An old question of P. A. Smith asks: If a finite group G acts smoothly on a closed homotopy sphere Σ with fixed set ΣG consisting of two points p and q, are the tangential representations Tp Σ and Tq Σ of G at p and q equal? Put another way: Describe the representations (V, W) of G which occur as (Tp ΣTq Σ) for Σ a sphere with smooth action of G and ΣG = p ∪ q. Under these conditions we say V and W are Smith equivalent (21) and write V ~ W. A stronger equivalence relation is also interesting. We say representations V and W are s-Smith equivalent if (V, W) = (Tp Σ, Tq Σ) and Σ is a semi-linear G sphere (23), i.e. ΣK is a homotopy sphere for all K and ΣG = p ∪ q. In this case we write V ≈ W.

One of the principal problems in low-dimensional topology concerns the question of whether all slice knots are ribbon(3),(4). It will be shown in this paper that this problem is closely related to geometric properties of discs properly embedded in the unit 4-ball in Euclidean space. The main result, Theorem 1·13, states that a knot is ribbon if and only if a representative within its isotopy class bounds an embedded minimal disc, and that this in turn happens if and only if a representative bounds a disc of sufficiently small total curvature.

Let G be an abelian discrete group, A a unital C*-algebra and an action of G on A, i.e. (A, G,) is a C*-dynamical system. Let K denote the kernel ker of and put R = G/K. The main purpose of this article is to determine the roles of K and R in the crossed product G A. This goal is achieved in Section 2, where we prove that G A is *-isomorphic to a twisted crossed product of R with C*(K) A with respect to the action 1 and a 2-cocycle related to the 2-cocycle determined by the extension G of R by K. Here is the obvious action of R on A.

Every finitary algebra is, of course, a topological algebra in the discrete topology. On the other hand, a topological Boolean σ-algebra is indiscrete; sequences (0, 0, …, x, x,…) with join x converge to (0, 0,…) with join 0, so 0 is in {x}−, and similarly x is in {0}−. For lattices with infinite joins but only finite meets, such as topologies, the matter is more interesting.

In this paper we attempt to determine the mod p cohomology rings of the extra special p-groups for an odd prime p. Quillen has calculated the mod 2 case [8]. In our case, its structure seems to be very complicated [6]. It seems reasonable that we consider the variety defined by the cohomology ring of even degree according to Serre [11]. The main result of this paper (Theorem 5.3) gives a subalgebra of the cohomology ring such that the inclusion homomorphism induces a homeomorphism between the varieties.

Let T be a measure space and m a finite measure on T. The space of p-Bochner integrable functions defined on T with values in a Banach space X is denoted by Lp(T, X). It is well known (1) that Lp (T, X) is a Banach space under the norm

A subspace E in a Banachh space F is said to be proximinal if for each xF there is at least one xE such that

The element y is called best approximant of x in E.

In this paper all 3-manifolds will be supposed to be compact, connected, oriented and without 2-spheres in the boundary.

Given a 3-manifold M we obtain a closed pseudomanifold M^ by capping off each boundary component of M with a cone. We prove that such an M^ is a covering of S3 branched over a subcomplex G of S3 which is independent of M, and such that S3 - G has free fundamental group on two generators. Hence M^ (and also M) can be represented by a transitive pair {σ, τ} of permutations in the symmetric group Σh on the set {1,2, …, h}, for some h. We show how to obtain {σ, τ} from a given Heegaard diagram of M.

In the previous papers in this series (see (5) and (6)) we used Brown-Peterson theory to obtain necessary conditions for the mod p cohomology of finite H-spaces to be U(M) algebras. In this paper we will use connective K-theory to strengthen the main results of (6). We should mention that we will be assuming a general knowledge of the previous two papers from this series. The definitions and notation from these papers will be used freely. (In particular, the reader is directed to § 2 of (5) for the definition of a U(M) algebra.)

A normed space E is said to be series immersed in a Banach space X if for every absolutely divergent series nxn in E there is a continuous linear mapping T from E into X such that nTxn diverges absolutely. The theorem of Dvoretzky and Rogers(1) implies that a normed space E is series immersed in a finite dimensional space if and only if E itself is finite dimensional. In (4) and (9) it was shown that E is series immersed in lp 1 p < if and only if E is isomorphic to a subspace of Lp() for some measure . In particular, E is isomorphic to an inner product space if and only if it is series immersed in a Hilbert space. The property of series immersion was further studied in the papers (7) and (8). The main results in these two papers are conditions on X under which E series immersed in X would imply E locally immersed in X, a condition slightly stronger (formally) than E finitely representable in X.

A linear connection on a differentiable manifold M defines a horizontal distribution on the tangent bundle T(M). Horizontal distributions on tangent bundles are of some interest even when they are not generated by connections. Much of linear connection theory generalizes to arbitrary horizontal distributions. In particular, there are generalized versions of Bianchi's two identities for the torsion T and curvature R

where C denotes the cyclic sum with respect to X, Y and Z. These identities are derived below by associating with a horizontal distribution a graded derivation of degree 1 in a graded Lie algebra of vertical forms on M. This approach reveals the fundamentally algebraic origin of the Bianchi identities.

In this paper we formulate and prove generalizations of a theorem of Lin [7]. Let X be a CW complex with base point x0. Define a free involution T on S∞×(X Λ X) by T (w, xΛy) = (−w, yΛx). The quadratic construction on X is the complex

This construction can be applied to spectra. A complete and thorough account will appear in the work on equivariant stable homotopy theory in preparation by L. G. Lewis, J. P. May, J. McLure and M. Steinberger. Some of the results are announced in [8].

In recent years there has been considerable progress in the study of certain linear maps of C*-algebras which preserve the natural partial ordering. The most tractable such maps, the completely positive ones, have proved to be of great importance in the structure theory of C*-algebras(4). However general positive (order-preserving) linear maps are (at present) very intractable. For example, there is no algebraic formula which enables one to construct a general positive map, even on the algebra of 3 3 complex matrices. It is therefore of interest to study conditions stronger than positivity, but weaker than complete positivity.

Let B denote the space of bounded measurable functions with period 2π. We will suppose throughout that f(x) ∈ B. All norms considered are essential sup norms. Let the Fourier series of f(x) be given by

Let D = (dnk) (n, k = 0, 1, …) be an infinite matrix.

Let Ln(f; x) be the D transform of the Fourier series of f(t) at t = x, i.e.

where Sk(x) = A0(x) + A1(x) + … + Ak(x). Let us write

In (4) a space C of generalized functions was defined which is rather larger than the simple space used to such effect by Lighthill in (3). At the core of C is the space C0 = T of test functions. These are entire (complex) functions f such that all derivatives of f and its Fourier transform F have order of magnitude not exceeding as x → ± ∞, where c is a positive number depending on the individual derivative concerned. If f, g∈ T, the inner product 〈f | g〉 is defined to be

Let be a maximal measure on a compact convex set K, K* be the state space of the space of all continuous functions f: KK ℝ which are affine in the first variable, 1 be the -algebra on K generated by the Baire sets and the compact extremal subsets of K, and = {BeK1}. Then

(i) For any fixed continuous function g:K ℝ and -almost all x in K, there is a closed face of K containing x on which g is constant.

(ii) The image of under the map :KK* defined by f, (x) = f(x, x) is the unique maximal measure on K* representing its barycentre

(iii) induces a measure on (eK) satisfying certain regularity conditions.

In [2, 4, 5, 6, 7] Calabi, Barbosa and Chern showedthat there is a 2:1 correspondence between arbitrary pairs of full isotropic (terminology as in [8]) harmonic maps ±φ:M→S2m from a Riemann surface to a Euclidean sphere and full totally isotropic holomorphic maps f:M→2m from the surface to complex projective space. In this paper we show, very explicitly, how to construct a similar one-to-one correspondence when S2m is replaced by some other space forms of positive and negative curvatures with their standard (indefinite) metrics obtained by restricting a standard (indefinite) bilinear form on Euclidean space to the tangent spaces. We get over a difficulty encountered by Barbosa of dealing with the zeros of a certain wedge product by a technique adapted from [8]. The case of complex projective space forms (indefinite complex projective and complex hyperbolic spaces) will be considered in a separate paper. Some further developments in classification theorems are given by Eells and Wood [8], Rawnsley[14], [15] and Erdem and Wood [10].

In [8], Fialkow classified Einstein hypersurfaces in indefinite space forms, when the Weingarten endomorphism is diagonalizable. Recently, [13]–[15], Magid completed this work when such endomorphism is not diagonalizable. On the other hand, Smyth [17], classified complex Einstein hypersurfaces in complex space forms. Since Barros & Romero, [1], have made a systematic study of indefinite Kāhlerian manifolds, in particular, indefinite complex space forms, we investigate here complex Einstein non-degenerate hypersurfaces in indefinite complex space forms. In our case, we cannot always diagonalize the Weingarten endomorphism Aassociated to a unit normal field because it is a self-adjoint endomorphism in an indefinite metric vector space, but we prove that, if A is not diagonalizable, one has either A2 = 0 and A ≠ 0 or A2 = −γ2I, γε. We obtain in Theorems 4.4 and 5.5 a full classification except in the case that the ambient space is flat and the Weingarten endomorphism A satisfies A2 = 0and A ≠0. Our classification is based on extending Smyth's results to show that, except in this last case, an Einstein hypersurface is locally symmetric. Using expressions for the curvature tensor to estimate the dimension of the isotropy group, together with Berger's list of symmetric spaces, [3], we then determine the possible Einstein hypersurfaces.

Motivated by problems in the analysis of spatial data, central limit theorems are developed for U-statistics whose kernels depend on the size of the observed sample. These theorems are then applied to obtain results for the interpoint distance statistic and the large angle statistic.

Infinite particle systems on lattices have been extensively studied in recent years. The main questions of interest concern the ergodic and limiting behaviour of these processes, and their relationship with the dimension of the underlying lattice. A comprehensive review is given by Durrett(6).

One of the more tractable of these processes is the voter model introduced by Clifford and Sudbury(3) and much studied since, see for example the monograph by Griffeath(8), or the papers by Harris(11), Holley and Liggett(13), Bramson and Griffeath(1) and (2) or, for a more general approach, Kelly(16).

In this paper we consider the case where the underlying spatial structure is finite and examine the transient behaviour of the voter process and also the infection process introduced by Williams and Bjerknes(21).

In this paper we consider the equilibrium configurations of a homogeneous, incompressible, isotropic elastic body subjected to a uniform dead load surface traction of magnitude T whose direction is normal to the surface of the body in the reference configuration, and to no other forces. We concentrate on homogeneous equilibrium solutions, that is those for which the deformation gradient F is constant, and we study their bifurcations and stability (with respect to an appropriate static criterion) as T varies. Since it turns out that the equations for homogeneous equilibrium solutions, and the stability properties that we consider of these solutions, are independent of the shape of the body in the reference configuration, we can suppose if desired that this shape is a cube. (See Fig. 1.1.)