NORMAL SUBGROUPS OF DIFFEOMORPHISM AND ...

More documents

Recommendations

Info

$(PUC - Rio de Janeiro ALOCA\307\303O.xlsx)$

compact support, and let A I be the subgroup of those that are isotopic to the identity by a compactly supported isotopy. Let κ : A K → π 0 Diff r +(S n ) be the homomorphism that takes a diffeomorphism f ∈ A K to the isotopy class of its extension to S n = R n ∪ {∞}. In particular, A I ⊆ Ker(κ). The group π 0 Diff r +(S n ) is easily seen to be abelian (by taking representative diffeomorphisms supported on disjoint balls), and if n ≠ 4 and r ≥ 1 it is known that it is isomorphic to θ n+1 , the finite abelian Kervaire-Milnor group of orientation preserving diffeomorphisms of S n up to h-cobordism [?]. Our main theorem is the following (with a stronger version in Theorem ?? below). Theorem 1.1 The groups κ −1 (η), where η varies over all the subgroups of π 0 Diff r +(S n ), are normal subgroups of A r +(R n ) if r ≥ 1. If r ≠ n + 1, they are the only proper normal subgroups. For homeomorphisms (r = 0), A I = A K is the only proper normal subgroup. In the 1960’s and early 1970’s, algebraic properties of homeomorphism and diffeomorphism groups were studied extensively in work of Anderson [?], Epstein [?], Herman [?], Thurston [?], and Mather [?, ?], among others. The fundamental result is Mather’s theorem [?, ?] that the group of C r diffeomorphisms of a smooth manifold M isotopic to the identity by isotopies with compact support is a simple group, provided that r ≠ dim(M) + 1. This theorem had been proven by Thurston [?] when r = ∞. (The case when r = 0 is fairly easy, and the case when r = dim(M) + 1 remains open.) These results are based on Epstein’s theorem [?] that under certain very general hypotheses, the commutator subgroup of a group of homeomorphisms is simple. In my lecture at the AMS Symposium on Pure Mathematics at Stanford University in August 1976, I presented theorems about groups of diffeomorphisms of R n . After the lecture, Wensor Ling, a graduate student working on his doctorate under Thurston at MIT and Princeton, mentioned that he had obtained similar results in his doctoral research. I decided to leave the publication of the results to the future Dr. Ling. Two years later, Dusa McDuff published a study [?] of the lattice of normal subgroups of the group of diffeomorphisms of an open manifold P that is the interior of a compact manifold with nonempty boundary. She referred to Ling’s preprint [?] and unpublished work of mine for the proof of a fundamental lemma stating that A 0 (P )/G i (in her notation, see [?] p. 353) is a simple group. Since the proof of this result and [?] have apparently never been published, I include its proof 2
here as Theorem ??. I also prove some new results for any open manifold M. All the results are valid for diffeomorphism groups of class C r , r = 1, 2, . . . , ∞ and also for homeomorphism groups (r = 0), except that r ≠ dim(M) + 1 in Theorem ?? since we use Mather’s theorem. I would like to thank Tomasz Rybicki for encouraging me to publish this paper after all these years. 2 Definitions and Results We recall some definitions from McDuff’s paper [?]. Let M be a connected n-manifold. For r = 1, 2, . . . , ∞, we set A = A(M) = Diff r (M), the group of all C r diffeomorphisms of M with the C r compact-open topology, and for r = 0 we let A = Homeo(M) be the group of homeomorphisms of M. Let A 0 be the identity component of A, so that it consists of the diffeomorphisms that are C r isotopic to the identity map I (or the homeomorphisms that are topologically isotopic to I, if r = 0). The support supp(g) of a diffeomorphism g ∈ A is the closure of the set {x ∈ M | g(x) ≠ x} and the support of an isotopy {g t | t ∈ [0, 1]} with g 0 = I is the closure of the union of the supports ∪ t∈[0,1] supp(g t ). All manifolds are connected. Theorem 2.1 Let h ∈ A 0 be a diffeomorphism isotopic to the identity map of an open (i.e., non-compact) manifold M such that supp(h) is not compact, and let H be any normal subgroup of A 0 containing h. Then the subgroup A I of A 0 consisting of all diffeomorphisms that are isotopic to the identity map of M by isotopies with compact support is contained in H. In the case that the manifold M is R n , we have the following stronger version of Theorem ??, using the same notation. Theorem 2.2 Let A = Diff r (R n ) be the group of all C r diffeomorphisms of R n , 0 ≤ r ≤ ∞. Then the subgroup of orientation preserving diffeomorphisms A + , which coincides with the subgroup A 0 of diffeomorphisms isotopic to I, and the groups κ −1 (η) where r ≥ 1 and η varies over all the subgroups of π 0 Diff r +(S n ) (or, equivalently, of θ n+1 , if n ≠ 4) are normal subgroups of A + and (except possibly if n = 4) also of A. If r ≠ n + 1, these are the only non-trivial normal subgroups of A 0 = A + and of A. For r = 0, the only proper normal subgroups of A 0 and A are A K = A I and A 0 . 3
Page 1: NORMAL SUBGROUPS OF DIFFEOMORPHISM
Page 5 and 6: Note that the diffeomorphism f is r
Page 7 and 8: Now, as in the Introduction, we con
Page 9 and 10: A 0 has index 2 in A and so it is a
Page 11 and 12: (straightening the corners, if nece
Page 13 and 14: commutes with T . By Proposition ??
Page 15: [12] W. Ling, Translations on M ×

NORMAL SUBGROUPS OF DIFFEOMORPHISM AND ...

Create successful ePaper yourself

Delete template?

Save as template?