Showing papers on "Equivariant map published in 1989"

Affine Hecke algebras and their graded version

Abstract: 0.1. Let H,o be an affine Hecke algebra with parameter v0 E C* assumed to be of infinite order. (The basis elements Ts E H,o corresponding to simple reflections s satisfy (Ts + l)(Ts v2c(s)) = 0, where C(S) E N depend on s and are subject only to c(s) = c(s') whenever s, s are conjugate in the affine Weyl group.) Such Hecke algebras appear naturally in the representation theory of semisimple p-adic groups, and understanding their representation theory is a question of considerable interest. Consider the "special case" where c(s) is independent of s and the coroots generate a direct summand. In this "special case," the question above has been studied in [1] and a classification of the simple modules was obtained. The approach of [1] was based on equivariant K-theory. This approach can be attempted in the general case (some indications are given in [5, 0.3]), but there appear to be some serious difficulties in carrying it out.

On the rigidity theorems of Witten

Abstract: In this paper we prove the rigidity theorems predicted by Witten in 1986, about the index of certain elliptic operators on manifolds with an S1 action [W]. Witten's insight was the culmination of an interesting interchange of ideas between him and Hopkins, Landweber, Ochanine, and Stong. For the detailed history, we refer the reader to [La]. The present account is essentially a reinterpretation of the second author's (Taubes' [T]) original proof of the theorem. The senior author's contribution was solely to notice that the rather densely written arguments of the original manuscript could be formulated in terms of the well known fixed point formulae of equivariant index theory and equivariant cohomology. In this context, the final proof then appears in direct lineage of the Atiyah-Hirzebruch theorem concerning the vanishing of the A genus of spin manifolds admitting a circle action [A-H] and of the even older idea of Lusztig concerning circle actions in the complex case. There remains, however, the beautiful, physics inspired novelty of connecting these techniques with elliptic function theory.

Infinite Loop G-Spaces Associated to Monoidal G-Graded Categories

Abstract: We construct a functor KG which takes each pair of monoidal G-graded categories (D,Df) to an infinite loop G-space KG(D,D'). When D'=D, its homotopy groups n%KG(D,D) coincide with the equivariant K-groups KnRepD of D. Applications include the simple construction of equivariant infinite deloopings of the maps BO(G}—^BPL(G}-^> BTop(G} between equivariant classifying spaces.

The Geometry of Spherical Space Form Groups

Abstract: In this volume, the geometry of spherical space form groups is studied using the eta invariant. The author reviews the analytical properties of the eta invariant of Atiyah-Patodi-Singer and describes how the eta invariant gives rise to torsion invariants in both K-theory and equivariant bordism. The eta invariant is used to compute the K-theory of spherical space forms, and to study the equivariant unitary bordism of spherical space forms and the Pinc and Spinc equivariant bordism groups for spherical space form groups. This leads to a complete structure theorem for these bordism and K-theory groups.There is a deep relationship between topology and analysis with differential geometry serving as the bridge. This book is intended to serve as an introduction to this subject for people from different research backgrounds.This book is intended as a research monograph for people who are not experts in all the areas discussed. It is written for topologists wishing to understand some of the analytic details and for analysists wishing to understand some of the topological ideas. It is also intended as an introduction to the field for graduate students.

Equivariant bifurcation theory and symmetry breaking

Abstract: A study is made of the failure of the Maximal Isotropy Subgroup Conjecture for the Weyl group seriesW(D) k . As part of the investigation, a general genericity and stability theorem is proved for bifurcation diagrams in equivariant bifurcation theory. As well, a concept of determinacy for equivariant bifurcation theory is introduced and it is shown that, for all compact Lie groupsG and absolutely irreducibleG-representationsV, G-equivariant bifurcation problems onV are finitely determined.

Helical geodesic immersions into complex space forms

Abstract: This paper consists of two parts. One is to construct a class of helical geodesic equivariant immersions of orderd(⩾3), which are neither Kaehler nor totally real immersions, into complex projective spaces. The other is to show the basic results about a helix in complex space forms.

Degree theory for equivariant maps. I

Abstract: A degree theory for equivariant maps is constructed in a simple geometrical way. This degree has all the basic properties of the usual degree theories and takes its values in the equivariant homotopy groups of spheres. For the case of a semifree 5'-action, a complete computation of these groups is given, the range of the equivariant degree is determined, and the general Slaction is reduced to that special case. Among the applications one recovers and unifies both the degree for autonomous differential equations defined by Fuller [F] and the S '-degree for gradient maps introduced by Dancer [Da]. Also, a simple but very useful formula of Nirenberg [N] is generalized (see Theorem 4.4(H)). 0. Introduction We expect this paper to be a reasonable example of the subject that we feel could be baptized as topological analysis. This term seems to us almost selfexplanatory. The gist would be to try to translate a problem from analysis into a problem in topology—or, to put it more gently, to try to use concepts and tools from topology in solving problems in analysis. Frequently, the nonvanishing of a topological invariant gives unexpected deep information on a certain question coming from analysis such as, say, the structure of the solution set of a given nonlinear equation. Thus, for example, ideas from singularity theory, Morse theory, Ljusternik-Snirel ' man category, index theories, topological degree theories, and their generalizations are more and more often used in analysis. This is particularly true with respect to what has been going on in nonlinear analysis during the past 15 years or so. We believe that it is not necessary here to stress any further the topics covered by topological analysis and we hope that this label will be accepted by enough supporters among nonlinearists to become familiar in the mathematical community. Our goal here is to carry on our program of research announced in [I.M.V] regarding the construction of a (generalized) degree theory for maps which are equivariant with respect to given representations of a compact Lie group T. Received by the editors July 18, 1987 and, in revised form, April 14, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 58B05, 58E07; Secondary 47H15, 54F45.

Median Unbiasedness and Pitman Closeness

Abstract: Under certain conditions, a median unbiased estimator of a parameter is shown to be the Pitman-closest estimator within a certain class of estimators. For location, scale, or location-scale families of distributions, the class of equivariant estimators possesses some desirable properties; the median-unbiasedness of equivariant estimators is explored in the characterization of the Pitman closeness property. Illustrative examples bear fruitful implications of this characterization.

Stability and equivariant maps

Abstract: Consider a linearized action of a reductive algebraic group on a projective algebraic varietyX over an algebraically closed field In this situation Mumford [1] defined the concept of stability for points ofX Given an equivariant morphismY→X we introduce a suitable linearization of the action onY and relate stability inY to stability inX In particular, we prove a relative Hilbert-Mumford theorem which says that stability inX andY can be tested simultaneously by 1-parameter subgroups It is hoped that this result will have applications to moduli problems In the caseY→X is a blowing up of a sheaf of ideals inX we give a simple explicit description of the properly stable and semi-stable loci inY Our relative Hilbert-Mumford theorem is used here in an essential way We also consider the following resolution problem introduced by Kirwan [2] The equivariant mapY→X is called a stable resolution if it is an isomorphism over the properly stable locus inX and every point ofY is either unstable or properly stable Kirwan [2] gave a canonical procedure for constructing a stable resolution over the complex numbers We show that a stable resolution can be obtained by resolving the singularities of a naturally defined subvariety ofX This gives an alternative (non-canonical) procedure for constructing stable resolutions

Equivariant Estimation in a Model with an Ancillary Statistic

Abstract: This paper reformulates a result of Hora and Buehler on best equivariant estimators to treat a model admitting an ancillary statistic. The approach itself was established by Pitman, Girshick and Savage and Kiefer, and expanded by Zidek. The model considered in this paper is assumed to be generated as an orbit under a group acting on the parameter space. The general result obtained here is applied to a model in the Nile problem, a model with a known variation coefficient, a circle model and the GMANOVA model, and best equivariant estimators (BEE's) are derived. In the first two models, the BEE's dominate the MLE's uniformly.

The Dynamics of Coupled Planar Rigid Bodies. II. Bifurcations, Periodic Solutions, and Chaos

Abstract: We give a complete bifurcation and stability analysis for the relative equilibria of the dynamics of three coupled planar rigid bodies. We also use the equivariant Weinstein-Moser theorem to show the existence of two periodic orbits distinguished by symmetry type near the stable equilibrium. Finally we prove that the dynamics is chaotic in the sense of Poincare-Birkhoff-Smale horseshoes using the version of Melnikov's method suitable for systems with symmetry due to Holmes and Marsden.

G-invariante Morse-funktionen

Abstract: If f is a Morse function on a smooth manifold M there exists a homotopy equivalence from M to a CW complex X such that the critical points of f with index λ are in a one-one correspondence to the λ-cells of X. In the equivariant case, a similar result holds for a special type of invariant Morse functions. In this paper we prove the existence of such special invariant Morse functions on compact smooth G-manifolds. As a consequence, any compact smooth G-manifold is homotopy equivalent to a G-CW complex. Other applications deal with the Euler number of the fixed point set and Morse inequalities in equivariant homology theory.

A further generalization of the segal conjecture

Abstract: THE Segal conjecture has been subjected to a number of usefulgeneralizations over the years [1,3,9,14,19]. We here give a still furthergeneralization which includes those in all of the cited papers. Weconsider an extension1->AT->G->F->1,where G is a compact Lie group, K is a (closed) normal subgroup, andthe quotient G/K = F is finite. There is a classifying F-space B(K; G) inthis situation. We shall prove that the generalized version of the Segalconjecture proven for stable F-cohomotopy in [1] remains valid for stableF-cohomotopy with coefficients in B(K; G). The case G = F x K, Kfinite, was studied in [9], where the close connection between the Segalconjecture and equivariant classifying spaces was first observed. Thispaper is the pushout over [9] of the unpublished preprints [14] and [19],which deal with the cases G finite and G = F x K, respectively.We introduce ideas in Section 1 by explaining the implications of ourresults for the calculation of equivariant stable maps between equivariantclassifying spaces. We state our main theorems in Section 2. We reducethe proofs of the theorems to questions about p -groups F and p-adiccompletion in Section 3. We prove the theorems when G is finite inSection 4. We also observe there that Carlsson's theorem [3] that theSegal conjecture for elementary Abelian p-groups implies the Segalconjecture for all finite p-groups is intrinsically a statement aboutequivariant classifying spaces. We prove our theorems when G is anextension of a torus by a finite group in Section 6. The proof proceeds byreduction to the case when G is finite. It depends on the dualization of acalculation of McClure [16], which may be of independent interest and isgiven in Section 5. We prove the general case of the theorems in Section7. The proof proceeds by reduction to the case handled in Section 6. Weuse results of Feshbach [7] to generalize some of our calculations inSection 8.

Spherical Varieties an Introduction

Abstract: When one studies complex algebraic homogeneous spaces it is natural to begin with the ones which are complete (i.e. compact) varieties. They are the “generalized flag manifolds”. Their occurence in many problems of representation theory, algebraic geometry, … make them an important class of algebraic varieties. In order to study a noncompact homogeneous space G/H, it is equally natural to compactify it, i.e. to embed it (in a G- equivariant way) as a dense open set of a complete G-variety. A general theory of embeddings of homogeneous spaces has been developed by Luna and Vust [LV]. It works especially well in the so-called spherical case: G is reductive connected and a Borei subgroup of G has a dense orbit in G/H. (This class includes complete homogeneous spaces as well as algebraic tori and symmetric spaces). A nice feature of a spherical homogeneous space is that any embedding of it (called a spherical variety) contains only finitely many G-orbits, and these are themselves spherical. So we can hope to describe these embeddings by combinatorial invariants, and to study their geometry. I intend to present here some results and questions on the geometry (see [LV], [BLV], [BP], [Lun] for a classification of embeddings).

Differential Structure of Étale Extensions of Polynomial Algebras

Abstract: The Jacobian Conjecture (see [BCW]) asserts that an etale polynomial map F: ℂ n → ℂ n is an isomorphism. In the course of some work with Haboush on group actions [BH] we observed that this is easily proved if F is equivariant for a reductive group acting on the two ℂ n ’s and having a dense orbit and a fixed point. One is thus tempted to try to prove the Jacobian conjecture by “feeding” such group actions into the picture. For example, let G = SL n (ℂ) act linearly on the target ℂ n . We may assume that F(0) = 0. In order to make F equivariant for a G-action, we must make G act on the source ℂ n by “F o G o F −1”. Of course this begs the issue, since we don’t know that F −1 exists. However F−1 exists analytically near 0. In particular we can pull back the action of the Lie algebra sl n(ℂ) as vector fields on ℂ n . Moreover the fact that the Jacobian matrix of F has a polynomial inverse implies that sl n (ℂ) pulls back to polynomial vector fields on the source ℂ n .

Equivariant completely bounded operators.

Abstract: An equivariant completely bounded linear operator between two C*-algebras acted on by an amenable group is shown to lift to a completely bounded operator between the crossed products that is equivariant with respect to the dual coactions A similar result is proved for coactions and dual actions It is shown that the only equivariant linear operators that lift twice through the action and dual coaction of an infinite group are the completely bounded ones 1 Introduction Let Φ: A -» B be a bounded linear map between C* -algebras, and let a: G —• A\x\A and β: G —> Auti? be actions of an amenable locally compact group G If Φ is a homomorphism and is equivariant—i e, Φ(as(a)) = βs(Φ(a)) for s e G, a e A —then it extends to a homomorphism Φ x i from the crossed product A xaG to B XJ^ G On the other hand, if C is another C*-algebra, then for any completely bounded map Φ: A —• B there is a bounded operator Φ x i: A® C -^ B ® C; indeed, by taking C to be the algebra 3£ of compact operators, we can see that the complete boundedness of Φ is necessary for this to be true We shall combine these results to prove that any equivariant completely bounded operator extends to a bounded map on the crossed product, formulate and prove the analogous results for crossed products by coactions of nonabelian groups, and investigate the extent to which complete boundedness is a necessary hypothesis We shall take similar approaches to the problems of lifting through actions and coactions First, we prove equivariant versions of Stinespring's theorem [23] on completely positive maps into 3§{%f), and we then show how to modify Wittstock's theorem [25] to write an equivariant symmetric completely bounded map into 3S{^) as a difference of equivariant completely positive ones From these we can use standard symmetrisation techniques to deduce that an equivariant completely bounded operator Φ: A —•ϊ%W) can be realized in the form Φ(a) = Tπ(a)V, where π is part of a covariant representation (π, U) of (A9G,ά) in some larger Hubert space We can then define

The cyclotomic trace and the K-theoretic analogue of Novikov's conjecture

Abstract: A trace construction, the cyclotomic trace, is given. It associates to algebraic K-theory of a group ring, or better to Waldhausen's A-theory, equivariant stable homotopy classes of the free-loop space of its classifying space. The cyclotomic trace detects the Borel classes in algebraic K-theory of the integers. It is used to prove, for a wide class of groups, that the K-theory assembly map is rationally injective. This is the K-theoretic analogue of Novikov's conjecture.

The instability of axisymmetric solutions in problems with spherical symmetry

Abstract: Among all possible equilibria that may bifurcate from the trivial state for one-parameter vector fields with $O(3)$-symmetry, one generically exists, whatever the (absolutely irreducible) representation of $O(3)$ is. This state is characterized by its group of symmetry, which includes rotations about a fixed axis, and for that reason is called “axisymmetric.” Recall that invariant spaces under irreducible representations of $O(3)$ have dimension $2l + 1$ and are generated by spherical harmonics $Y_m^l (\theta ,\phi )$, $ - l \leqq m \leqq l$. If l is even, the instability of the axisymmetric solutions follows from a theorem of Ihrig and Golubitsky [Phys. D (1984), pp. 1–33]. If l is odd, this theorem fails because it requires a condition on the quadratic part of the Taylor expansion of the equivariant vector field, but in that case it must have a zero quadratic part. However, the linearized vector field along an axisymmetric solution is diagonal in this basis and the computation of its eigenvalues is easy...

Equivariant spectral decomposition for flows with a -action

Abstract: Given a manifold M equipped with a free, properly discontinuous, cocompact Z-action, and a flow <£ on M which is Z-equivariant, we study the qualitative dynamics of <£. Under certain hypotheses on 4>, we show that the chain recurrent set of 4> has a decomposition which is the analogue, in the category of Z-equivariant flows, of Smale's spectral decomposition for recurrent sets of Axiom A flows. Given a compact manifold M, a flow on M, and a cohomology class a € H{M; Z), consider the Z-covering M-* M associated with a, and the lifted flow 4> on M. What can one say about the qualitative dynamics of 4> ? In a certain sense, this question is equivalent to the question of how the qualitative dynamics of are related to the cohomology class a. For example, in the Schwartzman-Fried theory of cross sections of flows (see [F2]), it is shown that has a cross-section Poincare dual to the cohomology class o if and only if ^ is a product flow, that is, M = 5 x R for some compact manifold 5, and is the flow

Hopf bifurcation in the presence of spherical symmetry: analytical results

Abstract: This paper considers a one-parameter family of vector fields equivariant under the orthogonal group $O(3)$, with an invariant fixed point. Hopf bifurcation of this family is studied assuming $O(3)$ acts on the eigenspaces belonging to each purely imaginary eigenvalue as an $l = 2$ representation. The dynamics are then reduced to a ten-dimensional center manifold, and the normal form of the vector field is explicitly given up to fifth order. The five different types of bifurcating periodic solutions, predicted geometrically by Golubitsky and Stewart, are derived analytically: a family of axisymmetric solutions, two types of rotating waves (one rotating at twice the speed of the other), a family of standing waves and a family of tetrahedral waves.The stability conditions are given for all these solutions. These stabilities depend on the three coefficients appearing at cubic order in the normal form and on one combination of three coefficients occurring at fifth order. All solutions can be stable except for ...

Equivariant K-Theory for Noetherian Rings

Abstract: A number of results are proved concerning the Quillen ^-theory K+(S*G) of the skew group ring S*G, where S is a Noetherian ring and G is a finite group of automorphisms of 5. Applications are given to the computation of AT-groups of group algebras and of equivariant /^-theory for affine varieties.

Grassmannians and Gauss Maps in Piecewise-Linear Topology

Abstract: Local formulae for characteristic classes.- Formal links and the PL grassmannian G n,k.- Some variations of the G n,k construction.- The immersion theorem for subcomplexes of G n,k.- Immersions equivariant with respect to orthogonal actions on Rn+k.- Immersions into triangulated manifolds (with R. Mladineo).- The grassmannian for piecewise smooth immersions.- Some applications to smoothing theory.- Equivariant piecewise differentiable immersions.- Piecewise differentiable immersions into riemannian manifolds.