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Finite Groups Acting on Smooth and Symplectic 4-Manifolds

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Finite Groups Acting on Smooth and Symplectic 4-Manifolds Finite groups acting on smooth and symplectic 4-manifolds Carles Sáez Calvo Aquesta tesi doctoral està subjecta a la llicència Reconeixement 4.0. Espanya de Creative Commons. Esta tesis doctoral está sujeta a la licencia Reconocimiento 4.0. España de Creative Commons. This doctoral thesis is licensed under the Creative Commons Attribution 4.0. Spain License. Departament de Matematiques` i Informatica` Doctorat en Matem`atiquesi Inform`atica Tesi Doctoral Finite groups acting on smooth and symplectic 4-manifolds Candidat: Carles S´aezCalvo Director de la tesi: Dr. Ignasi Mundet i Riera Agra¨ıments En primer lloc vull agrair al meu director, Ignasi Mundet i Riera, excel·lent matem`atic i excel·lent persona, per introduir-me a temes tant interessants com les accions de grups finits en varietats i la geometria simpl`ectica.Tamb´eli vull agrair la seva bona disposici´o a l’hora de compartir idees i resoldre dubtes. Ha estat tot un plaer aprendre i treballar amb ell durant aquests anys. Sense ell, aquesta tesi no hauria estat possible. En segundo lugar quiero agradecer a Teresa, que ha sido mi compa˜neraen todos los aspectos de la vida a lo largo de estos a˜nos. Ambos hemos compartido la experiencia de hacer una tesis doctoral, haciendo mejores los buenos momentos y m´asllevaderos los malos. Gracias por todos los viajes y excursiones que me han ayudado a desconectar de la tesis cuando lo necesitaba, y gracias tambi´enpor las interesantes conversaciones matem´aticasque hemos tenido. Sin Teresa, esta tesis tampoco habr´ıa sido posible. Muchas gracias por hacer mi vida mejor. Vull agrair al Centre de Recerca Matem`aticai a la BGSMath per totes les facilitats que han posat a la meva disposici´odurant aquests anys. Gr`aciesals doctorands de la UAB pel bon ambient de treball i pels bons moments que han ajudat a fer m´esagradable la realitzaci´od’aquesta tesi. Gr`acies tamb´ea la Nat`aliaCastellana per oferir-se a ser la meva mentora, encara que no hagi pogut exercir gaire com a tal. Un agra¨ıment especial a la gent del seminari de geometria de la UAB, on vaig aprender teoria de Seiberg–Witten, que juga un paper important en aquesta tesi. Gr`aciesa tota a la gent de la UB, en especial a l’Eduard Casas i a la Laura Costa per les facilitats i la llibertat a l’hora de fer classes. Gr`acies tamb´ea l’Eva Miranda i el seu grup, pel bon ambient i per conversacions interessants. Gracias a Francisco Presas y a Alvaro´ del Pino por las interesantes conversaciones e ideas sobre el contenido de esta tesis. Quiero agradecer tambi´ena Enrique Casanovas y al grupo de teor´ıade modelos de la UB, con quienes realic´euna tesis de m´aster,por introducirme en la investigaci´ony por el buen ambiente que siempre encontr´e. Finalmente, pero no por ello menos importante, quiero agradecer a mis padres y a mi familia por el apoyo incondicional que me han ofrecido a lo largo de mis a˜nosde estudio. 3 4 Contents Introduction 7 1 Preliminaries on group actions 17 1.1 Group actions . 17 1.2 Smooth actions . 20 1.3 The Serre spectral sequence . 24 1.4 Classifying spaces and group cohomology . 29 1.5 Equivariant cohomology . 30 1.6 Computation of equivariant cohomology . 33 ∗ r d 1.7 Computing H ((Z /p ) ; Z /n)......................... 36 1.8 Applications to actions on closed manifolds . 38 1.9 The Jordan property . 44 2 Preliminaries on smooth 4-manifolds 51 2.1 Preliminaries on 4-manifolds . 51 2.2 The Atiyah–Singer G-signature theorem . 53 2.3 Spinc-structures and Dirac Operators . 61 2.4 The Seiberg-Witten moduli spaces . 67 + 2.5 SW invariants for b2 (X) > 1 ......................... 73 + 2.6 SW invariants for b2 =1............................ 74 2.7 The generalized adjunction formula . 75 3 Preliminaries on J-holomorphic curves 79 3.1 Almost complex manifolds and J-holomorphic curves . 79 3.2 Local properties . 82 3.3 J-holomorphic curves in symplectic manifolds . 86 3.4 Moduli spaces of J-holomorphic curves . 87 3.5 Gromov compactness . 91 3.6 Gromov-Witten invariants . 94 3.7 Seiberg–Witten invariants in symplectic manifolds . 95 5 Contents 6 4 Which finite groups act on a given 4-manifold? 99 4.1 Main results . 99 4.2 Main ideas of the proofs . 102 4.3 First simplifications . 104 4.4 Abelian groups with a bound on the number of generators . 106 4.5 Linearization of finite group actions . 109 4.6 Surfaces and line bundles . 111 4.7 Non-free actions . 113 2 4.8 Diffeomorphisms normalizing an action of Z /p or Z /p .......... 118 4.9 Finite groups acting smoothly on 4-manifolds with b2 =0 ......... 121 4.10 Proofs of Theorems 4.1 and 4.2 . 132 4.11 Using the Atiyah–Singer G-signature theorem . 141 4.12 Automorphisms of almost complex manifolds: proof of Theorem 4.5 . 143 4.13 Symplectomorphisms: proof of Theorem 4.6 . 144 5 Symplectic actions on S2-bundles over S2 149 5.1 S2-bundles over S2 ............................... 149 5.2 Finite groups acting on S2 ........................... 152 5.3 Actions on line bundles over S2 ........................ 153 5.4 The trivial bundle . 155 5.5 The non-trivial bundle . 163 Introduction The main problem in the subject of finite (smooth) transformation groups is to determine which finite groups admit effective and smooth actions on a given smooth manifold X, and to determine the geometry of such actions. In precise terms, this means to give, for any smooth manifold X, a list of finite groups Gi and effective actions Φi,j : Gi ×X → X such that any effective and smooth action of a finite group on X is conjugated to exactly one of the actions Φi,j, where the conjugation is given by a diffeomorphism of X. A complete solution for this problem has only been achieved for a few manifolds (the paradigmatic example being S2, for which any effective action is conjugated to a linear action of a finite subgroup of SO(3)). In fact, it is extremely difficult to answer this question in full generality for any given manifold. Therefore, we restrict ourselves to a simpler question, which can be understood as a first step towards the full solution of the problem. We forget about the geometry of the actions and study the problem of determining which finite groups G (up to abstract isomorphism of groups) admit some effective and smooth action on X. This, although simpler than the full classfication problem, is extremely difficult in general, and again the full solution is only known in few cases. One possibility is to study a coarse version of that problem, by trying to prove theo- rems that impose restrictions on the finite groups that can act effectively and smoothly on a given manifold. That is, instead of trying to provide a complete list of finite groups that do admit effective and smooth actions on a manifold, we look for negative results that narrow as much as possible the list of finite groups admitting efective actions on a given manifold. Moreover, we want to maximize the class of manifolds for which this negative results apply. Hence, we will be especially interested in theorems applying to all smooth manifolds, or at least to some large class of manifolds, like the closed ones. A first result along these lines was provided by Mann and Su in 1963. They proved that for any closed smooth manifold X there exists a natural number r (depending only on X) such that, for any prime p, the rank of any p-elementary abelian group acting effectively on X is bounded by r (see Theorem 1.48 for the proof). In particular, there is no closed manifold that admits effective actions of all finite groups. This is in sharp contrast with the situation for general (not necessarily closed) man- ifolds. In [60], Popov provides examples of open 4-manifolds admitting effective and smooth actions of all finite groups (more generally, there exists an open 4-manifold in which any finitely presented group acts in a smooth and free way). Therefore, at least 7 Introduction 8 in dimensions greater than 3, there is no hope of proving any theorem imposing restric- tions on the class of finite groups acting on smooth manifolds that apply to all manifolds. Hence, we will restrict ourselves to the case of closed manifolds, for which we already know that some restrictions exist, by the theorem of Mann and Su. The first problem we consider in this thesis is related to the so-called Jordan property for diffeomorphism groups. A group G has the Jordan property (or simply, is Jordan) if there exists a constant C such that any finite subgroup G ≤ G has an abelian subgroup A ≤ G satisfying [G : A] < C. Hence, the Jordan property is a condition on the finite subgroups of an (infinite) group G. This condition can be thought of as stating that all finite subgroups of G are ”almost abelian”. To make this more explicit, one can define for any finite group G α(G) = min{[G : A] | A ≤ G is abelian}. α(G) therefore can be thought of as a complexity measure of G, where the complexity makes reference to how much non-abelian G is (note that α(G) = 1 if and only if G is abelian). With this notation, the Jordan property for G is equivalent to the existence of a constant C (depending only on G) such that α(G) < C for all finite G ≤ G.
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