Diffeomorphism Groups of Compact 4-manifolds are not always Jordan
aa r X i v : . [ m a t h . DG ] N ov Diffeomorphism Groups of Compact 4-manifolds are not always Jordan
Bal´azs Csik´os, L´aszl´o Pyber, Endre Szab´o
Abstract
We show that if M is a compact smooth manifold diffeomorphic to the total space of an orientable S bundle over the torus T , then its diffeomorphism group does not have the Jordan property, i.e., Diff( M )contains a finite subgroup G n for any natural number n such that every abelian subgroup of G n has indexat leat n . This gives a counterexample to an old conjecture of Ghys. By Jordan’s classical theorem (see, e.g., [1]), for any natural number n , there is a constant c n dependingonly on n , such that every finite subgroup of the group GL ( n, C ) has an abelian subgroup of index at most c n .We say that a group G has the Jordan property if there is a constant c depending only on the group G , suchthat every finite subgroup of G has an abelian subgroup of index at most c .Motivated by the Jordan theorem, E. Ghys conjectured that the diffeomorphism groups of compact smoothmanifolds have the Jordan property. As noted in [2], this conjecture was discussed in several talks of Ghys[3], but appeared in print for the first time in [4].Known examples of non-compact manifolds [5], [6] having diffeomorphism groups which do not possess theJordan property show that the compactness assumption in the conjecture is essential.The conjecture of Ghys was verified in several special cases. Zimmermann [6] proved the conjecture forcompact 3-manifolds and in a series of papers [7], [2], [8], [9], Riera proved the conjecture for tori, acyclicmanifolds, homology spheres, and manifolds with non-zero Euler characteristic.The goal of this paper is to prove the following theorem. Theorem 1. M is diffeomorphic to T × S or to the total space of a nontrivial smooth orientable S bundleover T , then the diffeomorphism group of M does not have the Jordan property. The proof uses some standard facts on smooth complex line bundles and sphere bundles over the torus, wesummarize in section 2.Then we apply ideas from algebraic geometry that appeared in Yu.G. Zarhin [10], where the same ideaswere used to prove that the group of birational automorphisms of the product of an elliptic curve and aprojective line over an algebraically closed field of characteristic zero does not have the Jordan property.We collect the necessary information on holomorphic line bundles and prove the main theorem in section 3.
Let T = R / Z be a 2-dimensional torus with a given complex structure, and fix a point o ∈ T . As theuniversal covering space of T is biholomorphically equivalent to C , we can identify T with the factor space Primary 57S17, Secondary 54H15.
Keywords and phrases:
Ghys’ conjecture, Jordan property, diffeomorphism group / Γ, where Γ is a rank 2 lattice in C acting on C by translations. Addition in C induces a commutativegroup operation + on T with neutral element o . For x ∈ T , we shall denote by T x : T → T , y x + y the translation by x .Smooth complex line bundles ξ = ( E π −→ T ) over T are classified by the first Chern class c ( ξ ) ∈ H ( T , Z ) ∼ = Z , or, equivalently, by the Chern number R T c ( ξ ) ∈ Z , where the integral is computedusing the orientation of T induced by the complex structure. Denote by ξ n = ( E n π n −→ T ) the complexline bundle with Chern number n ∈ Z . It is known that ξ n ⊗ C ξ m ∼ = ξ n + m , in particular ξ n is the n th tensorpower of ξ for all n > ξ is the trivial bundle.We can associate to any complex line bundle ξ a smooth complex projective line bundle P ( ξ ⊕ ξ ) withfibers diffeomorphic to C P ∼ = S . Denote by Y n the total space of the bundle P ( ξ n ⊕ ξ ). It is known thatnon-isomorphic sphere bundles over a surface can have diffeomorphic total spaces. According to Theorems1 and 2 in [11], the total spaces of two S -bundles η and η over T are diffeomorphic, if and only if theStiefel-Whitney classes w and w of them satisfy the following conditions:(i) w ( η ) = w ( η );(ii) either w ( η ) = w ( η ) = 0 or none of w ( η ) and w ( η ) is 0.As a corollary, the total spaces of all smooth sphere bundles over T belong to one of four different diffeomor-phism classes. Due to the presence of complex structures, the sphere bundles P ( ξ n ⊕ ξ ) are all orientable, sotheir first Stiefel-Whitney classes vanish. The second Stiefel-Whitney class w ( P ( ξ n ⊕ ξ )) ∈ H ( T , Z ) ∼ = Z is the mod 2 reduction of n , so we have the following Proposition 1. Y n is diffeomorphic to Y m if and only if n ≡ m (mod ). Consider the complex line bundle ξ . There are infinitely many holomorphic structures on ξ compatiblewith the given complex structure on T . Let us fix any of these holomorphic structures and equip ξ n withthe holomorphic structure for which the isomorphisms ξ n ⊗ C ξ m ∼ = ξ m + n become holomorphic isomorphism.Mumford [12] considered the subgroup H ( ξ n ) = { x ∈ T : T ∗ x ( ξ n ) ∼ = ξ n } of T , where ∼ = means the isomorphism of holomorphic line bundles. It is known that H ( ξ n ) is finite if andonly if ξ n is an ample line bundle, which is the case if and only if n > H ( ξ n ) defined by G ( ξ n ) = { ( x, φ ) : x ∈ H ( ξ n ) , φ : E n → E n is a biholomorphic map such that π n ◦ φ = T x ◦ φ } . The kernel of the natural homomorphism G ( ξ n ) → H ( ξ n ), ( x, φ ) x is C ∗ so we have a short exact sequence0 → C ∗ → G ( ξ n ) → H ( ξ n ) → . The key observation is that every holomorphic structure on the bundle ξ n induces a unique holomorphicstructure on the associated projective line bundle P ( ξ n ⊕ ξ ), hence a complex structure on Y n . For any( x, φ ) ∈ G ( ξ n ), φ extends uniquely to a biholomorphic map ˜ φ : Y n → Y n . The map ( x, φ ) ˜ φ provides anembedding of the group G ( ξ n ) into the diffeomorphism group of Y n . Combining this fact with Proposition1, we obtain the following Proposition 2.
The diffeomorphism group of Y n contains subgroups isomorphic to G ( ξ m ) for every m > satisfying m ≡ n mod . H ( ξ n ) and G ( ξ n ) was described in details by Mumford [12]. Recall those elementsof the description that are relevant for the proof of our main result. Theorem 2 (Mumford, § . The group H ( ξ n ) contains an (abelian) subgroup K such that H ( ξ n ) isisomorphic to the group K ⊕ ˆ K , where ˆ K is the multiplicative group of characters K → S ⊂ C ∗ .The group G ( ξ n ) is isomorphic to the group defined on the set C ∗ × ( K ⊕ ˆ K ) with the multiplication rule ( a, k ⊕ l ) · ( a ′ , k ′ ⊕ l ′ ) = ( a · a ′ · l ′ ( k ) , ( k + k ′ ) ⊕ ( l · l ′ )) . Theorem 3 (Mumford, Prop. 4 in § . If k is a positive integer, then H ( ξ kn ) = { x ∈ T : kx ∈ H ( ξ n ) } . In particular, the case n = 1 yields that H ( ξ k ) has at least k elements. We obtain as a corollary that if H ( ξ n ) has N elements, then N ≥ n , N is a square number, K has √ N elements. Moreover, if Z √ N < C ∗ denotes the cyclic subgroup of order √ N , then the subset G n = Z √ N × ( K ⊕ ˆ K ) ⊂ C ∗ × ( K ⊕ ˆ K ) is a finite subgroup of G ( ξ n ).Now we are ready to prove our main theorem. Proof of Theorem 1.
As it was pointed out Y.G. Zarhin [10], the index of any abelian subgroup of G n is atleast √ N ≥ n . By Proposition 1, M is diffeomorphic to Y m , where m is either 0 or 1. In both cases, Diff( M )contains an infinite sequence of finite subgroups G n , ( n > n ≡ m mod 2), having the property thatany abelian subgroup of G n has index at least n . This proves the theorem. The first author is supported by the Hungarian National Science and Research Foundation OTKA K112703.During the research he also enjoyed the hospitality of the Alfr´ed R´enyi Institute of Mathematics as a guestresearcher.The second author is supported in part by K84233.The third author is supported in part by OTKA NK81203, K84233, and by MTA R´enyi ”Lend¨ulet” Groupsand Graphs Research Group.The authors are indebted to Andr´as N´emethi and Andr´as Stipsicz for helpful and fruitful discussions.
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Invent. Math. , vol. 1, pp. 287–354, 1966.Bal´azs Csik´os, Institute of Mathematics, E¨otv¨os Lor´and University, Budapest, P´azm´any P. stny. 1/C,H-1117 Hungary.
E-mail address: [email protected]´aszl´o Pyber, Alfr´ed R´enyi Institute of Mathematics, Budapest, Re´altanoda u. 13-15, H-1053 Hungary,
E-mail address: [email protected] Szab´o, Alfr´ed R´enyi Institute of Mathematics, Budapest, Re´altanoda u. 13-15, H-1053 Hungary,