On fibrations with Grassmannian fibers

On fibrations with Grassmannian fibers Július Korbaš∗ To Peter D. Zvengrowski on his sixtieth birthday 1 Introduction Let F be either the field R of reals or the field C of complex numbers, and let F Gn,k be the Grassmann manifold of all k-dimensional vector subspaces in F n. In particular, RGn,1 = RP n−1 , real projective space of dimension n − 1, and CGn,1 = CP n−1 , complex projective space of complex dimension n− 1. Since one can identify F Gn,k with F Gn,n−k , we may suppose that 2k ≤ n in the sequel. There are situations, mainly in topology or geometry, where one needs to obtain cohomological information on the total space of a fibration (Hurewicz or Serre, a locally trivial fibration, or a fiber bundle) if the cohomological data of its fiber are given. Of course, in general, when the fiber is “wild enough”, one hardly can find anything reasonable about the cohomology of the total space. But in spite of that, there are certain results for some types of fibers. For instance, J. C. Becker and D. H. Gottlieb in [3, Corollary 9] proved that for any Hurewicz fibration p : E → B with fiber RP 2n (with E path connected, locally path connected, and semilocally 1-connected), the fiber inclusion i : RP 2n → E induces an epimorphism, i∗ : H ∗ (E; Z2 ) → H ∗ (RP 2n ; Z2 ), in Z2 -cohomology. In other words, here the fiber RP 2n is totally non-homologous to zero in E with respect to Z2 ; as is well known, then the Leray-Hirsch theorem applies, and one has that H ∗ (E; Z2 ) ∼ = H ∗ (RP 2n ; Z2 ) ⊗ H ∗ (B; Z2) as Z2 -vector spaces. For smooth fiber bundles, we proved the following more general result in [14]. The author was supported in part by Grants 1/4314/97 and 2/5133/98 of VEGA (Slovakia) Received by the editors October 1999. Communicated by Y. Félix. 1991 Mathematics Subject Classification : Primary: 55R20; Secondary: 57R19, 57R20. Key words and phrases : Fiber bundle; Hurewicz fibration; Serre fibration; Grassmann manifold; cohomology spectral sequence; orientable fibration; totally non-homologous to zero; derivation; Stiefel-Whitney class; Chern class; stable characteristic class; involution. ∗ Bull. Belg. Math. Soc. 8 (2001), 119–130 120 J. Korbaš Theorem A. ([14]). Let p : E → B be a smooth fiber bundle with E a closed connected manifold and with fiber the Grassmann manifold RGn,k (2 ≤ 2k ≤ n). If n is odd, then the fiber RGn,k is totally non-homologous to zero in E with respect to Z2 . The present paper is a kind of addendum to this theorem. I would like to thank Robert E. Stong for useful examples and comments. I am also grateful to Daniel H. Gottlieb, Shigeki Kikuchi, Parameswaran Sankaran, Fuichi Uchida and, last but not least, the referee for valuable comments. In particular, the latter brought to my attention the papers [4], [16], [17], [18] and proved Theorem B(2) for fibrations with fiber CG5,2 . 2 On fibrations with fiber CGn,k or with fiber RGn,2 Suppose that p : E → B is a locally trivial fibration (not necessarily smooth) with fiber F such that E and F are compact (we understand that this includes T2 ) and path connected, and B is locally path connected. It is known (see e.g. A. Borel [7]) that if the fundamental group π1 (B) acts trivially on H ∗ (F ; Z) (in other words, if the fibration p : E → B is orientable over Z), if H + (F ; Z) = {x ∈ H ∗ (F ; Z); deg(x) > 0} is generated by its elements of minimal positive degree, and if H ∗ (F ; Z) is torsionfree, then the fiber F is totally non-homologous to zero in E with respect to Z. In particular, this conclusion is true for locally trivial fibrations as above if the fiber is CP n . Indeed, recall that H ∗ (CP n ; Z) ∼ = Z[x]/(xn+1 ), where x = c1 (γ) ∈ H 2 (CP n ; Z) is the first Chern class of the canonical complex line bundle over CP n . More generally, one can consider Serre fibrations with fiber a homogeneous Kähler manifold (note that the complex flag manifolds U(n1 +· · ·+ns )/U(n1 )×· · ·× U(ns ), in particular the complex Grassmannians CGn,k = U(n)/U(k) × U(n − k), are homogeneous kählerian). Fibrations of this type (also with still more general fibers) were studied e.g. by A. Blanchard [4] (equivalently [5]), or by W. Meier [16], [17]: roughly speaking, the real or rational cohomology algebra of such a homogeneous kählerian fiber turns out to have no nontrivial derivations of negative degrees, therefore the Leray-Serre spectral sequence collapses at its E2 -term, and the fiber is totally non-homologous to zero with respect to R or Q if the fibration is orientable over R or Q, correspondingly. Similarly, H. Shiga and M. Tezuka in [21] study fibrations with special type of fibers for which they also are able to show that they have only trivial Q-derivations in negative degrees. More precisely, they study (Serre) fibrations p : E → B (under suitable hypotheses on E and B) with fiber G/U, where G is a compact connected Lie group and U is a closed connected subgroup having the same rank as G. The result is that if the fibration p is Q-orientable, then its fiber G/U is totally nonhomologous to zero in E with respect to the field Q. Another theorem of the same paper states that the result is also valid with coefficients Zq if q is a prime number not dividing the order of the Weyl group W (G). Note that here also one may in particular take CGn,k , or more generally U(n1 + · · · + ns )/U(n1 ) × · · · × U(ns ), in the rôle of G/U. Shiga and Tezuka remark at the end of their paper [21, p. 105] that the condition that the prime q should not divide the order of the Weyl group W (G) need not be On fibrations with Grassmannian fibers 121 best possible. Here is an indication (not mentioned by them): Gottlieb proved in [12, Corollary 7] that if p : E → B is a Zq -orientable Hurewicz fibration with fiber CP n such that n + 1 6≡ 0 (mod q), then the fiber CP n is totally non-homologous to zero in E with respect to Zq . Observe that the order of the Weyl group W (U(n)) is n! (see e.g. Husemoller [15]), hence the condition in the Shiga-Tezuka theorem [21, Theorem C] is more restrictive, namely (n + 1)! 6≡ 0 (mod q). We now show that for q = 2 and smooth fiber bundles, Gottlieb’s result can be strengthened to the first part of the following theorem; its second part will provide another indication that the above mentioned divisibility condition of Shiga and Tezuka might be too strong. Theorem B. (1) Let p : E → B be a smooth fiber bundle with E a closed connected manifold and with fiber the Grassmann manifold CGn,k (2 ≤ 2k ≤ n). If n is odd, then the fiber CGn,k is totally non-homologous to zero in E with respect to Z2. (2) Let p : E → B be a Serre fibration with E compact, with B a connected finite CW-complex, and with fiber either the complex Grassmannian CG2s+1,2 (s ≥ 2) or the real Grassmannian RG2s +1,2 (s ≥ 2). Then the fiber is totally non-homologous to zero in E with respect to Z2 . Remarks (a) Note that, as compared to the result of Gottlieb cited above, we do not have any orientability assumption in Theorem B(1). On the other hand, an obvious corollary is that every smooth fiber bundle considered in Theorem B(1) is Z2 -orientable if n is odd. But in fact, for q = 2, the Z2 -orientability assumption can be dropped from Gottlieb’s result, too. Indeed, since H 1 (CP n ; Z2 ) = 0 and H 2 (CP n ; Z2 ) has only one nonzero element, a, any degree preserving automorphism of the algebra H ∗ (CP n ; Z2 ) ∼ = Z2 [a]/(an+1) maps a to a, and is the identity. Therefore the action of the fundamental group of the base space on H ∗ (CP n ; Z2 ) is trivial. (b) The proof of Theorem B(1) will be geometrical in flavour, based on properties of tangent bundles and their characteristic classes. The proof of Theorem B(2) will be more cohomological in spirit, following the same line which was roughly outlined above when speaking about the papers by Blanchard and Meier. Note that the assumptions of Theorem B(2) can be relaxed. But if one wishes to state it under the weakest restrictions on E and B, then one should take care to distinguish whether one uses singular cohomology or cohomology with compact supports; in the latter case, B is necessarily compact (see Borel [6, §4(c)]). I gratefully acknowledge that Theorem B(2) with its proof is a modification and generalization of that what was suggested by the referee in case of fibrations with fiber CG5,2. After having seen an earlier version of Theorem B(2), P. Sankaran conjectured that the (originally present) assumption of Z2 -orientability of the fibration p : E → B might be deleted. This turned out to be true, as shown by Proposition 3. (c) Theorem B(1) has applications similar to those of Theorem A. For instance, when one is interested in the question of whether a given manifold can be the total space of a smooth fiber bundle with fiber RGn,k or CGn,k with n odd, then one can use Theorem A or Theorem B(1) for obtaining some necessary conditions (see [14] for examples of such conditions). Theorem B(2) can be applied in an analogous way. 122 J. Korbaš Note that for instance R. J. D. Ferdinands, W. Meier, and R. E. Schultz ([10], [11], [18], [20]) have studied a different but related question: When can a Grassmann manifold be the total space of a fibration? (d) In contrast to Theorem B(1), there are many smooth fiber bundles with fiber CGn,k (also with n odd) not totally non-homologous to zero with respect to Z or Zq , q 6= 2; more precisely, there are many such fiber bundles non-orientable over Z or Zq , q 6= 2. An example of this type will be presented in Remark (f) in Section 3. Hence, modifying Theorem B(1) by just changing the coefficients from Z2 to Z or Zq , q 6= 2, would not lead to a correct result. On the other hand, by the result of Shiga and Tezuka [21, Theorem C], the situation is different if we assume orientability: if the prime number q does not divide (n + 1)!, then for any Zq -orientable Serre fibration (in particular, Zq -orientable fiber bundle) with fiber CGn,k the fiber is totally non-homologous to zero with respect to Zq . As already remarked, the divisibility condition can probably be weakened. After some preparations, Theorem B(1) will be proved essentially along the same lines as Theorem A in [14]. 2.1 Preparations for the proof of Theorem B As is known (see e.g. Borel [6]), there is an isomorphism ϕ : H ∗ (RGn,k ; Z2) → H ∗ (CGn,k ; Z2 ) of the cohomology algebras which is doubling the degrees. More precisely, the algebra H ∗ (RGn,k ; Z2 ) is generated by the Stiefel-Whitney characteristic classes wi (ξk ) ∈ H i (RGn,k ; Z2 ) (i = 1, . . . , k) of the canonical k-plane bundle over RGn,k , the ring H ∗ (CGn,k ; Z) is generated (see e.g. A. Dold [9]) by the Chern classes ci (γk ) ∈ H 2i (CGn,k ; Z) (i = 1, . . . , k) of the canonical complex k-plane bundle over CGn,k , and the isomorphism ϕ maps wi (ξk ) ∈ H i (RGn,k ; Z2 ) to the mod 2 reduction of ci (γk ), hence to the Stiefel-Whitney class w2i (r(γk )) ∈ H 2i (CGn,k ; Z2 ). Here r(γk ) means the realification of γk . We shall just write w2i (γk ) instead of w2i (r(γk )) in the sequel. For the proof of Theorem B, we shall need the following. Lemma. If n is odd and 2k ≤ n, then the cohomology algebra H ∗ (CGn,k ; Z2 ) is generated by the Stiefel-Whitney classes w2i (CGn,k ) (i = 1, . . . , k) of the realification of the complex tangent bundle of CGn,k . Proof. We have 2k ≤ n so that (see Dold [9]) the Chern classes c1 (γk ), . . . , ck (γk ) in H ∗ (CGn,k ; Z) are algebraically independent in dimensions ≤ 2k. Similarly, the Stiefel-Whitney classes w1(ξk ), . . . , wk (ξk ) in H ∗ (RGn,k ; Z2 ) satisfy no algebraic relations in dimensions ≤ k, and then also their ϕ-images w2(γk ), . . . , w2k (γk ), are algebraically independent in dimensions ≤ 2k. Now let γk∗ be the conjugate bundle of the canonical vector bundle γk over CGn,k . Formally (as in A. Borel and F. Hirzebruch [8]) factorize the total Chern class c(γk ) = 1 + c1 (γk ) + · · · + ck (γk ) = k Y i=1 (1 + xi ), On fibrations with Grassmannian fibers 123 hence cj (γk ) is the j-th elementary symmetric function σj (x1 , . . . , xk ) in the variables x1, . . . , xk . Then c(γk∗) = k Y (1 − xi ), i=1 and by [8, p. 522] we have Y c(CGn,k ) · (1 − (xi − xj )2 ) = 1≤i<j≤k k Y (1 − xi )n . (∗) i=1 Therefore w(CGn,k ) · ( Y (1 + xi + xj ))2 = ( 1≤i<j≤k Since k Y k Y (1 + xi ))n (mod 2). i=1 (1 + xi ) = 1 + w2(γk ) + · · · + w2k (γk ) (mod 2), i=1 we obtain w(CGn,k ) · ( (1 + xi + xj ))2 = (1 + w2 (γk ) + · · · + w2k (γk ))n Y (mod 2). (∗∗) 1≤i<j≤k Now Q 1≤i<j≤k (1 + xi + xj ) 1+ (mod 2), being symmetric in x1, . . . , xk , is of the form X P2i (w2(γk ), . . . , w2k (γk )), i≥1 where P2i are Z2 -polynomials, P2i (w2(γk ), . . . , w2k (γk )) ∈ H 2i (CGn,k ; Z2 ). So (∗∗) implies w(CGn,k ) · (1 + X P2i (w2(γk ), . . . , w2k (γk )))2 = (1 + w2 (γk ) + · · · + w2k (γk ))n , i≥1 or, in other words, w(CGn,k ) = (1 + w2 (γk ) + · · · + w2k (γk ))n · (1 + X i P2i2 + X P2i4 + . . . ). i From this (recall that the Stiefel-Whitney classes w2 (γk ), . . . , w2k (γk ) satisfy no algebraic relations in dimensions ≤ 2k) then w2i(CGn,k ) = nw2i (γk ) + terms without w2i(γk ) for i = 1, . . . , k. Hence if n is odd, then w2i (CGn,k ) = w2i (γk ) + terms without w2i (γk ) for i = 1, . . . , k. Since the classes w2i (γk ) (i = 1, . . . , k) generate the algebra H ∗ (CGn,k ; Z2 ), an obvious induction (similar to that used in [14]) shows that the classes w2(CGn,k ), . . . , w2k (CGn,k ) also generate H ∗ (CGn,k ; Z2 ) if n is odd. This closes the proof of the lemma.  Having the lemma, we are now able to prove Theorem B(1). 124 J. Korbaš 2.2 Proof of Theorem B(1) Now, denoting by T M the tangent bundle of a manifold M, we have TE ∼ = p∗ (T B) ⊕ κ, where κ is the vector bundle along the fibers, and i∗(T E) ∼ = εdim(B) ⊕ T CGn,k , where T CGn,k is the realification of the complex tangent bundle of CGn,k . Here i : CGn,k → E is the fiber inclusion and εt is the trivial t-plane bundle. Hence i∗ (w2j (T E)) = w2j (CGn,k ) for j = 1, . . . , k. Since by the lemma w2(CGn,k ), . . . , w2k (CGn,k ) generate the algebra H ∗ (CGn,k ; Z2) if n is odd, one sees that i∗ : H ∗ (E; Z2 ) → H ∗ (CGn,k ; Z2 ) is an epimorphism in this case. This closes the proof of Theorem B(1). Once again we shall first derive some preparatory results, and then pass to the proof of Theorem B(2). 2.3 Preparations for the proof of Theorem B(2) As we already have recalled, the cohomology algebra H ∗ (RGn,2 ; Z2 ) is generated by the Stiefel-Whitney classes w1(ξ2 ), w2 (ξ2 ). But now we need the following detailed description (see e.g. H. Hiller [13, Theorem 1]). One has H ∗ (RGn,2 ; Z2 ) ∼ = Z2 [w1(ξ2 ), w2 (ξ2 )]/J (2, n − 2), where the ideal J (2, n − 2) is generated by the homogeneous relations f1,n−2 , f2,n−2 given by ! w1 (ξ2 ) 1 f1,n−2 = w2 (ξ2 ) 0 f2,n−2 !n−1 ! 1 . 0 Using this, we prove the following result on cup products. Proposition 1. In the cohomology algebra H ∗ (RG2s +1,2 ; Z2 ) (s ≥ 2) one has s+1 that w12 −5 (ξ2 )w2(ξ2 ) = 0 ∈ H d−1 (RG2s+1,2 ; Z2 ) ∼ = Z2 , where d = dim(RG2s+1,2 ) = s+1 2 − 2. Consequently, in the cohomology algebra H ∗ (CG2s +1,2; Z2 ) (s ≥ 2) one s+1 has that w22 −5 (γ2)w4 (γ2 ) = 0. Proof. In view of the isomorphism ϕ : H ∗ (RGn,k ; Z2 ) → H ∗ (CGn,k ; Z2 ), it is enough to verify the claim for H ∗ (RG2s +1,2; Z2 ). More precisely, we assert s that already w12 −1 (ξ2 )w2 (ξ2 ) vanishes (which then clearly implies the claim of the proposition since s ≥ 2). On fibrations with Grassmannian fibers 125 To prove this, observe that the relation f2,2s−1 in H ∗ (RG2s +1,2; Z2 ) is the element of the second row and the first column in the matrix !2 s w1(ξ2 ) 1 w2(ξ2 ) 0 . Then the following fact enables us to identify w12 f2,2s−1 in the algebra H ∗ (RG2s+1,2 ; Z2 ). Fact. s −1 (ξ2 )w2 (ξ2 ) with the relation Abbreviate a = w1 (ξ2 ), b = w2 (ξ2 ). Then for any positive integer s we have !2 s a 1 b 0 a2 + x(s) a2 −1 = s a2 −1 b x(s) s s ! for some element x(s) ∈ H ∗ (RG2s +1,2 ; Z2). This can readily be verified by induction on s, and the proof of Proposition 1 is complete.  Recall (see e.g. Meier [17]) that a (graded) derivation of degree s (s ∈ Z) in a commutative graded Z2 -algebra A is a linear map θ of degree s such that θ(a · b) = θ(a) · b + a · θ(b), where a, b are homogeneous elements of the algebra A. For any given n, let Der<n (A) denote the graded Z2 -vector space of all derivations in A of degree smaller than n. Recall further that the height of the Stiefel-Whitney class w1 (ξk ) in the algebra ∗ H (RGn,k ; Z2 ) (2k ≤ n) is height(w1 (ξk )) := sup{m; w1(ξk )m 6= 0}, and for 2s < n ≤ 2s+1 one has (see Stong [22])   n−1   if k = 1, height(w1 (ξk )) = 2 − 2 if k = 2 or if k = 3 and n = 2s + 1,    s+1 2 − 1 otherwise. s+1 For H ∗ (CGn,k ; Z2 ) (2k ≤ n), we then analogously define height(w2 (γk )), and due to the isomorphism ϕ : H ∗ (RGn,k ; Z2 ) → H ∗ (CGn,k ; Z2 ), the value of height(w2 (γk )) in H ∗ (CGn,k ; Z2 ) is the same as the above cited value of height(w1 (ξk )) in H ∗ (RGn,k ; Z2). Now we shall need the following. Proposition 2. For any s ≥ 2, one has Der<0 (H ∗ (CG2s +1,2 ; Z2)) = 0 and also Der<0 (H ∗ (RG2s +1,2; Z2 )) = 0. Proof. We know that the algebra H ∗ (CG2s +1,2 ; Z2 ) is generated by the StiefelWhitney classes w2 (γ2 ) and w4(γ2 ). Hence to show that a derivation is trivial, it is enough to verify that it vanishes on w2(γ2 ) and w4 (γ2 ). 126 J. Korbaš It is clear that the Z2 -cohomology of CGn,k vanishes in odd degrees. Therefore any of those elements in Der<0 (H ∗ (CG2s+1,2 ; Z2 )) which are nontrivially evaluated on w2 (γ2 ) must be of degree −2. Let us suppose that there exists a derivation θ s+1 with θ(w2(γ2 )) = 1 ∈ H 0 (CG2s +1,2; Z2 )) = Z2. Then of course w22 −1 (γ2 ) = 0, s+1 s+1 s+1 and therefore 0 = θ(w22 −1 (γ2 )) = (2s+1 − 1)w22 −2 (γ2 )θ(w2(γ2 )) = w22 −2 (γ2 ). s+1 But this is a contradiction, because w22 −2 (γ2 ) 6= 0 by the above mentioned height result. This means that each negative-degree derivation vanishes on the class w2 (γ2 ). Further, any of those elements in Der<0 (H ∗ (CG2s +1,2 ; Z2)) which are non-trivially evaluated on w4 (γ2 ) must be of degree −2 or of degree −4. Hence if θ is a non-vanishing derivation of degree −2, we must have θ(w4(γ2 )) = w2(γ2 ), because the cohomology group H 2 (CG2s +1,2; Z2 ) has just two elements, its generator being w2(γ2 ). Similarly, if θ̃ is a non-vanishing derivation of degree −4, then one has θ̃(w4(γ2 )) = 1 ∈ H 0 (CG2s +1,2; Z2 ). In either case we come to a contradiction. s+1 Indeed, we know by Proposition 1 that w22 −5 (γ2 )w4 (γ2 ) = 0. Hence, since s+1 s+1 both θ and θ̃ vanish on w2(γ2 ), we have 0 = θ(w22 −5 (γ2 )w4 (γ2 )) = w22 −4 (γ2 ), s+1 s+1 and 0 = θ̃(w22 −5 (γ2 )w4 (γ2 )) = w22 −5 (γ2 ). This contradicts the fact that the height of w2 (γ2) is 2s+1 − 2. For the real Grassmannians, the proof is an obvious modification of the proof given above for the complex case. The proof of Proposition 2 is complete.  Solving the Z2 -orientability question for certain fibrations, A. H. Back in [1] proved that if n is even and not congruent to 64 mod 192, then H ∗ (Gn+1,2 ; Z2) admits no nontrivial degree preserving automorphisms. To show that the fibrations considered in Theorem B(2) are Z2 -orientable, we shall prove the following. Proposition 3. Let p : E → B be a Serre fibration with B path connected and with one of the following spaces as fiber: (i) RGn,2 (n ≥ 5); (ii) CGn,2 (n ≥ 5). Then the fibration is Z2 -orientable. Proof. Those automorphisms of H ∗ (RGn,2 ; Z2 ) or of H ∗ (CGn,2 ; Z2 ) which correspond to the action of the fundamental group of the base space are induced by continuous maps. Hence all such automorphisms must commute with the Steenrod squares. To prove the claim, it is then enough to prove that the only degree preserving Z2 -algebra automorphism of H ∗ (RGn,2 ; Z2) or of H ∗ (CGn,2 ; Z2) (n ≥ 5 in both cases) that commutes with the Steenrod squares is the identity. Consider the real case. Suppose that f ∗ is an automorphism of H ∗ (RGn,2 ; Z2) (n ≥ 5) commuting with the Steenrod squares. Since in H 1 (RGn,2 ; Z2 ) the only nonzero element is w1 (ξ2 ), we must have f ∗ (w1 (ξ2 )) = w1(ξ2 ). Now H 2 (RGn,2 ; Z2) has three nonzero elements: w12 (ξ2 ), w2 (ξ2 ), and w12 (ξ2 ) + w2 (ξ2 ). We show that f ∗ (w2(ξ2 )) = w2(ξ2 ), which then yields the result. Indeed, first suppose that f ∗ (w2(ξ2 )) = w12 (ξ2 ). Then, using the Wu formula, we obtain that Sq 1(f ∗ (w2 (ξ2 ))) = 0 = f ∗ (Sq1 (w2(ξ2 ))) = f ∗ (w1(ξ2 )w2 (ξ2 )) = w13 (ξ2 ). On fibrations with Grassmannian fibers 127 But this is a contradiction: note that the classes w1 (ξ2 ), w2(ξ2 ) satisfy no algebraic relations in dimension 3 since n − 2 ≥ 3. Similarly, supposing that f ∗ (w2 (ξ2 )) = w12 (ξ2 ) + w2(ξ2 ), we derive that Sq 1(f ∗ (w2(ξ2 ))) = w1(ξ2 )w2 (ξ2 ) = f ∗ (Sq 1(w2 (ξ2 ))) = w13 (ξ2 ) + w1 (ξ2 )w2(ξ2 ), hence again w13 (ξ2 ) = 0, which is a contradiction. In the complex case, one proceeds analogously, replacing w1(ξ2 ) with w2 (γ2 ), w2(ξ2 ) with w4(γ2 ), and using the Steenrod square Sq2 instead of Sq1 . This completes the proof of Proposition 3.  2.4 Proof of Theorem B(2) It is known (it is enough to change Q to Z2 in Meier [16, Lemma 2.5], or cf. Shiga and Tezuka [21, Lemma 6.1]) that if p : E → B is a Z2 -orientable Serre fibration with fiber F (under suitable hypotheses on the spaces E, B, F , which are all satisfied in our case) such that Der<0 (H ∗ (F ; Z2)) = 0, then the Leray-Serre spectral sequence collapses, and the fiber is therefore totally non-homologous to zero with respect to Z2 . In view of this, Theorem B(2) now follows from Propositions 2 and 3. 3 Comments and observations In [14, Remark 3] we showed, using an example due to Robert E. Stong, that Theorem A cannot be extended to cover also all smooth fiber bundles having fiber RGn,k with n even and k odd ((n, k) 6= (2, 1)). Here we present another example, which we also owe to Stong, showing that Theorem A cannot be extended to cover all smooth fiber bundles with fiber RGn,k with n and k both even. Example Let k ≥ 2 and T : RG2k,k → RG2k,k be the involution sending D ∈ RG2k,k to its orthogonal complement in R2k . Let a : S m → S m , a(x) = −x, be the antipodal involution on the m-dimensional sphere; we shall suppose that m ≥ 1. Then the map RG2k,k × S m Sm p: −→ (= RP m ), T ×a a p([D, x]) = [x], defines a smooth fiber bundle with fiber RG2k,k . About this one can prove that the fiber RG2k,k is not totally non-homologous to zero in E with respect to Z2 . This can be done for instance in the following way. For any s ∈ S m , the map is : RG2k,k → RG2k,k × S m , T ×a is (D) = [D, s], is the inclusion of the fiber over [s] ∈ RP m . Now for any D ∈ RG2k,k one has is ◦ T (D) = [T (D), s] = [D, a(s)] = ia(s)(D), 128 J. Korbaš hence is ◦ T = ia(s). Since the sphere S m is path connected (and therefore is and ia(s) are homotopic), one has i∗s = i∗a(s) := i∗ for the induced homomorphisms in cohomology. Hence T ∗ ◦ i∗ = i∗ which means that the image of i∗ : H ∗(RG2k,k × S m /T × a; Z2) −→ H ∗ (RG2k,k ; Z2 ) must be contained in the set of elements which are invariant under T ∗ . On the other hand, the pullback T ∗(ξ) of the canonical k-plane bundle over RG2k,k is the complementary k-plane bundle ξ ⊥ . Since ξ ⊕ ξ ⊥ is the trivial 2k-plane bundle, one has w(ξ)w(ξ ⊥ ) = 1 for the total Stiefel-Whitney classes, and then T ∗(w2(ξ)) = w2(ξ ⊥ ) = w12 (ξ) + w2(ξ) 6= w2 (ξ). The latter is true, because k ≥ 2, and there are no relations among the classes w1(ξ), . . . , wk (ξ) in the Z2 -cohomology of RG2k,k up to dimension k. Hence w2 (ξ) ∈ / Im(i∗) which means that i∗ is not surjective. Therefore the fiber RG2k,k is not totally non-homologous to zero in the total space RG2k,k × S m /T × a as claimed. Gathering the present knowledge, the following conjecture seems plausible. Conjecture. Let p : E → B be a smooth fiber bundle with E a closed connected manifold and with fiber the Grassmann manifold RGn,k . If the fiber is not bordant to 0 (in other words (see [19] or [2]), if each power of 2 dividing n also divides k), then it is totally non-homologous to zero in E with respect to Z2 . Remarks (e) If the conjecture is valid, then the Leray-Hirsch theorem applies. As a consequence, a necessary condition for the validity of the conjecture is the following: p∗ : H ∗ (B; Z2) → H ∗ (E; Z2 ) is a monomorphism in case that each power of 2 dividing n also divides k. By [14, Theorem (a)], this necessary condition is fulfilled. (f) Let T : CGn,k → CGn,k be the involution induced by complex conjugation. Then one has T ∗(c1 (γk )) = −c1(γk ). So reasoning analogous to that in the above example shows that the fiber CGn,k of the fiber bundle CGn,k × S m −→ RP m T ×a (with m ≥ 1) is not totally non-homologous to zero with respect to Z or Zq , q 6= 2. This is the example promised in Remark (d) after Theorem B. (g) It seems natural to ask whether Theorem A or Theorem B(1) remains valid (without adding orientability or other assumptions) when one passes from smooth fiber bundles to continuous locally trivial fibrations or even to Hurewicz or Serre fibrations. Theorem B(2) provides a partial (positive) answer to this question. (h) Observe that the first part of the proof of Proposition 2, which shows that θ(w2(γ2 )) = 0, requires only that the smallest r for which w2 r (γ2 ) = 0 is odd. Therefore one can prove analogously that Der<0 (H ∗ (CP 2k ; Z2 )) = 0, and so one obtains another proof of the above cited Gottlieb’s result ([12, Corollary 7]) for q = 2. On fibrations with Grassmannian fibers 129 We also know (see the result on the height) that the smallest r for which w2 (γ3 ) = 0 is odd for CG2s+1,3 (s ≥ 3). But in this case I do not know if all odd-degree derivations vanish on w4 (γ3 ) and on w6(γ3 ). An inspection of Blanchard’s argument (see Meier [17, p. 474]) evokes a feeling that a “source of difficulties” in case of Z2 -coefficients, as compared to the situation with real or rational coefficients, is that for the manifolds CGn,k (4 ≤ 2k ≤ n) with (n, k) 6= (2s + 1, 2) there is no Z2 -version of the hard Lefschetz theorem. To see the latter, it is enough to look at the above cited height result for the generator w2(γ2 ) ∈ H ∗ (CGn,k ; Z2 ). (i) For a smooth closed connected manifold M and a stable characteristic class ψ (in the sense of Husemoller [15, Chap. 20, Sec. 2]), put ψ(M) := ψ(T M) ∈ H ∗ (M; R), where TM denotes the tangent bundle of M and R denotes a fixed coefficient ring. Using the same methods as in the proofs of Theorems A and B(1), it is clear that the following generalization can be proved. r Theorem C. Let F be a smooth closed connected manifold such that the characteristic classes ψj (F ) generate the cohomology ring H ∗ (F ; R). If p : E → B is a smooth fiber bundle with E a closed connected manifold and with fiber F , then the fiber inclusion i : F → E induces an epimorphism, i∗ : H ∗ (E; R) → H ∗ (F ; R), in cohomology with coefficients in the ring R. Bibliography [1] A. Back, Involutions on Grassmann manifolds, Thesis (Univ. of California), Berkeley, 1977; [2] V. Bartı́k, J. Korbaš, Stationary point free elementary abelian 2-group actions on Grassmannians – an elementary approach, Arch. Math. 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