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Positive Quaternion Kähler Manifolds Manuel Amann Positive Quaternion K¨ahlerManifolds 2009 Mathematik Positive Quaternion K¨ahlerManifolds Inaugural-Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften im Fachbereich Mathematik und Informatik der Mathematisch-Naturwissenschaftlichen Fakult¨at der Westf¨alischen Wilhelms-Universit¨atM¨unster vorgelegt von Manuel Amann aus Bad Mergentheim 2009 Dekan: Prof. Dr. Dr. h.c. Joachim Cuntz Erster Gutachter: Prof. Dr. Burkhard Wilking Zweiter Gutachter: Prof. Dr. Anand Dessai Tag der m¨undlichen Pr¨ufung: 10. Juli 2009 Tag der Promotion: 10. Juli 2009 Fur¨ meine Eltern Rita und Paul und meinen Bruder Benedikt When the hurlyburly's done, When the battle's lost and won. William Shakespeare, \Macbeth" Preface Positive Quaternion K¨ahlerGeometry lies in the intersection of very classical yet rather different fields in mathematics. Despite its geometrical setting which involves fundamental definitions from Riemannian geometry, it was soon discovered to be accessible by methods from (differential) topology, symplectic geometry and complex algebraic geometry even. Indeed, it is an astounding fact that the whole theory can entirely be encoded in terms of Fano contact geometry. This approach led to some highly prominent and outstanding results. Aside from that, recent results have revealed in-depth connections to the theory of positively curved Riemannian manifolds. Furthermore, the conjectural existence of symmetry groups, which is confirmed in low dimensions, sets the stage for equivariant methods in cohomology, homotopy and Index Theory. The interplay of these highly dissimilar theories contributes to the appeal and the beauty of Positive Quaternion K¨ahlerGeometry. Indeed, once in a while one may gain the impression of having a short glimpse at the dull flame of real mathematical insight. Quaternion K¨ahlerManifolds settle in the highly remarkable class of special geome- tries. Hereby one refers to Riemannian manifolds with special holonomy among which K¨ahlermanifolds, Calabi{Yau manifolds or Joyce manifolds are to be mentioned as the most prominent examples. Whilst the latter|i.e. manifolds with G2-holonomy or Spin(7)-holonomy|seem to be extremely far from being symmetric in general, it is probably the central question in Positive Quaternion K¨ahlerGeometry whether every such manifold is a symmetric space. This question was formulated in the affirmative in a fundamental conjecture by LeBrun and Salamon. It forms the basic motivation for the thesis. The conjectural rigidity of the objects of research seems to be reflected by the variety of methods that can be applied|especially if these methods are rather \far ii Preface away from the original definition”. For example, the existence of symmetries on the one hand contributes to the structural regularity of the manifolds; on the other hand does it permit a less analytical and more topological approach. The existence of various rigidity theorems and topological recognition theorems backs the idea that the structure imposed by special holonomy and positive scalar curvature is restrictive enough to permit a classification of Positive Quaternion K¨ahlerManifolds. We shall contribute some more results of this kind. Conceptually, the thesis splits into two parts: On the one hand we are interested in classification results, (mainly) in low dimensions, which has led to chapters2 and 4. On the other hand we extend the spectrum of methods used to study Positive Quaternion K¨ahlerManifolds by an approach via Rational Homotopy Theory. The outcome of this enterprise is depicted in chapter3. To be more precise, results obtained feature • the formality of Positive Quaternion K¨ahlerManifolds|which is obtained by an in-depth analysis of spherical fibrations in general, • the discovery and documentation (with counter-examples) of a crucial mistake in the existing “classification” in dimension 12, • new techniques of how to detect plenty of new examples of non-formal homoge- neous spaces, 20 24 • recognition theorems for HP and HP , • a partial classification result in dimension 20, n+4 • a recognition theorem for the real Grassmannian Grf 4(R ), which proves that the main conjecture, which suggests the symmetry of Positive Quaternion K¨ahler Manifolds, can (almost always) be decided from the dimension of the isometry group • and results on rationally elliptic Positive Quaternion K¨ahlerManifolds and rationally elliptic Joyce manifolds. Moreover, we depict a method of how to find an upper bound for the Euler characteristic of a Positive Quaternion K¨ahlerManifold under the isometric action of a sufficiently 1 large torus. This makes use of a classification of S -fixed-point components and Z2-fixed-point components of Wolf spaces which we provide. Furthermore, we reprove n+4 the vanishing of the elliptic genus of the real Grassmannian Gr4(R ) for n odd. The methods applied comprise Rational Homotopy Theory, Index Theory, elements from the theory of transformation groups and equivariant cohomology as well as Lie theory. Preface iii In the first chapter we give an elementary introduction to the subject, we recall basic notions and we recapitulate known facts of Positive Quaternion K¨ahlerGeometry, Index Theory and Rational Homotopy Theory. As we want to keep the thesis accessible to a larger audience ranging from Riemannian geometers to algebraic topologists, we shall be concerned to give an easily comprehensible and detailed outline of the concepts. In order to keep the proofs of the main theorems of subsequent chapters compact and comparatively short, we establish and elaborate some crucial but not standard theory in the introductory chapter already. We shall eagerly provide detailed proofs whenever different concepts are brought together or when ambiguities arise (in the literature). Chapter2 is devoted to a depiction of an error that was committed by Herrera and Herrera within the process of classifying 12-dimensional Positive Quaternion K¨ahler Manifolds. As this classification was highly accredited, and since not only the result but also the erroneous method of proof were used several times since then, it seems to be of importance to document the mistake in all adequate clarity. This will be done by giving various counter-examples for the different stages in the proof|varying from purely algebraic examples to geometric ones|as well as for the result, i.e. for the main tool used by Herrera and Herrera, itself. So we shall derive 7 5 1 Theorem. There is an isometric involution on Grf 4(R ) having Grf 2(R ) and HP as fixed-point components. The difference in dimension of the fixed-point components is not divisible by 4 which implies that the Z2-action is neither even nor odd. We shall see that this will contradict what was asserted by Herrera and Herrera. It is the central counter-example to an argument which resulted in the assertion that 2n the A^-genus of a π2-finite oriented compact connected smooth manifold M with an 1 effective smooth S -action vanishes. We even obtain Theorem. For any k > 1 there exists a smooth simply-connected 4k-dimensional 4k 1 ^ 4k 4k π2-finite manifold M with smooth effective S -action and A(M )[M ] 6= 0. This chapter is closely connected to the chaptersB andD of the appendix, where not only a more global view on the main counter-example is provided (cf. chapter B), but where we also discuss what still might be true within the setting of Positive Quaternion K¨ahlerManifolds (cf. chapterD)|judging from the symmetric examples. (The main tool of Herrera and Herrera was formulated in a more general context.) This will also serve as a justification for some assumptions we shall make in chapter4. In the third chapter we establish the formality of Positive Quaternion K¨ahler Manifolds. Theorem. A Positive Quaternion K¨ahlerManifold is a formal space. The twistor fibration is a formal map. For this we shall investigate under which circumstances the formality of the total space of a spherical fibration suffices to prove formality for the base space. An iv Preface application to the twistor fibration proves the main geometric result. The formality of Positive Quaternion K¨ahlerManifolds can be seen as another indication for the main conjecture, as symmetric spaces are formal. Besides, the question of formality seems to be a recurring topic of interest within the field of special holonomy. This discussion will be presented in a far more general context than actually necessary for the main result: We shall investigate how formality properties of base space and total space are related for both even-dimensional and odd-dimensional fibre spheres. Moreover, we shall discuss the formality of the fibration itself. It will become clear that the case of even-dimensional fibres is completely distinct from the one with odd-dimensional fibres, where hardly no relations appear at all. This topic will naturally lead us to the problem of finding construction principles for non-formal homogeneous spaces. Although a lot of homogeneous spaces were identified as being formal, there seems to be a real lack of non-formal examples. As we do not want to get too technical at this point of the thesis let us just mention a few of the most prominent examples we discovered: Theorem. The spaces Sp(n) SO(2n) SU(p + q) SU(n) SU(n) SU(p) × SU(q) are non-formal for n ≥ 7, n ≥ 8 and p + q ≥ 4 respectively. The techniques elaborated and the many examples will then also permit to find an (elliptic) non-formal homogeneous spaces in each dimension greater than or equal to 72. Apart from simple reproofs of the formality of Positive Quaternion K¨ahlerManifolds in low dimensions, the following sections are mainly dedicated to elliptic manifolds. We shall prove Theorem. An elliptic 16-dimensional Positive Quaternion K¨ahler Manifold is ratio- nally a homology Wolf space. Theorem. There are no compact elliptic Spin(7)-manifolds. An elliptic simply- connected compact irreducible G2-manifold is a formal space.
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