Showing papers on "Equivariant map published in 1979"

Transformation groups and representation theory

Abstract: The Burnside ring of finite G-sets.- The J-homomorphism and quadratic forms.- ?-rings.- Permutation representations.- The Burnside-ring of a compact Lie group.- Induction theory.- Equivariant homology and cohomology.- Equivariant homotopy theory.- Homotopy equivalent group representations.- Geometric modules over the Burnside ring.- Homotopy-equivalent stable G-vector bundles.

Monotone Equivariant Cluster Methods

Abstract: Monotone equivariant cluster methods are characterized within a certain order theoretic model for the subject. These methods are shown necessarily to be graph theoretic in nature, and the results used to construct a number of new agglomerative monotone equivariant techniques—some of them proving useful in connection with applications to biology, geology, and sociology. In that all of the techniques are intermediate between single linkage and complete linkage clustering, a characterization of the latter method is provided to be considered along with the well known characterization of the former method. Some of the difficulties inherent in obtaining a numerical measure of the optimality of a monotone equivariant cluster method are also examined.

An equivariant Wall obstruction theory

Abstract: Let G be a finite group. For a certain class of CW-complexes with a G-action which are equivariantly dominated by a finite complex we define algebraic invariants to decide when the space is equivariantly homotopy or homology equivalent to a finite complex.

Equivariant concordance of invariant knots

Abstract: The classification of equivariant concordance classes of high-dimensional codimension two knots invariant under a cyclic action, T, of order m has previously been reported on by Cappell and Shaneson [CS2]. They give an algebraic solution in terms of their algebraic K-theoretic F-groups. This work gives an alternative description by generalizing the well-known Seifert linking forms of knot theory to the equivariant case. This allows explicit algorithmic computations by means of the procedures and invariants of algebraic number theory (see the subsequent work [St], particularly Theorem 6.13). Following Levine [13], we define bilinear forms on the middle-dimensional homology of an equivariant Seifert surface B,(x, y) = L(x, i+(T;y)), for i = 1, . . ., m. Our first result (2.5) is that an invariant knot is equivariantly concordant to an invariant trivial knot if and only if there is a subspace of half the rank on which the B, vanish simultaneously. We then introduce the concepts of equivariant isometric structure and algebraic concordance which mirror the preceding geometric ideas. The resulting equivalence classes form a group under direct sum which has infinitely many elements of each of the possible orders (two, four and infinite), at least for odd periods. The central computation (3.4) gives an isomorphism of the equivariant concordance group with the subgroup of the algebraic knot concordance group whose Alexander polynomial, A, satisfies the classical relation 117. A(X9i)l = 1, where X is a primitive mth root of unity. This condition assures that the m-fold cover of the knot complement is also a homology circle, permitting the geometric realization of each equivariant isometric structure. Finally, we make an explicit computation of the Browder-Livesay desuspension invariant for knots invariant under an involution and also elucidate the connection of our methods with the results of [CS2] by explicitly describing a homomorphism from the group of equivariant isometric structures to the appropriate F-group. Introduction. The initial impetus and inspirational origins for this paper are found in the work of Santiago Lopez de Medrano on codimension two knots invariant under involutions [LdM1]-[LdM3] and in that of William Browder on homotopy lens spaces, particularly in the philosophical directions indicated in the last paragraphs of the introduction to his paper in the Proceedings of the 1969 Georgia Conference on the Topology of Manifolds [Br3]. Also very influential in the development were the works of H. Seifert [Sei], R. Received by the editors August 10, 1976. AMS (MOS) subject classifications (1970). Primary 57C45, 57D40, 57D85.

Some equivariant bordism theories vanish

Abstract: We show that in equivariant bordism theory using families of slice types it is possible for the theory to vanish. In fact we shall describe, for each finite abelian group G, families ~ of G slice types so that the theory G ~9 l , [ ~ ] is zero. In other words all G manifolds of type ~ are G boundaries. There are, in general, many such families. We shall describe one ~&(G) which has an interesting corollary. Let G 2 be the subgroup of G consisting of the identity and order 2 elements. If H is a subgroup of G such that H~G2#: G 2 then all G slice types with isotropy subgroup H belong to this family ~(G). It follows, in particular, that if M is a G manifold for which the subgroup G 2 acts without fixed points then M is a G boundary. This is a generalization of a result of Stong [4] which has been obtained by Khare in [1]. Further generalizations will be given later on (Lemmas 6 and 7) where in particular we obtain an unoriented version of a result of Ossa [3]. A precise formulation of the results will be given in subsequent sections. The proof of the main result uses ideas from the authors book [2], in particular generalizations of results in Sect. 4.5 of [2]. The main result (Theorem 1) of this paper makes sense in the case when G is a compact abelian lie group, the proof presented here does however have to be modified somewhat. We leave details for the interested.

Deformations of Holomorphic Seifert Fiber Spaces

Abstract: Let M be a holomorphic Seifert fiber space as constructed in Conner and Raymond [3] and let O M be the sheaf of germs of holomorphic vector fields on M. The purpose of this paper is to study the cohomology group H I(M, OM) and to describe the small deformations of M. Since M is given as a quotient of a principal torus bundle, it would be natural to use the cohomology of groups with coefficients in sheaves with operators to describe M and objects associated to it. These groups were introduced in Grothendieck [4] and also in Kodaira [7], Conner [2] and Conner and Raymond [3] by different approaches. In Sect. 1, we review these groups. Using the equivariant version of the Leray spectral sequence obtained in Sect. 1, we give a decomposition of H~(M, OM) into the direct sum of complex vector groups C, E, F and H in Sect. 2. In Sect's. 3 through 6, we study the geometric significance of the groups C, E, F and H. In Sect. 7, we compute these vector groups when the dimension of the base space is one and give some examples. It is possible to deal with r fibrations in a similar manner. It will be done in a forthcoming paper [14]. I would like to thank F. Raymond for many valuable conversations.

A converse of the Borel formula

Abstract: When an elementary Abelian p-group acts on a Zp-homology sphere (p a prime), it is known that the Borel formula must hold. Here we ask that the Borel formula hold and determine how this restricts, homologically, the type of space which can occur, assuming spherical fixed sets and connectedness. This is done by constructing a linear model of the action and an equivariant map to the model, the mapping cone of which yields certain homological information.

Homotopy classification of equivariant regular sections

Abstract: In his paper [2], Bierstone proves the equivariant Gromov theorem which is an integrability theorem for “open regularity condition” of equivariant sections of a smooth G-fibre bundle under the assumption that all orbit bundles of base manifold are non-closed. Here, we prove the result without his assumption under a nice “open regularity condition” which we call “G-extensible”.

Bayes and Equivariant Estimators of the Variance of a Finite Population

Abstract: : The problem of estimating the variance of a finite population is studied in a Bayesian framework. On the basis of the modern theoretical approach to sampling from finite populations and the special structure of the likelihood functions Bayes estimators of the population variance are derived. The structure of equivariant estimators is analyzed and Bayes equivariant estimators in the strict and the generalized sense are derived. Posterior and prior efficiency of the estimators is discussed. (Author)

Infinitesimal conditions for the equivariance of morphisms of fibered manifolds

Abstract: We generalize the usual definition of the Lie derivative to the case of a morphism between fibered manifolds which does not necessarily preserve the base. We prove that the vanishing of the Lie derivatives is a necessary and sufficient condition for the equivariance of a morphism of fibered manifolds under the action of a connected Lie group. 0. Introduction. In many problems in mathematical physics one considers a 'field', in effect, a section of a fibered manifold (which is very often a vector bundle) over a base manifold representing space-time, and a group of transformations of the base. To know whether the field is physically meaningful, one asks the classical question, whether the group of transformations is a 'symmetry group' (in alternate terminology, an 'invariance group') of the field, that is, does there exist a lifting of the action of the group on the base manifold to an action on the fibered manifold by automorphisms (if the fibered manifold is a vector bundle one generally requires that the group act by vector-bundle automorphisms) such that the section under consideration be invariant under that action of the group? More generally, one can ask the same question regarding a differential operator from one fibered manifold to another over the same base manifold. One must consider liftings of the group action to the two fibered manifolds: if there exist liftings such that the differential operator is equivariant with respect to the two lifted actions, the group is called a 'symmetry group' (or an 'invariance group') of the differential operator or of the associated system of partial differential equations. (For scalar differential operators or, more generally, for differential operators on tensor bundles, the word 'invariant' is usually reserved for 'invariant under the natural liftings' while the word 'covariant' can be used in the other cases.) In general, one assumes that the group is a Lie group, considers its corresponding infinitesimal transformations, and determines whether the section or the differential operator is 'infinitesimally invariant'. In the case of a section that means, does there exist a lifting of the Lie algebra of vector fields on the base to a Lie algebra of infinitesimal automorphisms of the fibered manifold such that the section is infinitesimally invariant, i.e., such that its Lie derivatives with respect to those vector fields defined by this lifting vanish? Received by the editors March 6, 1978 and, in revised form, June 13, 1978. AMS (MOS) subject classifications (1970). Primary 53A55; Secondary 50A15, 22A60.

On equivariant holomorphic imbeddings of Siegel domains to compact complex homogeneous spaces

Abstract: L e t D be a S iegel domain of the second k in d . I f A ut (D ) is so "small" that A ut(D ) = A f f (D ) , the domain D ca n b e equivariantly imbedded as an open set of complex projective space. In the case where A u t(D ) is the "largest", i.e., the domain D is symmetric, D can be also equivariantly imbedded as an open set of the hermitian symmetric space of compact type dual to D. Therefore it is natural to ask whether there exists an equivariant open imbedding o f a S iege l domain D to a compact complex homogeneous space M . In this paper, w e shall prove the following: (a) I f there exists an open equiv arian t imbedding o f D to M , then M must be a h erm itian symmetric space of com pact type (Theorem 8). (b) D can be equiv ariantly imbedded as an open set of P' (C ) if and only i f A u t(D ) = A f f (D ) o r D is holom orphically equ iva len t to a disk (Theorem 9). (c) There exists a Siegel domain which does not admit open equivarian t im bedd ings to compact complex homogeneous spaces (§ 6). Throughout this paper, w e use the fo llow ing notations: A ut (M ) means the group o f a ll holomorphic transformations of a complex manifold M . For a real vector space o r a rea l L ie algebra V , V denotes its complexification. W e denote by Gr (W , r) th e complex grassmann manifold consisting o f all r-dimensional subspaces o f a complex vector space W.

Theory of estimation of parameters in a linear regression scheme

Abstract: Fairly simple asymptotically optimal equivariant polynomial estimators of any degree k are constructed for the parameters of a standard linear regression scheme whose design matrix satisfies a certain additional condition. These estimators depend on the error distribution function only in terms of its first 2K moments. An explicit equation for an optimal equivariant quadratic estimator of parameters is also presented.

The equivariant surgery problem for involutions

Abstract: This paper describes a complete solution of the Z2-surgery problem where the dimension restrictions are less stringent then in [11] We have to investigate intersections of the fixed point set with spheres on which we have to do surgery This means that the algebra involved no longer need to be quadratic Because of this we have to define a new Wittgroup and compute it