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Showing papers in "Annals of Mathematics in 1977"





Journal ArticleDOI
TL;DR: In this paper, the authors studied the relation between the topological structure and the Riemannian structure of a complete, connected RiemANN manifold M of dimension n > 2.
Abstract: A basic problem in Riemannian geometry is the study of relations between the topological structure and the Riemannian structure of a complete, connected Riemannian manifold M of dimension n > 2. By a classical theorem of Myers [10] such a manifold is compact if the sectional curvature K of M satisfies K > a > O. More precisely, the diameter d(M) of M satisfies d(M) < zc/< 8 . After the pioneering work of Rauch [11] the following result, known as the sphere theorem, was proved first by Berger [1] in even dimensions and finally by Klingenberg [8] as stated.

298 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that the Adams-Novikov spectral sequence converges to the stable homotopy ring in a very specific way from periodic constituents, which can be described algebraically as the cohomology of the landweber-novikov algebra of stable operations in complex cobordism.
Abstract: The problem of understanding the stable homotopy ring has long been one of the touchstones of algebraic topology. Low dimensional computation has proceeded slowly and has given little insight into the general structure of 7ws(S0). In recent years, however, infinite families of elements of 7rs (S0) have been discovered, generalizing the image of the Whitehead J-homomorphism. In this work we indicate a general program for the detection and description of elements lying in such infinite families. This approach shows that every homotopy class is, in some attenuated sense, a member of such a family. For our algebraic grip on homotopy theory we shall employ S. P. Novikov's analogue of the Adams spectral sequence converging to the stable homotopy ring. Its E2-term can be described algebraically as the cohomology of the Landweber-Novikov algebra of stable operations in complex cobordism. In his seminal work on the subject, Novikov computed the first cohomology group and showed that it was canonically isomorphic to the image of J away from the prime 2. When localized at an odd prime p these elements occur only every 2(p 1) dimensions; so this first cohomology group has a periodic character. Our intention here is to show that the entire cohomology is built up in a very specific way from periodic constituents. Our central application of these ideas is the computation of the second cohomology group at odd primes. By virtue of the Adams-Novikov spectral sequence this information has a number of homotopy-theoretic consequences. The homotopy classes St, t > 1, in the p-component of the (2(p2 1)t 2(p 1) 2)-stem for p > 3, constructed by L. Smith, are detected here. Indeed, it turns out that all elements with Adams-Novikov filtration exactly 2 are closely related to the , family. The lowest dimensional elements of filtration 2 aside from the fi family itself are the elements denoted ej by Toda. The computation of the

277 citations




Journal ArticleDOI
TL;DR: In this paper, it was shown that the quasi-simple irreducible representations of a real Lie group whose Lie algebra is a real form of g define primitive ideals in U(g).
Abstract: Summary Let g be a semi-simple complex Lie algebra. We denote by U(g) its enveloping algebra. A (two-sided) ideal I of U(g) is called primitive if it is the kernel of an irreducible representation of U(g). For several reasons classifying such ideals is interesting. For instance, it is a well known result of Harish-Chandra that the quasi-simple irreducible representations of a real Lie group whose Lie algebra is a real form of g define irreducible representations of U(g), and thus, primitive ideals in U(g). The main result of this paper is the following.

176 citations



Journal ArticleDOI
TL;DR: In this paper, the moduli of degenerating families of conformally finite Riemann surfaces with signature may be studied using the Bers embedding T(G) of the Teichmuiller space.
Abstract: For several years it has been conjectured that the moduli of degenerating families of conformally finite Riemann surfaces with signature may be studied using the Bers embedding T(G) of the Teichmuiller space. In a previous paper ([2], see also Marden [17]) we examined a distinguished classthe regular b-groups-of Kleinian groups lying in AT(G). The augmented Teichmiiller space T(G) is the union of T(G) with the regular b-groups on its boundary.

147 citations


Journal ArticleDOI
TL;DR: The theory of regular singularities of differential equations with multiple variables has been studied in satisfactory form as discussed by the authors, although they appear in several important fields of mathematics and have become one of the most fundamental fields of analysis.
Abstract: The theory of ordinary differential equations with regular singularities has been well studied and has become one of the most fundamental fields of analysis. But the regular singularities of differential equations with several variables have not been studied in satisfactory form, although they appear in several important fields of mathematics. An example is the Laplacian of symmetric spaces of noncompact type. The typical one is the Laplacian

Journal ArticleDOI
TL;DR: In this paper, it was shown that the absolute neighborhood retracts (ANR's) are the image of a Q-manifold by a cell-like mapping if and only if whenever A is em-exponential.
Abstract: After Borsuk introduced the notion of absolute neighborhood retracts (ANR's) and J. H. C. Whitehead demonstrated that all ANR's have the homotopy types of cell-complexes [34], the question naturally arose as to whether compact (metric) ANR's must necessarily be homotopy-equivalent to finite cell-complexes. Borsuk expressly posed this conjecture in his address to the Amsterdam Congress in 1954 15], and over the ensuing years considerable progress was made (see [261), including (a) ANR's admitting "brick decompositions," by Borsuk [5], (b) the simply connected case, by de Lyra [21], (c) products with the circle, by M. Mather [22], (d) applications of Wall's obstruction to finiteness [29], (e) compact n-manifolds, by Kirby and Siebenmann [16], and (f) compact Hilbert cube manifolds and locally triangulable spaces, by Chapman [9]. The full problem remained open, however. In this paper the conjecture is settled positively by application of Hilbert cube manifold theory. Specifically, it is shown that each compact ANR is the image of some Hilbert cube manifold (Q-manifold) by a cell-like (CE) mapping. Such mappings, between ANR's, are always homotopy-equivalences [14], [17], [20], [28] (although for more general metric compacta they need not be shape-equivalences [27]), so that by appealing to (f) above, the conjecture is settled. In the process, a considerably stronger result is established, namely, that there exists a cell-like map f from a Q-manifold onto the ANR X whose mapping cylinder M(f) is itself a Q-manif old. The mapping cylinder collapse of M(f) to X provides a particularly nice CE-map of a Q-manifold to the ANR. The general outline of the proof is as follows: First, it is shown that the mapping cylinder of a CE-map from a Q-manifold to an ANR is always a Q-manifold. Second, this result is used to show that the ANR A is the image of a Q-manifold by a CE-mapping if and only if whenever A is em-


Journal ArticleDOI
TL;DR: In this article, it was shown that if K is a local field and V is a smooth K-variety, then V(K) is a K-analytic manifold.
Abstract: Let K be a field and A be a K-algebra. For a variety V, defined over K, we shall let V(A) denote the set of A-rational points of V. In case A is a locally compact topological ring, V(A) has a natural locally compact Hausdorff topology induced by the topology on A (see Weil [25: App. III]); in the sequel we shall assume V(A) endowed with this topology. If K is a local field (i.e., a non-discrete locally compact field) and V is a smooth K-variety, then V(K) is a K-analytic manifold. Moreover, when V is a K-group, V(K) is a K-analytic group. Let G be a connected semi-simple affine algebraic group defined, isotropic and almost simple over a local field K of arbitrary characteristic. Let G = G(K). Let G+ be the normal subgroup of G generated by the K-rational points of the unipotent radicals of parabolic K-subgroups of G. The object of this paper is to prove the following:

Journal ArticleDOI
TL;DR: In this paper, the Harish-Chandra modules associated to a connected semisimple Lie group G having finite center have been studied and structural information about the category of irreducible Harish chandra modules has been given.
Abstract: study of the category of Harish-Chandra modules over a real semisimple Lie algebra equipped with a Cartan involution. In this paper we prove some very general theorems about tensor products of finite dimensional modules with Harish-Chandra modules. These theorems yield new structural information about the category of Harish-Chandra modules, as well as new information on the classification, construction, and properties of the infinite dimensional irreducible Harish-Chandra modules. We study the Harish-Chandra modules associated to a connected semisimple Lie group G having finite center. We let q, be the Lie algebra of G,

Journal ArticleDOI
TL;DR: The Thompson group of G as mentioned in this paper is a Chevalley group of odd characteristic, M11, M12 or SP6(2), which is a generalization of the M11-M12-SP6-2 group.
Abstract: THEOREM I. Let G be a finite group with F*(G) simple. Let z be an involution in G and K a subnormal subgroup of CQ(z) such that K has nonabelian Sylow 2-subgroups and z is the unique involution in K. Assume for each 2-element k C K that kG n C(z) C N(K) and for each g e C(z) N(K), that [K, Kg]

Journal ArticleDOI
TL;DR: In this paper, an extensional set theoretic formalism B, a subsystem of Zermelo set theory based on intuitionistic logic, is introduced, which provides a set-theoretic foundation for constructive analysis.
Abstract: We introduce an extensional set theoretic formalism B, a subsystem of Zermelo set theory based on intuitionistic logic, which provides a set theoretic foundation for constructive analysis which is strikingly analogous to the usual set theoretic foundations for ordinary analysis. We prove that B has the same elementary (LL?) consequences as first order arithmetic, and that every arithmetic consequence of B is a consequence of Peano arithmetic. We indicate a definitional translation of B into an intensional theory of predicates, which in turn has a natural interpretation in the informal theory



Journal ArticleDOI
TL;DR: In this paper, it was shown that the homeomorphism group of a compact Q-manifold is an ANR and that the nerve of a suitably nice open cover of a complex is simple homotopy equivalent to the complex.
Abstract: In this paper, we prove that the homeomorphism group, H(M), of a compact Q-manifold is an ANR. Results of Geoghegan and Torunczyk then show that H(M) is an 12-manif old. As by-products of the proof, we obtain a CE approximation theorem for 12-manifolds, a Vietoris theorem for simple homotopy theory (generalizing the result that a CE map between complexes is simple), and a proof that the nerve of a suitably nice open cover of a complex is simple homotopy equivalent to the complex.


Journal ArticleDOI
TL;DR: In this article, the authors studied 9P(X) when X is an arbitrary polyhedron and showed that there exists a homotopy equivalence 9P (X x Q) -QWh(X), where Q denotes the Hilbert cube.
Abstract: map 9P(X) QWh(X) which is highly connected; and that, in general, there exists a homotopy equivalence 9P(X x Q) -- QWh(X) where Q denotes the Hilbert cube. The latter equivalence depends partly on results of T.A. Chapman [5]. In this paper we study 9P(X) when X is an arbitrary polyhedron. Our

Journal ArticleDOI
TL;DR: In this paper, it was shown that the geodesic and horocycle flows satisfy a simple commutation relation, and the proof of the latter is shown to be topologically mixing, with respect to the unique invariant Borel probability measure.
Abstract: Extensive work has been done in understanding the ergodic properties of the classical horocycle flow for a compact connected, orientable surface of constant negative curvature ([9], [10], [11], [15], [23]). Such a flow may be viewed as a one-parameter subgroup action of a compact homogeneous space. Many of the results are based on this observation; in particular the Lie group structure and group representations play an important role. The basic goal of our approach is to give completely different proofs (of these results) which are valid in the variable negative curvature case as well and are more dynamic in nature. Apparently, in this case one does not have the Lie group structure and cannot use representation theory. The spirit of our approach (which goes back to Hedlund) is to view horocycle flows as "expanded flows"-i.e., as continuous flows whose orbits are permuted and expanded by another flow; for horocycle flows, the "other" flow is the geodesic flow. In all of our proofs, we exploit the interdependence between these two flows. We first prove (in Section 3) that every continuous parametrization of a horocycle flow is topologically mixing; for this, we use the well-known minimality of the horocycle flow. Next (in Section 4) we prove that for every horocycle flow there is a large class of continuous reparametrizations which are measure theoretically mixing, with respect to the unique invariant Borel probability measure (uniqueness was established earlier ([9], [21])). For this, we use the same idea as in proving topological mixing, but instead of minimality exploit unique ergodicity of horocycle flows.* The class of parametrizations, for which we are able to prove measure theoretic mixing, includes the classical horocycle parametrizations, unit speed parametrizations and special uniformly expanding parametrizations (whose existence was proved earlier ([21])). With the latter parametrization the geodesic and horocycle flows satisfy a simple commutation relation; and the proof of An abstract of a preliminary version of this paper was published in AMS Notices, October, 1975 # 728-614. * Ergodicity would suffice, but using only this one does not get the stronger conclusionLemma 4.5.

Journal ArticleDOI
TL;DR: The first-order definability of the first order theory of arithmetic was shown to be undecidable by Jockusch and Soare as mentioned in this paper, where the jump operator was used to obtain a strong result that the firstorder theory is recursively isomorphic to the truth set of second-order arithmetic.
Abstract: Let be the semilattice of degrees of recursive unsolvability. The main result of this paper is that the first-order theory of is recursively isomorphic to the truth set of second-order arithmetic (Corollary 5.6). We also obtain a strong result concerning first-order definability in where j is the jump operator (Theorem 3.12). The structure of has been investigated strenuously by Kleene and Post [12], Spector [29], Sacks [20], Lerman [15] and a host of others. The first-order theory of has been commented upon from time to time by various authors including Jockusch and Soare [10], Miller and Martin [17], Rogers [19], Shoenfield [24], [26] and Stillwell [30]. Our main result can be regarded as a refinement of the theorem of Lachlan [13] that the first-order theory of is undecidable. Like Lachlan we use initial segments, but we combine them with the jump operator (Theorem 2.1). Our curiosity about the subject of this paper was first awakened in 1969 by Gerald E. Sacks who asked whether the first-order theory of is hyperarithmetical (see also Problem 70 in [5]). We are also grateful to Carl G. Jockusch, Jr. for timely expressions of interest in this work. We use c to denote the set of nonnegative integers {0, 1, 2, ... }. Letters such as i, j, k, m, n denote elements of co. We write 20 for the set of totally defined, {0, 1}-valued functions on w. Letters such as f, g, h denote elements of 20. We write f 0 g for the unique function h such that h(2n) = f(n) and h(2n + 1) = g(n) for all n e w. The jump of f e 20 is f* which is again an element of 2w. Finite iterates of * are defined by f(') = f and f(nPl) = (f (n))*


Journal ArticleDOI
TL;DR: In this article, a new construction of the P(A)2 Euclidean (quantum) field theory was given, and a structure analysis of this theory was proposed, where the strong Gibbs variational equality holds, for all states constructed so far for a given P.
Abstract: We give a new construction of the P(A)2 Euclidean (quantum) field theory and propose a structure analysis of this theory. Among our results are: (1) For any polynomial P bounded from below, we construct two Euclidean states (expectations) P,?, not necessarily distinct, which satisfy all Osterwalder-Schrader axioms including clustering and obey the DobrushinLanford-Ruelle (DLR) equations for P. (2) Equality p,+ = , - holds if and only if the pressure awe) corresponding to the polynomial P(x) - px is differentiable at e = 0, and in this case the state p,+ is independent of a large class of different (in particular classical) boundary conditions. (3) All P(0)2 expectations thus far constructed are locally absolutely continuous with respect to the free field Gaussian expectations with LI RadonNikodym derivatives, for all p < oo. (4) The strong Gibbs variational equality holds, for all states constructed so far for a given P.

Journal ArticleDOI
TL;DR: In this paper, it was shown that every element of K not in the field of rational functions has a unique infinite continued fraction expansion f = a. + 1/(al + 2/(a2 + * * * )), or more briefly f = [a0; a, a, a2, * *.
Abstract: 1. Statement of results We let F be the finite field with two elements, and let K be the field F((x')) of all formal power series in x-' over F. Every element of K not in the field of rational functions F(x) has a unique infinite continued fraction expansion f = a. + 1/(al + 1/(a2 + * * *)), or more briefly f = [a0; a,, a2, * *.* ],

Journal ArticleDOI
TL;DR: In this article, it was shown that if there exists a topological isomorphism v: G1 G2 such that U(gJ = rgl 1 for every g, e G1.
Abstract: THEOREM. If there exists a topological isomorphism v: G1 G2, there corresponds a unique isomorphism of fields z: Q, I Q2 such that U(gJ = rgl 1 for every g, e G1. An analogous theorem for algebraic number fields was proved in [6], though Q1 = Q2 was assumed there. One-to-one correspondence of prime divisors is essentially due to Neukirch [3], [4]. Most of the arguments in Sections 1 and 2 are also valid in the number field case.