Showing papers on "Equivariant map published in 1984"

Equivariant stable homotopy and Segal's Burnside ring conjecture

Abstract: Analogous results were proved later for KO, in the generality of compact Lie groups, by Atiyah and Segal [8], and for KFq, the algebraic K-theory spectrum associated to the finite field Fq, by Rector [29] using Quillen's [26] computation of iT*(KFq). In each case, the answer involves an appropriate completed representation ring of G, and the cohomology theory in question is constructed from the permutative category of finite dimensional vector spaces over a field (see [31]). If one considers cohomology theories constructed from other permutative categories, one expects to find analogous computations in terms of a "completed representation ring" of G in the given category, appropriately defined. In particular, stable cohomotopy, Ts*, is constructed from the category of finite sets [31], and for this category the analogue of the representation ring is a well-known object, the Burnside ring A(G) [13]. A(G) is a commutative ring with augmentation, so one may speak of A(G), the completed Burnside ring. Moreover, there is

A Survey of Trace Forms of Algebraic Number Fields

Abstract: Every finite separable field extension F/K carries a canonical inner product, given by trace(xy). This symmetric K-bilinear form is the trace form of F/K.When F is an algebraic number field and K is the field Q of rational numbers, the trace form goes back at least 100 years to Hermite and Sylvester. These notes present the first systematic treatment of the trace form as an object in its own right. Chapter I discusses the trace form of F/Q up to Witt equivalence in the Witt ring W(Q). Special attention is paid to the Witt classes arising from normal extensions F/Q. Chapter II contains a detailed analysis of trace forms over p-adic fields. These local results are applied in Chapter III to prove that a Witt class X in W(Q) is represented by the trace form of an extension F/Q if and only if X has non-negative signature. Chapter IV discusses integral trace forms, obtained by restricting the trace form of F/Q to the ring of algebraic integers in F. When F/Q is normal, the Galois group acts as a group of isometries of the integral trace form. It is proved that when F/Q is normal of prime degree, the integral form is determined up to equivariant integral equivalence by the discriminant of F alone. Chapter V discusses the equivariant Witt theory of trace forms of normal extensions F/Q and Chapter VI relates the trace form of F/Q to questions of ramification in F. These notes were written in an effort to identify central problems. There are many open problems listed in the text. An introduction to Witt theory is included and illustrative examples are discussed throughout.

Prerequisites (on equivariant stable homotopy) for Carlssons's lecture

Abstract: Cambridge CB2 ISB ENGLAND §i. Introduction. Three things might be done to help those who wish to understand Carlsson's work on Seqal's Burnside Ring Conjecture [8]. First, one might attempt a general exposition about Seqal's Burnside Ring Conjecture, both in its non-equivariant and in its equivariant forms. Secondly, one might explain the results which Gunawardena, Miller and myself have obtained by calculation for the case G = (Zp) n. (This is relevant because Carlsson uses these results.) Thirdly, one might attempt a general introduction to equivariant stable homotopy. In the lecture I gave in Aarhus, I tried to say something on all three topics, but for lack of time I was forced to omit an important part of what I had prepared. In this published text I shall omit the first and second topics, and try to do better justice to the third.

Equivariant triviality theorems for hilbert c*-modules

Abstract: The purpose of this paper is to give an exposition of the various triviality theorems, the equivariant version of a result due to L. Brown, and a simplification of the proof of Kasparov's triviality theorems.

Structural stability of equivariant vector fields on two-manifolds

Abstract: A class of vector fields on two-dimensional manifolds equivariant under the action of a compact Lie group is defined. Properties of openness, structural ability, and density are proved. Introduction and statement of results. Let G be a compact Lie group acting smoothly on a smooth compact connected two-dimensional manifold M. In this paper we define a subset of the space %rG(M) of C, r > 1, equivariant vector fields on M which correspond to the Morse-Smale vector fields studied by Peixoto, Palis, Smale [17,11,15] and others (see [13]). We prove that this subset of equivariant vector fields, which we call G-Morse-Smale vector fields, is open, and each GMorse-Smale vector field X is equivariantly structurally stable in £rc(M). That is, if Y E ?irc(M) is near X then there exists an equivariant homeomorphism of M that sends trajectories of X into trajectories of Y. Also, we prove that it is dense in £rc(M), with some exceptions; the most important exception corresponds to the fact that the density of the Morse-Smale vector fields for nonorientable 2-manifolds in the Cr topology, with r s* 2, is still an open question (see Gutierrez [6]). Our results generalize those of Peixoto [17]. The definition of G-Morse-Smale vector fields given here is different from that given by Field [3,5] in that it allows the presence of graphs in the nonwandering set. In this way, we have enough vector fields to get density, along with structural stability. However, in higher dimensions the situation is unclear since no results on structural stability have been proved so far. We believe that the concept of "modulus of stability" (Melo [10],Palis [12]) seems to be more appropriate to the equivariant framework. Examples of equivariant vector fields on 3-manifolds with modulus of stability equal to one are given in [14]. In fact, this phenomenom appears even locally. A G-equivariant vector field X on M defines a G X R action on M (see §1). A critical element of X is a compact G X R orbit. We require that it be normally hyperbolic. Of course, normal hyperbolicity does not imply local structural stability, but it may provide modulus of stability. We may impose an additional condition on the G-action (see §§1,2), in order to get local Received by the editors February 24, 1982 and, in revised form, May 24, 1983. 1980 Mathematics Subject Classification. Primary 58F10; Secondary 57E1S.

Total minimality of the unitary groups

Abstract: Dierolf and Schwanengel ([-7]) showed that if X is an infinite diescrete space, then the group F(X) of all bijections f : X ~ X provided with the topology of pointwise convergence is a totally minimal topological group. That was the first example of a (totally) minimal group which is not precompact. Other examples of such groups can be found in [8] (see also [21, w 2]). Infinite groups which does not admit non-discrete Hausdorff group topologies were constructed first by Shelah ([22], assuming CH), and then by Hesse ([14]), and A. Ol'ghanski~ (who noted that a quotient of the Adian's group A(m,n) has the property in question, cf. [1, w Note that all the examples mentioned above are examples of non-Abelian groups. Prodanov ([19]) established the totally minimal Abelian groups are precompact, and recently Prodanov and the author ([20]) proved that all minimal Abelian groups are precompact. The main purpose of the present paper is to show that the unitary group of every real or complex Hilbert space provided with the strong operator topology is a totally minimal topological group. This result gives an affirmative answer to a question posed by I. Prodanov. In Sects. 2 and 3 we study the equivariant (with respect to the action of the unitary group) compactifications of the unit sphere S of an infinite-dimensional Hilbert space. It is shown in Sect. 2 that the unit ball endowed with the weak topology is the greatest equivariant compactification of S. This fact is used in Sect. 3 to describe all equivariant compactification of S. The main theorem is proved in Sect. 4. The proof uses the scheme of the proof of [7, (1)], a generalization of which is discussed in Proposition 4.6.

Bordism of Diffeomorphisms and Related Topics

Abstract: Bordism groups of orientation preserving diffeomorphisms -- Report about equivariant Witt groups -- The isometric structure of a diffeomorphism -- The mapping torus of a diffeomorphism -- Fibrations over S1 within their bordism class and the computation of ?* -- Addition and subtraction of handles -- Proof of Theorem 5.5 in the odd-dimensional case -- Proof of Theorem 5.5 in the even-dimensional case -- Bordism of diffeomorphisms on manifolds with additional normal structures like Spin-, unitary structures or framings; orientation reversing diffeomorphisms and the unoriented case -- Application to SK-groups -- Miscellaneous results: Ring structure, generators, relation to the inertia group.

2-sphere bundles over compact surfaces

Abstract: Closed 4-manifolds which fiber over a compact surface with fiber a sphere are classified, and the fibration is shown to be unique (up to diffeomorphism). It is well known that there are at most two orientable 4-manifolds which fiber over a given compact surface with fiber the 2-sphere S2. (There is exactly one if the surface has nonempty boundary, and two if it is closed.) If the orientability condition is dropped, then the situation becomes more involved. In particular the (mod 2) intersection pairing is no longer sufficient to distinguish among the mani- folds that arise. One must also consider the ?Tl-action on g2 and the peripheral structure. The purpose of this note is to classify all 4-manifolds (orientable or not) which are total spaces of S2-bundles over compact surfaces. We shall work in the smooth category. Since Diff(S2) deformation retracts to 0(3), we may assume that all bundles that arise have 0(3) as structure group. Along the way it is shown that the bundle structures are unique. That is, if any two 4-manifolds, fibered as above, are diffeomorphic, then there is a fiber preserving diffeomorphism between them which is orthogonal on fibers. Our interest in S2-bundles arose in the study of Lie group actions (in particular of S0(3)) on 4-manifolds. The results obtained here are used in the equivariant classification of such actions (MP).

Finite group actions and nonseparating 2-spheres

Abstract: We extend the splitting theorem of Meeks-Yau for finite group actions on three-manifolds to include manifolds containing nonseparating 2-spheres, and give applications to branched covers of links 0 Introduction The purpose of this note is to point that the splitting theorem for finite group actions on three-manifolds of Meeks-Yau [4, Theorem 9] can be extended in an appropriate manner to include manifolds with S' X S2 summands We then give applications to branched covers of links, proving several geometrically "obvious" result The splitting theorem basically says that a finite group action on a compact, orientable 3-manifold with no S X s2 summands splits, modulo permuting homeomorphic summands, as the equivariant connected sum of the actions on the irreducible summands (As stated in [4], homotopy sphere summands are not permitted, but this restriction is no longer necessary See ?1) This is false when SI X S2 summands are present-even for involutions-as one can see in [1 and 2] In [2], however, Kim and Tollefson show that all involutions can be built up from involutions on irreducible 3-manifolds in a simple fashion by finding a suitable collection of 2-spheres along which an involution can be split into simpler actions This is possible to do by cut and paste methods essentially because intersections arising from an involution are easily visualized In a similar manner, Gordon and Litherland [1] were able to prove a Z2-equivariant loop theorem For actions of more general groups, these methods get hopelessly complicated However, minimal surface methods enables Meeks and Yau to overcome these problems and find invariant collections of spheres and discs We show below that the presence of nonseparating 2-spheres offers no real problem, and that the results of Kim and Tollefson can be generalized to arbitrary groups All manifolds will be oriented, and all actions smooth I should like to thank the referee for several useful suggestions 1 Splitting actions Let G be a finite group The most direct way to go from G-actions on irreducible 3-manifolds to G-actions on arbitrary 3-manifolds seems to be the following: Let X,,,Xn be orientable, irreducible 3-manifolds, and suppose Received by the editors November 2, 1981 Presented at the special session on 3-manifolds, 1981 summer meeting of the AMS, Pittsburgh 1980 Mathematics Subject Classification Primary 57S17; Secondary 57M 12 'This material is based upon work partially supported by the National Science Foundation under Grant No MCS-8002730 ?1 984 American Mathematical Society 0002-9939/84 $100 + $25 per page

Continuous equivariant images of lattice-actions on boundaries

Abstract: Let G be a connected semisimple Lie group with finite center and R-rank at least 2. For any parabolic subgroup P of G the quotient space G/P (equipped with the G-quasi-invariant measure class) is called a boundary of G. In [9] G. A. Margulis proved that if F is any irreducible lattice in G then any measurable factor (equivariant image) of the non-singular F-action on any boundary G/P is again of the same form-viz., it is measurably isomorphic to the F-action on a boundary G/Q, where Q is a parabolic subgroup containing P. This result happens to be a major ingredient of his proof of finiteness of either the cardinality or the index of any normal subgroup of F as above. Here and in the sequel, by an irreducible lattice, we mean a lattice F such that FF is dense for any non-central normal subgroup (in particular, G can have no compact factors). The purpose of this note is to establish the following topological analogue of the above result. (See also Remark 3.2 for a generalisation.)

Chapter VIII The Equivariant Loop Theorem for Three-Dimensional Manifolds and a Review of the Existence Theorems for Minimal Surfaces

Abstract: Publisher Summary The chapter discusses the equivariant loop theorem for three-dimensional manifolds that is needed in settling the Smith conjecture and reviews the existence theorems for minimal surfaces. The equivariant version of the loop theorem says that there are a finite number of properly embedded disks in M that satisfy the required properties. The loop theorem respects the action of the group G in a suitable manner. The chapter puts a metric on M so that the group G acts isometrically and so that ∂M is convex with respect to the outward normal. Then with respect to this metric, the existence of an immersed disk D 1 is demonstrated, in M whose boundary ∂ D 1 , represents a nontrivial element in π 1 ( S ) and whose area is minimal among all such disks. The chapter describes Morrey's solution for the plateau problem in a general Riemannian manifold. The existence theorem for manifolds with boundary proves that M is homogeneously regular if M is a subdomain of another homogeneously regular manifold N that has no boundary.

Chapter IX Group Actions on R3

Abstract: Publisher Summary The chapter discusses the group actions on R 3 The Smith conjecture has many equivalent forms, and each of these forms has various consequences and generalizations The Smith conjecture is a structure theorem about symmetries of the product of a compact surface with an interval Here the interval may be closed or open The usual Smith conjecture is equivalent to proving the smooth Z n actions on S 2 × [0, 1] are conjugate to actions that preserve the product structure This generalized Smith conjecture represents the belief that all the symmetries of the product of a compact surface with an interval actually arise from the symmetries of the surface extended trivially to the product structure The chapter presents how to apply minimal surfaces to study the generalized Smith conjecture on S 2 × I , where I is an open or closed interval A geodesic version of the loop theorem and a new equivariant version of the loop theorem has been proved

On the construction of nonequatorial minimal hyperspheres in Sn(1) with stable cones in Rn+1

Abstract: Within the framework of equivariant differential geometry, we outline the construction of some imbedded minimal hyperspheres of Sn(1) and show that many of them have a stable cone in Rn+1. The statement of these and related results are given.

Morse Theory for Flows in Presence of a Symmetry Group.

Abstract: : Different ways of studying an equivariant flow are investigated and, in particular, the equivariant Morse theory for flows is described This theory requires results on the cohomology of classifying spaces for finite groups which are also described here

Hecke Functors and the Equivariant Dold-Thom Theorem

Abstract: We recall that the Dold-Thom theorem [3] asserts that the weak homotopy type cf a topological abelian group is determined by its homotopy groups, and hence the homotopy category cf topological abelian groups with respect to weak equivalences is equivalent to the category cf graded abelian groups. In this note we consider the equivariant version. We restrict ourself to the case cf finite group actions. Let G be a finite group, and let k be a commutative ring with unit. A topological &[G]module is a topological abelian group M as well as a &[G]-module such that the bilinear map

Asymptotic risk comparison of some estimators for bivariate normal covariance matrix

Abstract: For Selliah's(5) and Stein's(3) loss functions, Sharma and Krishnamoorthy (6) have considered orthogonal equivariant estimators W. In thispaper, we obtain asymptotic risk expressions for \ and make a numerical comparison with those for Haff's(2) and Sugiura-Fujimoto (7) estimators. It is observed that W is uniformly better than Haff'sestimator and better than Sugiura-Fuji- moto estimator except in a small region of the parameter space.