isibang/ms/2011/2 April 18th, 2011 http://www.isibang.ac.in/e statmath/eprints Realizing cohomology classes as Euler classes Aniruddha C Naolekar Indian Statistical Institute, Bangalore Centre 8th Mile Mysore Road, Bangalore, 560059 India REALIZING COHOMOLOGY CLASSES AS EULER CLASSES ANIRUDDHA C NAOLEKAR s ± Abstract. For a space X let Ek(X), Ek(X) and Ek (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). We s ± prove some general facts about the set Ek(X), Ek(X) and Ek (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. 1. Introduction Given a topological space X, we address the question of which classes x 2 Hk(X; Z) can be realized as an Euler class e(») of an oriented k-plane bundle » over X. We also look at the question of which integral cohomology classes are spherical. It is convenient to make the following de¯nition. s De¯nition 1.1. For a space X and an integer k ¸ 1, the sets Ek(X), Ek(X) and ± Ek(X) are de¯ned to be k Ek(X) = fe(») 2 H (X; Z) j » is an oriented k-plane bundle over Xg s k Ek(X) = fe(») 2 H (X; Z) j » is a stably trivial k-plane bundle over Xg ± k k ¤ Ek(X) = fx 2 H (X; Z) j there exists f : S ¡! X with f (x) 6= 0g: 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 classes are of in¯nite 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], [13]). Besides being a natural question to study, a part of the motivation for studying ± the sets Ek and Ek comes from the following result. Theorem 1.2. ([5], Theorem 1.3, p. 378) Let » be an oriented k-plane bundle over a CW -complex X. If the Euler class e(») is spherical, then the holonomy group of any Riemannian connection on » acts transitively on the sphere bundle S(») of ». ¤ There are isolated results in the literature about the set Ek(X) (see [5], [6], [13]). The most general result about realizing cohomology classes as Euler classes seems to be the following theorem of Guijarro, Schick and Walschap [5]. Theorem 1.3. ([5], Theorem 1.6, p. 379) Given k; n 2 N with k even, there is an integer N(k; n) > 0 such that for every CW -complex X of dimension n and every 2000 Mathematics Subject Classi¯cation. 57R20, 19M20. Key words and phrases. Euler class, Spherical class. 1 2 ANIRUDDHA C NAOLEKAR cohomology class x 2 Hk(X; Z), there is an oriented k-plane bundle » over X with e(») = 2N(k; n) ¢ x. ¤ Thus, in our notation, under the hypothesis of the above theorem 2N(k; n)Hk(X; Z) ⊆ Ek(X). Note that, with the hypothesis as in the above theorem, we are guaranteed ± ± that (for k even) if Ek(X) 6= ;, then Ek(X) \ Ek(X) 6= ;. Thus the existence of a spherical class implies that some spherical class is also an Euler class of some ori- ented 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) and Ek(X) for the cases when X is projective space, a product of spheres and the Dold manifold P (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 ¯rst describe the sets ± s Ek(X), Ek(X) and Ek(X) when X 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 the 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 computaion of the sets Ek(X), Ek(X) and Ek(X) when X = Sm £ Sn is a product of spheres. It is a classical result of Milnor [8] and n n n Atiyah-Hirzebruch [1] that if n 6= 2; 4; 8 is even, then En(S ) = 2H (S ) (see also [13] Theorem 1.2, for a geometric proof of this fact). In particular, in these cases a generator of Hn(Sn; Z) is never an Euler class. The main theorem of this section is the following. Theorem 1.4. Let X = Sm £ Sn. s (1) If m; n ´ 3 (mod 8), then Ek(X) = 0 for 1 · k < m + n and Em+n(X) = m+n Em+n(X) = 2H (X; Z). (2) If n is even, and n 6= m, then ½ 2Hn(X; Z) if n 6= 2; 4; 8 E (X) = n Hn(X; Z) if n = 2; 4; 8 s n+1 (3) If m = 1 and n ´ 5 (mod 8), then En+1(X) = En+1(X) = 2H (X; Z). ± s Finally, in Section 3.12 we discuss the computation of the sets Ek, Ek and Ek when X is the Dold manifold P (m; 1). The computations depend upon the existence of certain canonical bundles bundles over the Dold manifolds (see [11], [12]). It follows from the computations that the assumption that k is even in Theorem 1.3 is essential. The main theorem of this section is the following. ± Theorem 1.5. Let m > 1 be an odd integer. Then Em(P (m; 1)) 6= ;, Em(P (m; 1)) 6= ± 0 and Em(P (m; 1)) \ Em(P (m; 1)) = ;. COHOMOLOGY CLASSES AS EULER CLASSES 3 2. Generalities s ± In this section we prove some general facts about the sets Ek;Ek and Ek. Throughout, we follow the notations in [9]. Recall that if » is an oriented k-plane bundle over X, then its Euler class e(») is k k an element of H (X; Z). Let u» 2 H (E(»);E(»)0; Z) be the Thom class of » and k 2k ' : H (X; Z) ¡! H (E(»);E(»)0; Z) be the Thom isomorphism, then the Euler ¡1 class e(») of » is by de¯nition e(») := ' (u» ^ u»). If k is odd, then the (graded) commutativity of the cup product shows that 2e(») = 0. For an oriented k-plane bundle », the mod 2 reduction of the Euler class e(») equals the top Stiefel-Whitney class wk(») of ». s k Given a space X, we have inclusions Ek(X) ⊆ Ek(X) ⊆ H (X; Z) and in 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 2 Ek(X) and hence the set ± Ek(X) is non empty. On the other hand, as Ek(X) consists of spherical classes, ± 0 2= 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) = f0g, which will simply be denoted by E1(X) = 0). n We ¯rst note what is known about the set Ek(S ). We begin by recalling the following result. Theorem 2.1. [1], [8], ([13], Theorem 1.2) If n 6= 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 6= 2; 4; 8, then n n n n En(S ) ⊆ 2H (S ; Z). The set Ek(S ) for the spheres can now be described com- pletely. 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 arbi- n n n trary degrees and as the Euler class is natural, it follows that En(S ) = H (S ; Z) if n = 2; 4; 8. It is well known (see [9]) that if n is even, then the Euler class of the tangent bundle of Sn is twice a generator. Thus if n is even, then ev- ery element of 2Hn(Sn; Z) is the Euler class of some n-plane bundle over Sn. In n n n other words, 2H (S ; Z) ⊆ En(S ). Together with Theorem 2.1, this implies that n n n En(S ) = 2H (S ; Z) if n is even and n 6= 2; 4; 8. Thus, in this case, the generator of Hn(Sn; Z) does not occur as an Euler class of some bundle over Sn. We begin by making some easy observations.