical classes. Clearly, spherical 2010 M a t h e m a t i c s Subject Classification: Primary57R20;Secondary 19M20. K e y w o r d s: Euler class, spherical class. Part of the work was supported by IISER-Pune. ANIRUDDHA C NAOLEKAR classes are of infinite order and are indecomposable as elements of the integral cohomology ring. In recent times there has been some interest in understanding the sets Ek (see [5], [6], [14]). Besides being a natural question to study, part of the motivation ◦ for studying the sets Ek and Ek comes from the following result. º Ì ÓÖÑ 1.2 ([5, Theorem 1.3, p. 378]) Let ξ be an oriented k-plane bundle over a CW-complex X. If the Euler class e(ξ) is spherical, then the holonomy group of any Riemannian connection on ξ acts transitively on the sphere bundle S(ξ) of ξ. There are isolated results in the literature about the set Ek(X)(see[5],[6], [14]). The most general result about realizing cohomology classes as Euler classes seems to be the following theorem of Guijarro, Schick and Walschap [5]. º ∈ N Ì ÓÖÑ 1.3 ([5, Theorem 1.6, p. 379]) Given k, n with k even, there is an integer N(k, n) > 0 such that for every CW-complex X of dimension n and every cohomology class x ∈ Hk(X; Z), there is an oriented k-plane bundle ξ over X with e(ξ)=2N(k, n) · x. Thus, in our notation, under the hypothesis of the above theorem k 2N(k, n)H (X; Z) ⊆ Ek(X). Note that, with the hypothesis as in the above theorem, we are guaranteed that ◦ ◦ (for k even) if Ek (X) = ∅,thenEk (X) ∩ Ek(X) = ∅. Thus the existence of a spherical class implies that some spherical class is also the Euler class of some oriented k-plane bundle. In this paper we shall show that the hypothesis that k is even is essential (see Theorem 1.5 below). ◦ s In this paper we shall compute the sets Ek(X), Ek (X)andEk(X) for the cases when X is a projective space, a product Sm × Sn of spheres for certain values of k, m, n (see Theorem 1.4 below) and the Dold manifold P (m, 1) with m>1. The paper is organized as follows. ◦ s In Section 2 we discuss some general properties of the sets Ek, Ek and Ek. Section 3 contains the computational part of the paper. We first describe the ◦ s sets Ek(X), Ek (X)andEk(X)whenX is a projective space. The description of these sets when X is the real projective space is arrived at by looking at certain canonical bundles over X. The case when X is a complex projective space follows from a general result that we prove about spaces whose cohomology ring is generated by the second cohomology (see Proposition 3.2). ◦ s Section 3.3 deals with the computation of the sets Ek(X), Ek (X)andEk(X) when X = Sm × Sn is a product of two spheres with restrictions on k, m, n (see Theorem 1.4 below for the precise statement). It is a classical result of Milnor [9] n n n and Atiyah-Hirzebruch [1] that if n =2 , 4, 8iseven,thenEn(S )=2H (S ; Z) (see also [14, Theorem 1.2] for a geometric proof of this fact). In particular, in 950 REALIZING COHOMOLOGY CLASSES AS EULER CLASSES these cases a generator of Hn(Sn; Z) is never an Euler class. The main theorem of this section is the following. m n º × Ì ÓÖÑ 1.4 Let X = S S . (1) If m, n ≡ 3(mod8),thenEk(X)=0for 1 ≤ k<m+ n and s m+n Em+n(X)=Em+n(X)=2H (X; Z). (2) If n is even, and n = m,then 2Hn(X; Z) if n =2 , 4, 8 En(X)= Hn(X; Z) if n =2, 4, 8 s n+1 (3) If m =1and n ≡ 5(mod8),thenEn+1(X)=En+1(X)=2H (X; Z). m+n Note that the conclusion 2H (X; Z)=Em+n(X) in the cases (1) and (3) in the above theorem are not true in general. Indeed, if X = S3 × S5, 8 then the inclusion 2H (X; Z) ⊆ E8(X) is strict. We shall make a more general observation later (see Example 2.10 below). ◦ Finally, in Section 3.9 we discuss the computation of the sets Ek, Ek and s Ek when X is the Dold manifold P (m, 1) with m>1. We give a complete description of the sets Ek(P (m, 1)) except in the case when m is even and k = m + 2 (see Proposition 3.13). The computations depend upon the existence of certain canonical bundles over the Dold manifolds (see [12], [13]). It follows from our computations that the assumption that k is even in Theorem 1.3 is essential. The main theorem of this section is the following. º Ì ÓÖÑ 1.5 Let m>1 be an odd integer. Then we have ◦ ◦ Em(P (m, 1)) = ∅,Em(P (m, 1)) =0 and Em(P (m, 1)) ∩ Em(P (m, 1)) = ∅. 2. Generalities s ◦ In this section we prove some general facts about the sets Ek, Ek and Ek . Throughout, we follow the notations in [10]. Recall that if ξ is an oriented k-plane bundle over X, then its Euler class e(ξ) k k is an element of H (X; Z). Let uξ ∈ H (E(ξ),E(ξ)0; Z) be the Thom class of ξ k 2k and ϕ: H (X; Z) −→ H (E(ξ),E(ξ)0; Z) be the Thom isomorphism, then the −1 Euler class e(ξ)ofξ is by definition e(ξ):=ϕ (uξ uξ). If k is odd, then the (graded) commutativity of the cup product shows that 2e(ξ)=0.Foran oriented k-plane bundle ξ, the mod 2 reduction of the Euler class e(ξ)equals the top Stiefel-Whitney class wk(ξ)ofξ. s k Given a space X,wehaveinclusionsEk(X) ⊆ Ek(X) ⊆ H (X; Z)andin general all the inclusions can be strict. The set Ek(X) is inverse closed as changing the orientation changes the sign of the Euler class. Also, 0 ∈ Ek(X) 951 ANIRUDDHA C NAOLEKAR ◦ and hence the set Ek(X) is non empty. On the other hand, as Ek (X) consists of ◦ spherical classes, 0 ∈/ Ek (X). It is well known that a real line bundle is orientable if and only if it is trivial. Thus for any space X the set E1(X) is trivial (meaning E1(X)={0}, which will simply be denoted by E1(X)=0). n We first note what is known about the set Ek(S ). We begin by recalling the following result. º Ì ÓÖÑ 2.1 ([1], [9], [14, Theorem 1.2]) If n =2, 4, 8 is even, then the Euler class of any n-plane bundle over Sn is an even multiple of a generator of Hn(Sn; Z). In particular, the above theorem implies that if n is even and n =2 , 4, 8, then n n n n En(S ) ⊆ 2H (S ; Z). The set Ek(S ) for the spheres can now be described completely. n Example 2.2. If n is odd then clearly En(S ) = 0. It is known that Euler classes (of the underlying real bundle) of the canonical (complex, quaternionic and octonionic) line bundles over S2 = CP1, S4 = HP1 and S8 = OP1 are generators of H2(S2; Z), H4(S4; Z), and H8(S8; Z) respectively. Since there are maps f : Sn −→ Sn of arbitrary degrees and as the Euler class is natural, it n n n follows that En(S )=H (S ; Z)ifn =2, 4, 8.