An H P 2 -bundle over S 4 with nontrivial A ^ -genus
aa r X i v : . [ m a t h . A T ] J u l AN HP -BUNDLE OVER S WITH NONTRIVIAL ˆ A-GENUS
MANUEL KRANNICH, ALEXANDER KUPERS, AND OSCAR RANDAL-WILLIAMSAbstract. We explain the existence of a smooth H P -bundle over S whose total spacehas nontrivial ˆ A -genus. Combined with an argument going back to Hitchin, this answersa question of Schick and implies that the space of Riemannian metrics of positive sectionalcurvature on a closed manifold can have nontrivial higher rational homotopy groups. In view of applications to spaces of Riemannian metrics with positive curvature, therehas been recent interest in constructing smooth fibre bundles over spheres whose totalspace has nontrivial ˆ A -genus. In their work on the space of positive scalar curvaturemetrics, Hanke–Schick–Steimle [HSS14, Corollary 1.6] showed that such bundles existfor every dimension of the base sphere. However, as noted on page 337 loc. cit., theirmethod does not yield bundles with an explicit description of the fibre, though they areable to show that the fibre may be chosen to carry a metric of positive scalar curvatureusing a theorem of Stolz. For applications to spaces of metrics with positive sectionalor Ricci curvature it is desirable to have examples with fibre carrying such a metric. In[BEW20, p. 3999] (see also [Sch14, Section 9]) it is said that this “seems to be a very difficultproblem”: we offer the following solution. Theorem.
There exists a smooth oriented fibre bundle H P → E → S with ˆ A ( E ) , .Remark. The argument we give also shows that this fibre bundle may be assumed to havea section with trivial normal bundle (see Remark 2), and provides analogous H P n -bundlesover S for all even n ≥ (see Remark 3). It can certainly be extended further.The standard Riemannian metric д st on H P has positive sectional curvature, so pullingback д st along orientation-preserving diffeomorphisms yields a map (−) ∗ д st : Diff ( H P ) −→ R sec > ( H P ) ⊂ R Ric > ( H P ) ⊂ R scal > ( H P ) from the group of diffeomorphisms of H P in the smooth topology to the spaces of Rie-mannian metrics on H P having positive sectional, Ricci, or scalar curvature. By an argu-ment of Hitchin [Hit74] (see for instance [BEW20, p. 3999] for an explanation of this), thetheorem has the following corollary, which answers a question of Schick [OWL17, p. 30]and provides an example as asked for in [BEW20, Remark 2.2]. Corollary.
The induced map π ((−) ∗ д st ) ⊗ Q : π ( Diff ( H P ) ; id ) ⊗ Q −→ π (R scal > ( H P ) ; д st ) ⊗ Q is nontrivial, so in particular π (R sec > ( H P ) ; д st ) ⊗ Q , and π (R Ric > ( H P ) ; д st ) ⊗ Q , . Proof of the Theorem.
Smooth H P -bundles over S (together with an identificationof the fibre over the basepoint with H P ) are classified by π ( BDiff ( H P )) , so our taskis to show that the morphism ˆ A : π ( BDiff ( H P )) → Q assigning an H P -bundle E → S the ˆ A -genus of the total space is nontrivial. This morphism factors over the map π ( BDiff ( H P )) → π ( B g Diff ( H P )) induced by the canonical comparison map Diff ( H P ) → g Diff ( H P ) to the block diffeomorphism group of H P . This follows for instance from[ERW14, Theorem 1], but there is also a more direct argument: via the canonical iso-morphism π ( BDiff ( H P )) (cid:27) π ( Diff ( H P ) ; id ) , the morphism ˆ A : π ( BDiff ( H P ) → Q isgiven by mapping a diffeomorphism ϕ : D × H P → D × H P that is the identity on the boundary and commutes with the projection to S to the ˆ A -genus of the glued manifold D × H P ∪ ϕ ∪ id D × H P . This description of the morphism makes clear that it factorsthrough the map π ( Diff ( H P ) ; id ) → π ( Diff ∂ ( D × H P )) (cid:27) π ( g Diff ( H P ) ; id ) that onlyremembers the underlying isotopy class of ϕ .We thus have to show nontriviality of the composition π ( BDiff ( H P )) −→ π ( B g Diff ( H P )) ˆ A −→ Q . It suffices to check this after rationalisation, which makes the first map surjective:
Lemma.
The map π ( BDiff ( H P )) ⊗ Q −→ π ( B g Diff ( H P )) ⊗ Q surjective.Proof. Choosing an embedded disc D ⊂ H P , we consider the commutative square BDiff ∂ ( D ) BDiff ( H P ) B g Diff ∂ ( D ) B g Diff ( H P ) whose horizontal maps are induced by extending (block) diffeomorphisms of D that arethe identity on the boundary to H P by the identity. The claim follows by showing that thethird rational homotopy group of the right vertical homotopy fibre vanishes for which wenote that, since H P is -connected, the square is -cartesian by Morlet’s lemma of disjunc-tion [BLR75, Corollary 3.2, p. 29], so it suffices to show that the third rational homotopygroup of the left vertical map is trivial. Since π i ( B g Diff ∂ ( D n )) (cid:27) π Diff ∂ ( D n + i − ) (cid:27) Θ n + i vanishes rationally as the group Θ n + i of homotopy ( n + i ) -spheres is finite, the claimfollows from π ( BDiff ∂ ( D )) ⊗ Q = which holds by [RW17, Theorem 4.1]. (cid:3) Given the lemma, we are left to show that the map ˆ A : π ( B g Diff ( H P )) → Q is nontriv-ial, which we shall do after precomposition with the map π ( hA ut ( H P )/ g Diff ( H P ) ; id ) −→ π ( B g Diff ( H P )) induced by the inclusion of the homotopy fibre of the comparison map B g Diff ( H P ) → BhAut ( H P ) , where hAut ( H P ) is the topological monoid of homotopy automorphismsof H P . Considering this homotopy fibre is advantageous since the h -cobordism theoremprovides an isomorphism π ( hAut ( H P )/ g Diff ( H P )) (cid:27) S ∂ ( D × H P ) to the structure group of D × H P relative to ∂ D × H P in the sense of surgery theory(see [Wal99] for background on surgery theory, especially Chapter 10), which in turn fitsinto the surgery exact sequence of abelian groups0 = L ( Z ) ∂ −→ S ∂ ( D × H P ) η −→ N ∂ ( D × H P ) σ −→ L ( Z ) (cid:27) Z featuring the surgery obstruction map σ from the normal invariants N ∂ ( D × H P ) to the L -group L ( Z ) (cid:27) Z . The standard smooth structure on D × H P provides an isomorphism N ∂ ( D × H P ) (cid:27) [ S ∧ H P + , G / O ] , where [− , −] stands for based homotopy classes and G / O is the homotopy fibre of the mapBO → BG classifying the underlying stable spherical fibration of a stable vector bundle.As BG has trivial rational homotopy groups, the map [ S ∧ H P + , G / O ] −→ [ S ∧ H P + , BO ] = g KO ( S ∧ H P + ) is rationally an isomorphism. Furthermore the Pontrjagin character gives an isomorphismph (−) = ch (− ⊗ C ) : g KO ( S ∧ H P + ) ⊗ Q ∼ −→ É i ≥ e H i ( S ∧ H P + ; Q ) = u · Q [ z ]/( z ) , N HP -BUNDLE OVER S WITH NONTRIVIAL ˆ A-GENUS 3 where u ∈ e H ( S ; Q ) denotes the cohomological fundamental class, and z ∈ H ( H P ; Q ) is the usual generator. Therefore for any triple ( A , B , C ) ∈ Q there exists a nonzero λ ∈ Z and a normal invariant n ∈ N ∂ ( D × H P ) whose underlying stable vector bundle ξ hasph ( ξ ) = λ · u · ( A + Bz + Cz ) . Since S ∧ H P + has no nontrivial cup-products amongelements of positive degree, we have ph i ( ξ ) = (− ) i + /( i − ) ! · p i ( ξ ) and hence(1) p ( ξ ) = λA · u p ( ξ ) = − λB · u · z p ( ξ ) = λC · u · z . To evaluate the surgery obstruction map σ , recall that a normal invariant n with un-derlying stable vector bundle ξ is represented by a degree 1 normal map(2) ν M ν D × H P ⊕ ξM D × H P , ˆ ff where ∂ M = ∂ D × H P and f and ˆ f restrict to the identity maps on the boundary.Here ν (−) denotes the stable normal bundle of a manifold. The surgery obstruction isunchanged by gluing into M and D × H P a copy of D × H P along the identification oftheir boundaries with ∂ D × H P , and extending f and ˆ f trivially, giving rise to a degree1 normal map to f ′ : M ′ → S × H P . The surgery obstruction may then be expressed interms of the signatures of these manifolds, as σ ( n ) = (cid:0) sign ( M ′ ) − sign ( S × H P ) (cid:1) . The signature of S × H P is trivial, and that of M ′ may be computed in terms ofthe Hirzebruch signature theorem as the evaluation ∫ M ′ L ( T M ′ ) of the L -class. As f ′ hasdegree 1 and pulls back ν S × H P ⊕ ξ to the stable inverse of T M ′ , we have(3) sign ( M ′ ) = ∫ M ′ L ( T M ′ ) = ∫ S × H P L ( T S ) · L ( T H P ) · L (− ξ ) . The first terms of the total L -class are given as L = + p + · p − p + · p − · p p + · p + · · · , which we combine with p ( T H P ) = + z + z from [Hir53, Satz 1] to compute L ( T S ) = L ( T H P ) = + · z + z L (− ξ ) = + λ (− A · u + B · ( u · z ) − C · ( u · z )) and thus 8 σ ( n ) = sign ( M ′ ) = λ (− A + B − C ) . It follows that for each triple ( A , B , C ) ∈ Q satisfying A − B + C = λ ∈ Z and a degree 1 normal map as in (2) with f a homotopy equivalenceand with ξ having Pontrjagin classes as in (1). This gives a smooth block H P -bundlestructure on the composition M ′ f ′ −→ S × H P π −−→ S giving rise to a class in π ( B g Diff ( H P ) , so it remains to evaluate ˆ A ( M ′ ) . As in (3), we getˆ A ( M ′ ) = ∫ M ′ ˆ A ( T M ′ ) = ∫ S × H P ˆ A ( T S ) · ˆ A ( T H P ) · ˆ A (− ξ ) , which we combine with the formula for the first terms of the total ˆ A -classˆ A = − p + − · p + · p + − · p + · p p − · p + · · · to computeˆ A ( T S ) = A ( T H P ) = − · z ˆ A (− ξ ) = + λ (− A · u + B · ( u · z ) + C · ( u · z )) from which we conclude ˆ A ( M ′ ) = λ ( B + C ) . MANUEL KRANNICH, ALEXANDER KUPERS, AND OSCAR RANDAL-WILLIAMS
As there are clearly triples ( A , B , C ) ∈ Q satisfying − A + B − C = B + C , , this finishes the argument. Remark . A fibre bundle π : E → S constructed in this way is fibre homotopy equivalentto the trivial bundle π : S × H P → S , and under this fibre homotopy trivialisation wehave p ( T E ) = · ( ⊗ z ) − λA · ( u ⊗ ) . Thus p ( T E ) = − λA · ( u ⊗ z ) , and so ∫ E p ( T E ) = − λA and ∫ E ˆ A ( T E ) = λ ( B + C ) . This argument therefore guarantees the existence of a 2-dimensional subspace of thegroup π ( B Diff ( H P ))⊗ Q , detected by the characteristic numbers ∫ E p ( T E ) and ∫ E ˆ A ( T E ) . Remark . The Theorem can be slightly strengthened: we may in addition assume thatthe smooth H P -bundle π : E → S admits a smooth section with trivial normal bundle,which may be helpful for fibrewise surgery constructions.To see this, note that a fibre bundle π : E → S as constructed above is fibre homotopyequivalent to the trivial bundle π : S × H P → S , so it admits a smooth section s : S → E corresponding to a trivial section of the trivial bundle. By the description of p ( T E ) in theprevious remark we have s ∗ p ( T E ) = − λA · u . If we choose ( A = , B = , C = ) ,which is a triple whose surgery obstruction vanishes, then the corresponding bundle has s ∗ p ( T E ) = A ( E ) ,
0. As the tangent bundle of S is stably trivial, it follows thatthe normal bundle of s ( S ) ⊂ E has trivial first Pontrjagin class. As p : π ( BSO ( )) → Z is injective, this implies that this normal bundle is trivial. Remark . The argument can be generalised to prove that for any even n ≥ H P n -bundle E n + → S with ˆ A ( E ) ,
0, so the Corollary holds for H P n as well.Indeed, the application of Morlet’s lemma only required the fibre to be at least 8-dimensional and 3-connected. Similar to the above, one argues that for any pair ( A , C ) ∈ Q there exists a nonzero λ ∈ Z and a normal invariant n ∈ N ∂ ( D × H P n ) with underlyingstable vector bundle ξ such that p ( ξ ) = λA · u p i ( ξ ) = < i < n , p n + ( ξ ) = λ ( n + ) ! (− ) n C · u · z n . Using that the coefficient of z n in L ( T H P n ) is 1 by the Hirzebruch’s signature theorem, wesee that the surgery obstruction satisfies 8 σ ( n ) = λ (− A + h n + ( n + ) ! (− ) n + C ) where h n is the coefficient of p n in the total L -class. In particular, we can always find pairs ( A , C ) with C , σ ( n ) =
0. As the coefficient of z n in ˆ A ( T H P n ) vanishes, because H P n admits a metric of positive scalar curvature, it holds ˆ A ( E ) = λa n + ( n + ) ! (− ) n + C , a n the coefficient of p n in the total ˆ A -class, which is easily proved to be non-zerousing [Hir95, Ch. 1.§1 (10)]. Remark . If one is willing to instead consider H P n -bundles over S m for n sufficientlylarge compared with m , then one may replace the appeal to the Lemma by the moreclassical [BL82, Corollary D], which implies that the map π m ( hAut ( H P n )/ Diff ( H P n ) ; id ) ⊗ Z [ ] −→ π m ( hAut ( H P n )/ g Diff ( H P n ) ; id ) ⊗ Z [ ] is (split) surjective as long as 4 m − H P n (so3 m < n suffices, by [Igu88]). See also [Bur79, Theorem 1]. One must still produce anappropriate element of S ∂ ( D m × H P n ) , which may be approached as in Remark 3. Acknowledgements.
We thank Thomas Schick for encouraging us to write this note,and Johannes Ebert for a comment which reminded us of [Bur79] and [BL82]. The firstand third author were supported by a Philip Leverhulme Prize from the Leverhulme Trust.The third author was also supported by the ERC under the European Union’s Horizon2020 research and innovation programme (grant agreement No. 756444).
N HP -BUNDLE OVER S WITH NONTRIVIAL ˆ A-GENUS 5
References [BEW20] B. Botvinnik, J. Ebert, and D. J. Wraith,
On the topology of the space of Ricci-positive metrics , Proc.Amer. Math. Soc. (2020), 3997–4006.[BL82] D. Burghelea and R. Lashof,
Geometric transfer and the homotopy type of the automorphism groups ofa manifold , Trans. Amer. Math. Soc. (1982), no. 1, 1–38.[BLR75] D. Burghelea, R. Lashof, and M. Rothenberg,
Groups of automorphisms of manifolds , Lecture Notes inMathematics, Vol. 473, Springer-Verlag, Berlin-New York, 1975, With an appendix (“The topologicalcategory”) by E. Pedersen.[Bur79] D. Burghelea,
The rational homotopy groups of Diff ( M ) and Homeo ( M n ) in the stability range , Alge-braic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978), Lecture Notes in Math., vol.763, Springer, Berlin, 1979, pp. 604–626.[ERW14] J. Ebert and O. Randal-Williams, Generalised Miller-Morita-Mumford classes for block bundles and topo-logical bundles , Algebr. Geom. Topol. (2014), no. 2, 1181–1204.[Hir53] F. Hirzebruch, Über die quaternionalen projektiven Räume , S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss. (1953), 301–312 (1954).[Hir95] ,
Topological methods in algebraic geometry , Classics in Mathematics, Springer-Verlag, Berlin,1995, Reprint of the 1978 edition.[Hit74] N. Hitchin,
Harmonic spinors , Advances in Math. (1974), 1–55.[HSS14] B. Hanke, T. Schick, and W. Steimle, The space of metrics of positive scalar curvature , Publ. Math. Inst.Hautes Études Sci. (2014), 335–367.[Igu88] K. Igusa,
The stability theorem for smooth pseudoisotopies , K -Theory (1988), no. 1-2, vi+355.[OWL17] Mini-workshop: Spaces and moduli spaces of Riemannian metrics , Oberwolfach Rep. (2017), no. 1,133–166, Abstracts from the mini-workshop held January 8–14, 2017, Organized by F. Thomas Farrelland Wilderich Tuschmann.[RW17] O. Randal-Williams, An upper bound for the pseudoisotopy stable range , Math. Ann. (2017), no. 3-4,1081–1094.[Sch14] T. Schick,
The topology of positive scalar curvature , Proceedings of the International Congress ofMathematicians—Seoul 2014. Vol. II, Kyung Moon Sa, Seoul, 2014, pp. 1285–1307.[Wal99] C. T. C. Wall,
Surgery on compact manifolds , second ed., Mathematical Surveys and Monographs,vol. 69, American Mathematical Society, Providence, RI, 1999, Edited and with a foreword by A. A.Ranicki.
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