Showing papers in "Annals of Mathematics in 1971"

The Spectrum of an Equivariant Cohomology Ring: II

Abstract: Let G be a compact Lie group (e.g., a finite group) and let HG= H*(BG, Z/pZ) be its mod p cohomology ring. One knows this ring is finitely generated, hence upon dividing out by the ideal of nilpotent elements it becomes a finitely generated commutative algebra over the field Z/pZ. It is the purpose of this series of papers to relate the invariants attached to such a ring by commutative algebra to the family of elementary abelian p-subgroups of G. For example we prove a conjecture of Atiyah and Swan to the effect that the Krull dimension of the ring equals the maximum rank of an elementary abelian p-subgroup. Another result, which will appear in part II, asserts that the minimal prime ideals of the ring are in one-one correspondence with the conjugacy classes of maximal elementary abelian p-subgroups. Actually the theorems of the series are formulated more generally for the equivariant cohomology ring of a G-space X, defined by the formula

Iterated Cohen Extensions and Souslin's Problem

Abstract: We can characterize the real line, up to order isomorphism, by the following list of properties: R is order complete, order dense, has no first or last elements, and contains a countable dense subset. (One shows first that the countable dense subset is order isomorphic to the rationals, Q, and then that the ordered set is isomorphic to the Dedekind completion of its dense subset.) Souslin raised the question as to whether the "countable dense subset" condition could be replaced by the following consequence [15]: (*) Every disjoint family of non-empty open intervals is countable.' We use SH (Souslin's Hypothesis) to denote the following proposition: Every order complete order dense linearly ordered set satisfying ( * ) contains a countable dense subset. We use ZFC to denote Zermelo-Fraenkel set theory (including the axiom of choice). In [16], Tennenbaum constructed models of ZFC in which SH is false. Moreover, in one of these models the continuum hypothesis (CH) is false, while in another one, the generalized continuum hypothesis (GCH) is true. Thus SH is independent of the usual axioms of set theory. (This result is due, independently, to Jech [7].)2

A structural stability theorem

Abstract: precise definition of structural stability given in ? 5 is somewhat stronger; it demands that the conjugacy 9q can be found within an arbitrary CO neighborhood of the identity when g is sufficiently close to f.) The idea of this definition is that qualitative properties of structurally stable diffeomorphisms are unchanged by small C' perturbations. The definition (or rather an analogous one for flows) was proposed by Andronov and Pontrjagin [2] in 1938. In this paper we prove that certain geometric conditions are sufficient for strutural stability. Our result is the following

On Fraisse's order type conjecture

Abstract: ing from the embeddability relation between order types, define a quasi-order to be a reflexive, transitive relation. Throughout this paper, the letters Q and R will range over quasi-ordered sets and classes. Various quasi-ordered spaces will be defined; in each case we will use the symbol i<(,, of members of Q, 9i, j < (o: i < j and qj ? qj, equivalently, (ii) every descending sequence of members of Q is finite, and every antichain of members of Q is finite. Thus, in these terms, the first of the two theorems listed in the introduction reads: the class OR is wqo under the embeddability relation. Well-quasi-orderings were first studied by Higman in [5], where the equivalence of the two definitions (immediate from Ramsey's theorem) was observed. If q E Q, let Qq ={r E Q: q S r}. From part (i) in the definition of wqo (which will be the version of wqo used from now on) we have immediately the following Induction principle for well-quasi-orderings: If a proposition F(Q) is true This content downloaded from 157.55.39.17 on Wed, 31 Aug 2016 04:08:56 UTC All use subject to http://about.jstor.org/terms

A topological characterisation of the affine plane as an algebraic variety

Abstract: This investigation arose from an attempt to answer the following question, raised by M. P. Murthy: If X is an affine algebraic variety over a field k such that X x A' e A3, where An denotes the affine n-space over k, is X isomorphic to A2? It was hoped that when k = C, just the hypotheses that X is a non-singular, affine, rational and contractible surface would imply that X C2. This, however, is not true as we show by a counter-example. We are unable to decide if the variety Y of this counter-example also satisfies the condition Y x C C3.** On the other hand, we do prove the following positive result.

On the Plancherel Formula and the Paley-Wiener Theorem for Spherical Functions on Semisimple Lie Groups

Abstract: One of the difficult points in the proof of Harish-Chandra's Plancherel formula for spherical functions on a semisimple Lie group is to show that an appropriate inversion formula exists for the Fourier transform f f, and that this inversion formula holds for sufficiently many functions in the space of spherical square-integrable functions on G. Briefly, let f be a square integrable spherical function on G for which the Fourier transform f(X)= 5f(x)cp_(x)dx is well-defined. Here, as in [5], qA is the elementary (zonal) G spherical function corresponding to the parameter X. The problem is to show that for f in a L2-dense subspace of square-integrable spherical functions, the following inversion formula holds:

Orbits of linear algebraic groups

Abstract: Let G be a linear algebraic group and let p: G GL(V) be a rational representation of G. When G is linearly reductive, D. Mumford has shown that if a point x C V has 0 in the Zariski-closure cl (Gs x) of its orbit, then there exists a one-parameter subgroup X: Gm G such that x(a) x 0 as a 0 (Theorem (4.1)). (See ? 2 for notation and definitions.) Suppose that G and p :are defined over a field k and that x C Vk. It has been conjectured by Mumford (based on a stronger conjecture of J. Tits see [13, p. 64]) that, when k is perfect, X can be chosen to be defined over k. More generally, one can ask when a linear algebraic k-group G has the following property:

On Automorphic Forms on GL 2 and Hecke Operators

Abstract: In [4], Jacquet and Langlands showed that if an irreducible unitary representation of the general linear group of degree 2 over the adele ring of an A-field' is an irreducible constituent of the representation on the space of cuspidal functions, then it is decomposable, i.e., it is a tensor product of irreducible representations of local components of the group. In the present paper, we prove the semi-simplicity of the algebra generated by some Hecke operators operating on a certain space of cusp forms. As a consequence of this, we show that two such irreducible constituents are equivalent if their local factors are equivalent for almost all places including all archimedean places. In more detail, let F be an A-field, and FA (resp. FAX) the adele ring (resp. the idele group) of F. We put GF= GL2(F) and GA = GL2(FA). We know that the center ZA of GA is isomorphic to FAX. For a unitary character X of the idele class group FA /Fs, let L'(GF\GA, X) denote the space of measurable functions on GA satisfying

Minimal Immersions of Spheres into Spheres

Abstract: In this paper we announce a qualitative description of an important class of closed n-dimensional submanifolds of the m-dimensional sphere, namely, those which locally minimize the n-area in the same way that geodesics minimize the arc length and are themselves locally n-spheres of constant radius r; those r that may appear are called admissible. It is known that for n = 2 each admissible r determines a unique element of the above class. The main result here is that for each n ≥ 3 and each admissible r ≥ [unk]8 there exists a continuum of distinct such submanifolds.

Duality and Subgroups

Abstract: The second author proved, in [27], a duality theorem for general locally compact groups as a generalization of the so-called Tannaka duality theorem for compact groups. In the proof, the regular representation plays an essential role. In order to clarify the role of the regular representation in duality theory, the first author gave, in [23], a characterization of group algebras in terms of a Hopf-von Neumann algebra and showed that the abelian von Neumann algebra L-(G) of all essentially bounded functions on a locally compact group G together with the co-multiplication 6, the involution j and the Haar measure f involves the duality principle for the given group G. In this paper we shall examine, by showing a correspondence between closed subgroups and subalgebras of the Hopf-von Neumann algebra {L-(G), a, j}, how this duality principle works for subgroups. For a locally compact abelian group G, the Pontryagin duality theorem shows that there is a beautiful and complete one-to-one correspondence between a closed subgroup H of G and a closed subgroup H I of the dual group G such that

Interpolation and Embedding in the Recursively Enumerable Degrees

Abstract: By a degree is meant a degree of recursive unsolvability. A degree is recursively enumerable (r.e.) just if it contains a recursively enumerable subset of N, the set of non-negative integers. Two infinite injury priority arguments are presented, in ? 2 and ? 3, which are generalizations of parts of Sacks' proof that the r.e. degrees are dense [9]. These results are in a form sufficiently flexible to admit a variety of applications. It is found, for example, that a degree which is a minimal r.e.-uniform upper bound for a sequence of degrees must be the join of a finite subsequence. By way of contrast, any non-recursive r.e. degree is a minimal upper bound for some strictly ascending sequence of r.e. degrees. It is also found that if a, b are r.e. and a < b, then any countable partial ordering is embeddable in the r.e. degrees between a and b with joins preserved whenever they exist. In ? 4 it is shown that if a' = 0' then there is always such an embedding which also preserves greatest and least elements when they exist. The proof of the latter result is a finite injury priority argument which is more closely related to two splitting theorems of Sacks [8, Th. 2 of ? 5 and Th. 2 of ? 61 than to the preceding results. Our basic notation follows Kleene [4], Kleene and Post [5], and Sacks [8]. A convenient property of Kleene's T-predicates is that if Tn(e, x1, * * * x ,nq y) or Tn(p e xi, ...,xny) holds then U(y)

On seminorms and probabilities, and abstract Wiener spaces

Abstract: In real Hilbert space H there is a finitely additive measure n on the ring of sets defined by finitely many linear conditions, which is analogous to the normal distribution in the finite-dimensional case. This has been examined from various points of view in Gelfand and Vilenkin [11], Gross [12], Segal [19], as well as by many earlier authors. Now, if the Hilbert space is “completed” in any of number of ways, this “cylinder set measure” extends to an actual Borel measure on the completed space. For example, if we define | x | T = P Tx P, where T is a Hilbert-Schmidt operator with trivial nullspace, then the completion of H with respect to the norm | • | T is such a completion. A more subtle example is the following. Let H be realized explicity as L 2(0, 1). Define a norm on H by \(|f|={\rm sup}_{0\leqq t \leqq 1}|\left|\int^{{t}}_{0}f\left(s\right)ds\right|.\) Then the completion of H with respect to this is isomorphic to the continuous functions on [0, 1] with suo norm and vanishing at zero, via the map which sends f to the function g whose value at t is \(\int^{{t}}_{0}f\left(s\right)ds, 0\leqq t \leqq 1.\) The finitely additive measure n is then realized as Wiener measure. Other natural norms give rise to realizations of n as Wiener measure on the Holder–continuous function, exponent α < α < 1/2. This sort of phenomenon has been investigated abstractly by L. Gross in [14]. He shows there (Theorem 4) that if H is completed with respect to any of his measurable seminorms, as defined in Gross [13], then n gives rise to a country additive Borel measure on the Banach space obtained frome H by means of the seminorm.

Formal Cohomology: III. Fixed Point Theorems

Abstract: Let (R, (wi)) be a complete discrete valuation ring of characteristic 0 with quotient field K and finite residue class field k = GF(q). In our previous papers, [1] and [2], (which we shall refer to as FC I and FC II), we constructed the formal cohomology groups, Hi(A; K), of certain smooth k algebras A and developed several of their basic properties. The Frobenius endomorphism F: X Xq of A operates on the groups H'(A; K). In this paper we shall prove a fixed point theorem for the action of F, which generalizes results of Dwork and Reich and leads via techniques of Dwork to a new proof of the rationality of the zeta function of a k-variety. Applying the same techniques to the study of Fod, where j is an automorphism of A of finite order, we shall obtain a new and rather simple proof of the rationality of the Artin L-functions. Let A be a w. c. f. g. algebra over (R, (wr)). (See FC I for terminology.) A Frobenius endomorphism of A is a lifting of F: A d A, where A = A/wA. Such liftings exist provided A is very smooth. Suppose then that A is very smooth, that A is pure n dimensional and that F is a fixed Frobenius endomorphism of A. By FC I, Th. 8.6 Cor. 1, p 216, F induces a bijection F* on H(A, K). In fact Corollary 2 of the same theorem constructs an operator A: D(A) D(A) such that A* = qn(Fh)on H(A; K). The main object of this paper is to show that qlfF~l behaves like a compact operator on H(A, K) and that the alternating sum of the traces of (qnfF;l)s on the Hi(A; K) is equal to the number of points of Spec A rational over GF(qs). (It is likely that the H'(A; K) are always finite dimensional but we have only been able to prove this in general for i < 1.) In outline the paper goes as follows. Section 1 introduces a class of linear operators on vector spaces over K, the "nuclear operators". The axiomatics here are essentially the Riesz-Serre decomposition theorem for compact endomorphisms of a Banach space. In particular, we are able to define the trace and characteristic power series of a nuclear operator. Section 2 studies "Dwork operators". If M is a finite A-module, a Dwork