Let us first assume k≥1, the case k=0 being postponed to the end of the proof.
We construct P is in several steps. Letα≥k/4+2 be a natural number, let either β= α+1 orβ=α+2 and n=α+β. We consider the trivial fibration
φ0: P0=Sk×S4α−k×S4β→Sk
with (path connected) total space of dimension 4n. By choosingα appropriately we can assume in addition that the fiber S4α−k×S4β ofφ0is l-connected.
We first apply the following result of surgery theory, in which τ: P0→Bπ1(P0) denotes the classifying map of the universal covering.
Theorem5.1 ([7, Theorem 6.5]). — LetL∈
j>1H4j(P0;Q)be a class such that τ∗
L∩ [P0]
=0∈H4n−4∗
Bπ1(P0);Q .
Then there is some non-zero integer R and a homotopy equivalence f : P1→P0 of closed smooth manifolds such that the L-polynomials of P0and P1satisfy the equation
L(P1)=f∗
L(P0)+RL .
Sketch of proof. — As we need a slight modification of the argument later, we sketch the proof. For more details, proofs or references concerning some of the statements below we refer the reader to [7].
At first we identify the set of topological normal invariants on P0with[P0,G/Top], which we equip with the Abelian group structure so that the surgery obstruction is a group homomorphism. There is an isomorphism
We need to compute the surgery obstruction for this normal invariant f . The surgery obstruction the Bass-Heller-Swan splitting theorem, so that we can in fact choose R2=1.
Finally it follows from [24] that for each normal invariant f ∈ [P0,G/Top]some multiple of f lies in the image of the canonical map [P0,G/O] → [P0,G/Top]. Hence we find a non-zero multiple R3 of R2R1 and a smooth normal invariant f ∈ [P0,G/O] which satisfies(f)=R3LL(P0)−1and whose surgery obstruction vanishes. Performing surgery along this normal invariant yields a homotopy equivalence f :P1→P0 so that
the stated equation holds with R=8R3.
We apply this result to the following situation: Let ek∈Hk
of smooth closed manifolds so that all homogeneous components of the Hirzebruch
We will show that a manifold with the above properties automatically has non-zeroA-genus. For jˆ ≥1 we denote byλLj ∈Q the coefficient of the degree 4j Pontryagin
where Bj denotes the j-th Bernoulli number, defined by the relation (cf. [19, p. 281]) x
Proof. — According to [19, Problem 19-C] we have equations L(t)d(t/L(t))
This gives (setting x=12√
t in the relation defining the Bernoulli numbers) A(t)ˆ d(t/Aˆ(t))
dt =1+1
2· ∞
j=1
(−1)j−1 Bj
(2j)!tj.
This implies the second equation. The first equation appears explicitly in [19,
Prob-lem 19-C] or can be obtained by similar methods.
This lemma implies that all λLj andλAj are different from zero. By our choice of β=α+1 orβ=α+2, it follows recursively that all the Pontryagin classes of P1vanish except possibly pα, pβ, and pn, and that pα and pβ as well as pα·pβ are non-zero.
Let us write the degree 4n-components of the L- and theA-polynomial of an arbi-ˆ trary vector bundle with vanishing pj for j=α, β,n as
Ln(pα,pβ,pn)=λLn ·pn+μL·pαpβ
Aˆn(pα,pβ,pn)=λAn ·pn+μA·pαpβ
with rational numbersμL, μA. Proposition5.3. — We have
μL λLn =μA
λAn
and hence the following implication for a 4n-dimensional connected oriented manifold M with pj(M)=0 for j=α, β,n:
If pα(M)·pβ(M)=0 and L(M)=0, thenA(M)ˆ =0.
Proof. — This is a calculation in universal characteristic classes. To keep the no-tation transparent, assume that Eα,Eβ are vector bundles with total Pontryagin classes 1+pα and 1+pβ, in particular Ln(Eα)=0=Ln(Eβ). By the multiplicativity of the total Pontryagin class and the universal L- andA-polynomials we obtainˆ
pn(Eα⊕Eβ)=pα·pβ, pα(Eα⊕Eβ)=pα, pβ(Eα⊕Eβ)=pβ. This implies
λLn +μL
·(pα·pβ)=Ln(Eα⊕Eβ)
=Ln(Eα)+Ln(Eβ)+Lα(Eα)·Lβ(Eβ)
=λLα·λLβ·(pα·pβ),
and hence
λLn +μL=λLαλLβ. An analogous computation shows
λAn +μA=λAαλAβ,
so that altogether we obtain 1+μL
λLn =λLαλLβ
λLn , 1+μA
λAn =λAαλAβ λAn .
From Lemma5.2it follows that 1+μL
λLn = −2(22α−1−1)(22β−1−1) 22n−1−1
λAαλAβ λAn
= −2(22α−1−1)(22β−1−1) 22n−1−1
μA λAn +1
.
Because−2(22α−12−2n−11)(2−2β−11 −1) =1 the conclusion follows.
If k=1 then the map φ0◦f : P1→S1
is homotopic to the projection map of a smooth fiber bundle. This follows from Far-rell’s obstruction theory over the circle [8, Theorem 6.4]. Note that in the case at hand φ0◦f induces an isomorphism of fundamental groups, so that the kernel of the π1 -ho-momorphism is the trivial group. By [8, Remarks on p. 316], the fibering obstructions vanish and one only has to check that the universal covering of P1is homotopy equiva-lent to a finite CW-complex. But it is homotopy equivaequiva-lent to the universal covering of P0
and therefore has this property.
If k≥2 the mapφ0◦f will usually not be homotopic to a fiber bundle projection and a more complicated construction is needed. The theory which is relevant for the following discussion was developed by Casson [4] and Hatcher [12].
Recall that for any continuous map f : X→Y of topological spaces, the homotopy fiber L of f is the fiber of the map
Ef →Y (x, γ ) →γ (1) where
Ef =
(x, γ )|x∈X, γ: [0,1] →Y, γ (0)=f(x) .
Let b∈Sm be the north pole, viewed as base point in Sm. Let V be a closed smooth manifold V equipped with a smooth map f : V→Sm such that the homotopy fiber is simply-connected. Casson [4, Section 1] defines such a map to be a pre-fibration if the point b∈Sm is a regular value, V\f−1(b)is simply-connected and the canonical inclusion
f−1(b)→L, v →
v, γ:t →f(v)
of the point-preimage into the homotopy fiber of f is a weak homotopy equivalence. In this case the smooth manifold F=f−1(b)is called the fiber of(V,f).
Proposition 5.4. — Let R=0 be the number appearing in the construction of f :P1→P0
after the proof of Theorem5.1. If we construct the homotopy equivalence f :P1→P0using the number 2R instead of R, then the map φ0◦f : P1→Sk constructed above is homotopic to a pre-fibration φ1: P1→Sk.
Proof. — After applying a homotopy to φ0◦f we obtain a map g for which the value b∈S4k is regular with some fiber F. By [4, Lemma 4] we can assume that F and P1\F are simply connected. By [4, Lemma 2 and the proof of Theorem 1] the inclusion from F into the homotopy fiber of g is a degree one normal map. From [4, Theorem 1 and p. 497] we know that the obstruction for g to be homotopic to a pre-fibration is given by the (simply-connected) surgery obstruction of this degree one normal map. The surgery obstruction groups in odd dimensions being trivial, we have to distinguish between the cases where k is divisible by 4 and where k is congruent 2 modulo 4.
In the case where k is divisible by 4, the surgery obstruction is an integer which (up to a multiple) is given by the difference of signatures. The homotopy fiber L of g is homotopy equivalent to S4α−k×S4β, so that its signature is zero. On the other hand, by Hirzebruch’s signature formula we obtain the signature of F (which is framed in P1) by
σ (F)=
L(P1),[F] ,
which is zero since by our choices ofα andβthere is no L-class in the relevant degree.
In the case where k is congruent 2 modulo 4, the surgery obstruction group is cyclic of order 2. In the notation of the proof of Theorem5.1, this surgery obstruction is given by the composite group homomorphism
[P0,G/Top] → S4α−k×S4β,G/Top
→L4n−k(Z)∼=Z/2
where the first map is given by restriction. It follows that this obstruction becomes zero if the number R appearing in Theorem5.1is multiplied by 2.
In general the mapφ1is not homotopic to an actual (smooth) fiber bundle: By the theory developed in [4] there is an obstruction lying in a homotopy group of a certain concordance space. Our goal is to show that (P1, φ1)can be replaced by some (P2, φ2) which fibers over Sk.
Casson [4, Section 4] calls two pre-fibrations f : V→Sm and f: V→Sm equiva-lent if there is a diffeomorphism V→Vcompatible with f and f in a neighborhood of f−1(b). This implies that the fibers of f and f are diffeomorphic. Let F denote this com-mon fiber. Following Casson, equivalence classes of pre-fibrations are classified as follows:
Let A(F)˜ be the simplicial group of block diffeomorphisms of F. Recall from [12, p. 5]
that the k-simplices inA(F)˜ are given by diffeomorphisms of F×k restricting to diffeo-morphisms of F×τ for each face ofk. An element inπk(A˜(F),id)is represented by a self-diffeomorphism of F×Dk which is the identity in a neighborhood of F×Sk−1, and two elements agree if and only if they are concordant by a concordance which keeps a neighborhood of F×Sk−1 fixed. Thinking of Dk as the northern hemisphere Dk+⊂Sk, such an f extends by the identity map to a self-diffeomorphism f of F×Sk. We may use fto glue two copies of F×Dk+1along their common boundaries to obtain a closed man-ifold V which comes with a canonical projection to Sk+1, collapsing F×Dk+ to a point, which is easily seen to be a pre-fibration which we call V(x).
In the following we will suppress the obvious base point id in the notation of ho-motopy groups.
Proposition5.5. — Let F be simply connected and of dimension at least 6, and let k≥2. Then the rule x →V(x)defines a one-to-one correspondence betweenπk−1(A(F))˜ and equivalence classes of pre-fibrations over Sk with fiber F.
Proof. — This is [4, Lemma 6], once one has identifiedπk(A(F))˜ with what Casson calls Dk(F). The latter group is given by the diffeomorphisms of F×Sk keeping a neigh-borhood of F×Dk− pointwise fixed, modulo concordance keeping F×Dk− pointwise fixed. (Here Dk− is the lower hemisphere.) There is a canonical map Dk(F)→πk(A(F))˜ given by restriction of a diffeomorphism to F×Dk+. Its inverse is given by extending a
diffeomorphism of F×Dk+by the identity.
For a pre-fibration f : V→Sk with fiber F, which we assume to be simply con-nected and of dimension at least 6, we denote by h(V,f)∈πk−1(A(F))˜ the corresponding element. We call this the characteristic element of the pre-fibration. We now formulate a condition when a pre-fibration is equivalent to a fiber bundle.
Let A(F)be the simplicial group of diffeomorphisms of F, where the k-simplices are given by diffeomorphisms of F×k which are compatible with the projection tok. Note that A(F)is a simplicial subgroup ofA(F), which is in general not normal.˜
If x∈πk(A˜(F))lies in the image of πk(A(F)), then V(x)is obtained (up to equiv-alence) by gluing two copies of V×Dk along a diffeomorphism V×Sk−1 which acts fiberwise over Sk−1. It follows that the projection V(x)→Sk is a smooth fiber bundle.
Using the exact sequence πk
A(F)
→πkA(F)˜ ψ
−
→πkA(F),˜ A(F) we obtain:
Proposition5.6. — Let f : V→Sk be a pre-fibration with simply connected fiber F of dimen-sion at least 6 with characteristic element h(V,f)∈πk−1(A(F)). If˜ ψ (h)=0∈πk−1(A(F),˜ A(F)), then f : V→Sk is (as a pre-fibration) equivalent to a smooth fiber bundle.
For a closed smooth manifold K we consider the pre-fibration φ1◦pr1: P1×K→Sk (p,x) →φ1(p)
with fiber F×K. Thenψ (h(φ1◦pr1))is the image ofψ (h(φ1))under the map of homo-topy groups induced by the map
A(F),A(F)
→A(F×K),A(F×K)
which sends a diffeomorphismω: F×k→F×ktoω×id: F×k×K→F×k×K.
Proposition5.7. — For a closed smooth manifold K with vanishing Euler characteristicχ (K) and r>0, the induced map
πrA(F),˜ A(F)
→πrA˜
F×Kr ,A
F×Kr is equal to 0. Here Kr denotes the r-fold Cartesian product of K with itself.
Proof. — We consider the Hatcher spectral sequence [12, Proposition 2.1]
E1pq=πq C
F×Ip
=⇒πp+q+1A(F),A(F) .
By [12, Proposition in Appendix I on p. 18], multiplication with K induces the zero map πj
C
F×Ip
→πj C
F×Ip×K
for each j, becauseχ (K)=0. It follows that in the filtration ofπr(A(F),˜ A(F)) induced by the E∞-term of the Hatcher spectral sequence the map induced by the product with K reduces the filtration degree of each element by one. This implies the assertion of
Proposition5.7.
Thus let K be a closed max(l,1)-connected smooth manifold withAˆ(K)=0 and χ (K)=0. (For instance, we may use the construction at the beginning of this section for appropriate values of k,αandβ.) Applying Lemma5.7to our previous construction we conclude that the pre-fibration
φ1◦pr1: P1×Kk−1→Sk
is equivalent to a smooth fiber bundle φ2: P2→Sk. Because A(K)ˆ =0 we still have A(Pˆ 2)=0.
It remains to prove that we can assume that we have a smooth section s: Sk→P2
with trivial normal bundle. First notice that by choosingα and l large enough, there is a smooth section s: Sk →P2, of which we would like to show that its normal bundle is trivial.
Again the case k =1 is the simplest one. In fact a real bundle over the circle is trivial if and only if its first Stiefel-Whitney class vanishes, and the first Stiefel-Whitney class of the normal bundle agrees withw1(P0)=0.
In the case where k ≥2 we argue as follows. Since we are in the stable range, the normal bundle is classified by an element inπk(BO). If k≡3,5,6,7 modulo 8, the group πk(BO) is zero, so that the normal bundle is automatically trivial. In the cases where k=4l is divisible by 4, we haveπk(BO)=Z and non-trivial bundles over Sk may be detected by their l-th rational Pontryagin classes. The l-th rational Pontryagin class of the normal bundle is equal to s∗(pl(P2))∈H4l(S4l;Q). But by the above construction of P2this class is equal to 0, so that the normal bundle is trivial in this case, too.
The remaining cases are k≡1 or 2 modulo 8, in which caseπk(BO)=Z/2. We do not see a general reason why the normal bundle should be trivial in this case, but we describe a procedure how to change the pre-fibration φ1: P1→Sk so that the normal bundle of the embedded Skin P1becomes trivial. Since P2is diffeomorphic to P1×Kk−1, this will imply that the normal bundle of the embedding Sk →P2is also trivial.
Recall that by Casson’s classification result, if dim F≥ 6 and F is simply con-nected any pre-fibration over Sk with fiber F is equivalent to one of the form V(x), with x∈πk−1(A(F)). If F is k-connected and at least˜ (k+2)-dimensional, then there is an em-bedding Sk →V(x)such that the composite map Sk→V(x)→Sk is homotopic to the identity, and any two such embeddings are isotopic. We call such an embedding simply the embedding of Sk into V(x) and denote by νx: Sk →BO the classifying map of its normal bundle.
Lemma5.8. — Suppose that F is k-connected and at least(k+2)-dimensional (where k≥2), and let x,y∈πk−1(A(F)). Then the normal bundle of the embedding of S˜ k into V(x+y)is classified V(x+y)can be written as a “fibered connected sum”
V(x+y)∼=
We use this identification to describe the embedding of Sk into V(x+y). To do that, let us first describe the embedding ixof Sk into V(x). Choose a base point∗ ∈F. On
the lower hemisphere Dk− the embedding ix is given by the inclusion i−: Dk−→Dk−×F at the base point. The composite
Sk−1 i−→− Sk−1×F−→α Sk×F
extends continuously to a map Dk+→Dk+×F (coning off and using a null homotopy of the map prF◦α◦i−: Sk→F). The extension may be approximated by a smooth embedding i+in such a way that i+and i−together define the embedding of Sk into V(x).
Similarly one obtains a choice of the embedding iy of Sk into V(y) which is the inclusion at the base point on the upper hemisphere. Denoting the embedding at the base point by j:Sk−1× [0,1] →F×Sk−1× [0,1], it follows that
ix|Dk+∪j∪iy|Dk−: Dk+∪
Sk−1× [0,1]
∪Dk−
→
F×Dk+
∪α
F×Sk−1× [0,1]
∪β
F×Dk− defines the embedding of Sk into V(x+y).
The normal bundle of this embedding is given byνx|Dk+ on Dk+and byνy|Dk−on Dk−, while on Sk−1× [0,1]the normal bundle is trivialized. It follows that the classifying map νx+yfactors as
Sk→Sk∨Sk−−→νx∨νy BO
where the first map is the pinch map. But this composite defines the sumνx+νy in the homotopy groupπk(BO)so that the first statement follows.
Finally notice that V(x+y)is obtained from V(x)V(y)by cutting out two copies of F×Dk and gluing along the resulting F×Sk−1 boundaries, defining a parametrized connected sum. Hence there is a standard bordism between these V(x+y)and the dis-joint union V(x)∪V(y). This implies the second statement.
Suppose now that in our situation, the normal bundle of Sk in P1=V(x)is non-trivial. Asπk(BO)has order 2, replacing P1by P1=V(2x)yields a pre-fibration with the same fiber F, whose section has trivial normal bundle and such that 2A(Pˆ 1)= ˆA(P1)=0.
This completes the proof of Theorem1.4in the case k≥1.
If k=0 we can use the same construction as for case k ≥1, but starting with P0=S4α×S4β. With Theorem5.1, Lemma5.2and Proposition5.3we find an N(0,l)so that for all n≥N(0,l)there is a 4n-dimensional l-connected closed spin manifold F with Aˆ(F)=0. Now we define E as the disjoint union of two copies of the spin manifold F.
The surjective map to S0is constant on each component.
Acknowledgements
We thank Boris Botvinnik and Peter Landweber for helpful comments on an earlier version of this paper and the anonymous referees for carefully reading our manuscript.
Their numerous thoughtful remarks helped us to improve the exposition and to eliminate a blunder in the proof of Proposition1.9(b).
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B. H.
Institut für Mathematik, Universität Augsburg, 86135 Augsburg, Germany hanke@math.uni-augsburg.de
T. S.
Mathematisches Institut,
Georg-August-Universität Göttingen, 37073 Göttingen, Germany
schick@uni-math.gwdg.de W. S.
Mathematisches Institut, Universität Bonn, 53115 Bonn, Germany steimle@math.uni-bonn.de
Manuscrit reçu le 4 décembre 2012 Manuscrit accepté le 5 février 2014 publié en ligne le 1 mars 2014.