Showing papers on "Equivariant map published in 1986"

K-Theory for Operator Algebras

Abstract: I. Introduction To K-Theory.- 1. Survey of topological K-theory.- 2. Overview of operator K-theory.- II. Preliminaries.- 3. Local Banach algebras and inductive limits.- 4. Idempotents and equivalence.- III. K0-Theory and Order.- 5. Basi K0-theory.- 6. Order structure on K0.- 7. Theory of AF algebras.- IV. K1-Theory and Bott Periodicity.- 8. Higher K-groups.- 9. Bott Periodicity.- V. K-Theory of Crossed Products.- 10. The Pimsner-Voiculescu exact sequence and Connes' Thorn isomorphism.- 11. Equivariant K-theory.- VI. More Preliminaries.- 12. Multiplier algebras.- 13. Hilbert modules.- 14. Graded C*-algebras.- VII. Theory of Extensions.- 15. Basic theory of extensions.- 16. Brown-Douglas-Fillmore theory and other applications.- VIII. Kasparov's KK-Theory.- 17. Basic theory.- 18. Intersection product.- 19. Further structure in KK-theory.- 20. Equivariant KK-theory.- IX. Further Topics.- 21. Homology and cohomology theories on C*-algebras.- 22. Axiomatic K-theory.- 23. Universal coefficient theorems and Kunneth theorems.- 24. Survey of applications to geometry and topology.

Equivariant Stable Homotopy Theory

Abstract: The last decade has seen a great deal of activity in this area. The chapter provides a brief sketch of the basic concepts of space-level equivariant homotopy theory. It also provides an introduction to the basic ideas and constructions of spectrum-level equivariant homotopy theory. The chapter also illustrates ideas by explaining the fundamental localization and completion theorems that relate equivariant to nonequivariant homology and cohomology. To retain the homeomorphism between orbits and homogeneous spaces one shall always restrict attention to closed subgroups. The class of compact Lie groups has two big advantages: the subgroup structure is reasonably simple (nearby subgroups are conjugate), and there are enough representations (any sufficiently nice (7-space embeds in one).

Refining binomial confidence intervals

Abstract: A method for refining an equivariant binomial confidence procedure is presented which, when applied to an existing procedure, produces a new set of equivariant intervals that are uniformly superior. The family of procedures generated from this method constitute a complete class within the class of all equivariant procedures. In certain cases it is shown that this class is also minimal complete. Also, an optimally property, monotone minimaxity, is investigated, and monotone minimax procedures are constructed.

Steady-state mode interactions in the presence of 0(2)-symmetry

Abstract: This paper studies the multiple bifuraction phenomenon of steady-state mode interactions in the presence of 0(2)-symmetry. For such problems the flow on the centre manifold is determined by a vector field in or that is equivariant under an action of 0(2). The action is related to the wave numbers of the unstable modes. The unfolded normal forms for these equivariant bifurcation problems admit primary bifurcations to single-mode solutions, secondary bifurcations to mixed-mode solutions and, in some instances, tertiary bifurcations to travelling and standing waves. The bifurcation behaviour depends crucially on the wave numbers. For small wave numbers, the mixed-mode solutions encounter subordinate saddle-node bifurcations.

equivariant function spaces and characteristic classes

Abstract: We determine the structure of the homology of the BeckerSchultz space SGiS1) ~ Q(CP^° AS1) of stable Sx-equivariant self-maps of spheres (with standard free S1 -action) as a Hopf algebra over the DyerLashof algebra. We use this to compute the homology of BSGiS1). Along the way, we give a fresh account of the partially framed transfer construction and the Becker-Schultz homotopy equivalence. We compute the effect in homology of the '^-transfers\" CPf AS1^ Q((BZpn) + ), n > 0, and of the equivariant J-homomorphisms SO -> Q(RP£°) and U -» Q(CP°° A S1). By composing, we obtain U —> QS° in homology, answering a question of J. P. May. Introduction. Let ii be a compact Lie group admitting a finite-dimensional orthogonal representation W such that H acts freely on the unit sphere sW. H must thus be S1, S3, the normalizer of S1 in S3, or one of a known list [13] of finite groups with periodic cohomology, including (as subgroups of S3) the cyclic and generalized quaternion groups. Let EndB(sW) denote the space of ü-equivariant continuous self-maps of sW. By joining with the identity map we obtain inclusions EndH(s(nW)) C EndH(s((n+l)W)), and we write G(H) for the direct limit. The homotopy type of G(H) was determined by J. C. Becker and R. E. Schultz [2], and turns out to be independent of W. If we write SG(H) for the component of G(H) containing the identity map (so SG(H) = G(H) if H is connected), then in [3], Becker and Schultz (see also [9] in case H is finite) enrich the composition product in SG(H) to an infinite-loop space structure. The classifying space BSG(H) classifies oriented spehrical fibrations with a fiber-preserving ü-action modelled on s(nW), stabilized by forming fiberwise joins with the trivial if-fibration with fiber sW. In this paper we determine the modp homology of SG(S1) and of BSG(S1) as Hopf algebras over the Dyer-Lashof algebra. Along the way, we compute the effect in homology of the \"forgetful\" maps SG(S1) —> 5G(Zpn) and of the equivariant J-homomorphisms Jz, : SO -» SG(Z2) and jSi : U -+ SG(S1). The starting point for our analysis is the study of certain \"transfer\" maps. §1 is devoted to an account of the construction and general properties of these maps. In §2 we study certain transfers t associated to an inclusion K C H of compact Lie groups. If E is a smooth principal H-space, then t:(E/HY\" ^Q((E/Ky«), where ÇB is the vector-bundle obtained by mixing E J, E/H with the adjoint representation of H on its Lie algebra, the superscript denotes formation of the Thom space and QX is the enveloping infinite loop space of X. Received by the editors August 12, 1980 and, in revised form, June 15, 1985. 1980 Mathematics Subject Classification. Primary 55R40; Secondary 55R91, 55Q50, 55R12. ©1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page

Equivariant intersection forms, knots in ⁴, and rotations in 2-spheres

Abstract: We consider the problem of distinguishing the homotopy types of certain pairs of nonsimply-connected four-manifolds, which have identical three-skeleta and intersection pairings, by the equivariant isometry classes of the intersection pairings on their universal covers. As applications of our calculations, we: (i) construct distinct homology four-spheres with the same three-skeleta, (ii) generalize a theorem of Gordon to show that any nontrivial fibered knot in S4 with odd order monodromy is not determined by its complement, and (iii) give a more constructive proof of a theorem of Hendriks concerning rotations in two-spheres embedded in threemanifolds. 0. Introduction. In this paper we are interested in several sorts of \"twists\" on low-dimensional manifolds, and their relationships. We consider three situations: (i) In constructing four-manifolds by performing surgery on simple loops, one makes a framing choice. The two possible choices are related by the twist coming from mx(SO(3)) = Z2. How does the choice affect the homotopy type of the resulting manifold? (ii) At most two knots in S4 have the same complement, the possible difference given by a twist in gluing a regular neighborhood of the knot to its exterior. How does the choice affect the knot type (a relative version of (i))? (iii) If a three-manifold M03 has 3M0 = S2, one may define the rotation p in S2 (see §1). Is p = idwo (rel9A/0)? It turns out that by considering these situations from the point of view of intersection forms on four-manifolds, we can give fairly complete answers (for certain cases of (i) and (ii)). For instance, we prove Corollary 3.5. There exist (infinitely many pairs of) homology A-spheres which have the same 3-skeleton but distinct homotopy type. Concerning (ii), we have the following generalization of a result of C. McA. Gordon [8, Proposition 4.2]: Theorem 6.2. Let K be any nontrivial fibered knot in S4 with odd order monodromy. Then K is not determined by its complement. Received by the editors May 15,1985. 1980 Mathematics Subject Classification. Primary 57N13; Secondary 57N10, 57Q45. ' This research was partially supported by NSF Grant MCS-82-01045. ©1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page 543 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

A topological proof of the equivariant Dehn lemma

Abstract: An elementary topological proof is given for a completely general version of the Equivariant Dehn Lemma, in the spirit of the original proof of the nonequivariant version due to C. D. Papakyriakopolous in 1957.

Secondary bifucations of a thin rod under axial compression

Abstract: We study the post-buckling behavior of a prismatic rod with rectangular cross-section, under axial compression. We employ the nonlinear rod theory stated in [2]. Let $\delta $ be the difference between the sides of the cross-section. When $\delta e 0$ there are two distinct eigenvalues which coalesce when $\delta = 0$ and the cross-section becomes a square. By employing equivariant singularity theory [4], we unfold the bifurcation problem corresponding to $\delta = 0$ and study also the case $\delta e 0$. We obtain bifurcation diagrams with second bifurcations when $\delta e 0$.

Bifurcation problems with O(2)⊕Z2 symmetry and the buckling of a cylindrical shell

Abstract: In this paper we employ equivariant singularity theory to study the postbuckling behavior of a cylindrical shell under axial compression, obtaining some results about the existence of secondary bifurcations and how they are connected to each other. The basic idea, first employed by Bauer, Keller and Reiss in [1], and then coupled with singularity theory by Schaeffer and Golubitsky in [16] and [17] and by Buzano in [4], consists in unfolding a multiple eigenvalue, obtained by forcing two eigenvalues to coalesce by varying the geometric parameters of the shell. This approah is made possible by a general analysis of bifurcation problems invariant with respect to the symmetries of the cylinder i.e. with respect to the group O(2)⊕Z2.

Multiple critical points of invariant functionals and applications

Abstract: (7 in a closed subspace V of L2(R; R’) which is invariant under VF, where FE C1(2v, $3) is a strictly convex function and L is an unbounded self-adjoint operator on V with no essential spectrum. We assume that VF(0) = 0, so that u = 0 is a solution of (*). We also assume, without loss of generality, that F(0) = 0. Loosely speaking, we shall see that the number of nontrivial solutions of (*) is related to the number of eigenvalues of L “crossed by 2F(u)/luj’” as Iu] varies from 0 to x, provided that (*) is equivariant with respect to some group action, so that Lusternik-Schnirelman theory can be used. We apply this theory to the “dual action” associated with (*), which was introduced by Clarke and Ekeland [5] for Hamiltonian systems. The abstract framework and main results are presented in Section 2. In Section 3 we consider two applications. First we consider the nonlinear Dirichlet problem -Au = g(rc) in R u=O on aR, for which it is classical to use the Z,-action when g is odd [3]. When R is a disc in R2, the symmetry of the domain was used in [7] instead of a (possible) symmetry of the nonlinearity. In this case, a natural S’-action is provided by the rotations in R. In theorem 3 we extend the multiplicity result of [7]. Moreover the use of the dual action simplifies the proof. It is

Global Results on Continuation and Bifurcation for Equivariant Maps

Abstract: This paper gives an extension for equivariant maps of our previous results concerning the study of global branching phenomenae for parameter dependent maps. Using elementary point set topology, we study the class of Γ - epi maps and give a detailed description of the structure, local dimension and global behaviour of the set of solutions for equations involving equivariant maps. A new degree theory for S1- maps is sketched and our results are applied to global continuation and global bifurcation in presence of symmetries.

Equivariant morse theory for flows and an application to the N-body problem

Abstract: A l'aide de l'indice de Conley et de la cohomologie equivariante, on deduit certaines inegalites de type Morse pour un flot equivariant par rapport a l'action d'un groupe topologique compact

The set of balanced orbits of maps of ¹ and ³ actions

Abstract: Suppose that the group G = S1 or G = S3 acts freely on a space X and on a representation space V for G. Let f: X -p V. The paper studies the size of the subset of X consisting of orbits over which the average of f is zero. The result can be viewed as an extension of the Borsuk-Ulam theorem. 1. The average of a map. Let f be a map from S5 to RW. The classical Borsuk-Ulam theorem says that the set Af = {x E 5" I fx = f(-x)} is nonempty. The formula f(-x) fx may be viewed as the average of f at the point x, with respect to the antipodal Z2-actions on the source space nsn, and on the target space RW. Thus the Borsuk-Ulam theorem can be expressed by saying that for any map f: Sn -* RW there is a point where the average of f (with respect to the antipodal actions) is zero. The average can be defined for any map of a G-space into a representation space, provided that the transformation group G admits a Haar integral, as is the case for compact groups. A theorem proved by Liulevicius in [5] can be expressed as follows: If G is a nontrivial compact Lie group acting freely on Sm and freely and orthogonally on the unit sphere in a representation space V of dimR V < m then for any map f: Sm -* V there exists an x E Sm where the average of f is zero. (1.1) DEFINITION. Let X be a G-space and let V be a finite-dimensional representation space of G. Let f: X -+ V be a (continuous) map.Then the average of f is the map Avf: X -* V defined by (Avf)x = f g-f(gx) dg. We note the following properties: (1.2) For any map f: X -* V, Avf: X -* V is an equivariant map. (1.3) If f: X -V is equivariant, then Avf = f. 2. The set of balanced points. (2.1) DEFINITION. Let X be a G-space and let V be a finite-dimensional representation space of G. A map f: X -* V is said to be balanced at a point x E X if (Avf )x = 0. (We will also say then that x is a balanced point of f.) Let Af denote the set of points of X where f is balanced. Then Af is an invariant subset of X; it is Received by the editors October 31, 1984 and, in revised form, April 26, 1985. 1980 Mathematics Subject Classification. Primary 55M20, 55N25, 55R40. Kev words and phrases. Average, equivariant map, equivariant cohomology, characteristic class, index. ?D1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page

Forgetful homomorphisms in equivariant k-theory

Abstract: The purpose of this note is to analyse the surjectivity of the forgetful homomorphism f ( G , X] : KG(X)-*K(X\\ which gives some useful information about lifting group actions in stable vector bundles. Here G is a compact connected Lie group and X is a compact G-space such that K%(X) is finitely generated over R(G). Moreover let T denote a maximal torus of G throughout this paper. It is known that if ^(G) is torsion free, then the homomorphism a(G, T) : R(T}-*K(G/T] which is interpreted as /(G, G/T) via the isomorphism KG(G/T)^R(T) is surjective (cf. [5], [6]). We shall use a theorem which Pittie [6] presented to prove this fact. In Section 1 we shall give a sufficient condition for the surjectivity of /(G, X) for G a torus (Theorem 1) and further we shall prove that if ^i(G) is torsion free and /(T, X) is surjective, then /(G, X) is also surjective (Theorem 2). Section 2 consists of applications of the preceding theorems to actions on homotopy complex projective spaces, pseudo-linear G-spheres and complex quadrics. In Section 3 we shall give a generalized form of Theorem 2 for the case when Tor Tr^G^O (Theorem 5) and using some results due to Hodgkin we shall obtain examples of actions of these groups. In the last section we shall prove that if a(G, T) is surjective, then ^(G) is torsion free (Theorem 6).

R-spaces associated with a Hermitian symmetric pair

Abstract: The linear isotropy representation of a Riemannian symmetric pair (G,K) is defined as the differentialof the left action of K on GJK at the origin. Every orbit of the linear isotropy representation of (G,K) is called an R-space associated with {G,K), which is an important example of equivariant homogeneous Riemannian submanifolds in a Euclidean sphere (See Takagi-Takahashi [2] and TakeuchiKobayashi [3]). This paper is concerned with the linear isotropy representation of a Hermitian symmetric pair (G,K). Its restriction to the center of K defines an S1-action on the associated i?-spaces. We determine all i?-spaces associated with Hermitian symmetric pairs (G,K) on which the semisimple part of K acts transitively. In particular,we know allirreducible Hermitian symmetric pairs such that each of the associated i?-spaces has such a property. This result is utilizablefor the classificationof orthogonal transformation groups by their cohomogeneity (See the forthcoming paper [4] concerned with this problem in low cohomogeneity). The authors are profoundly grateful to Professor Ryoichi Takagi for his helpful suggestion and criticalreading of a primary manuscript.

Equivariant homology decompositions

Abstract: This paper presents some results on the existence of homology decompositions in the context of the equivariant homotopy theory of Bredon. To avoid certain obstructions to the existence of equivariant Moore spaces oc- curring already in classical equivariant homotopy theory, most of the work of this paper is done "over the rationals." The standard construction of homology decompositions by Eckmann and Hilton can be followed in the present equi- variant context until it is necessary to produce appropriate k'-invariants. For these, the Eckmann-Hilton construction uses a certain Universal Coefficient Theorem for homotopy sets. The relevant extension of this to the equivariant situation is an equivariant Federer spectral sequence, which is developed in §2. Using this, we can formulate conditions which imply the existence of the desired k'-invariants, and hence the existence of the homology decomposition. The conditions involve a certain notion of projective dimension. For one ap- plication, equivariant homology decompositions always exist when the group has prime order.

Equivariant bundles and cohomology

Abstract: Let G be a topological group, A an abelian topological group on which G acts continuously and X a G-space. We define "equivariant cohomol- ogy groups" of X with coefficients in A, HlG(X\ A), for i > 0 which generalize Graeme Segal's continuous cohomology of the topological group G with coeffi- cients in A. In particular we have H^(X; A) ~ {equivalence classes of principal (G, A)-bundles over X}. We show that when G is a compact Lie group and A an abelian Lie group we have for i > 1 H^(X;A) ~ HZ{EG Xq X;tA) where tA is the sheaf of germs of sections of the bundle (X x EG x A)/G —» (X x EG)/G. For i = 1 and the trivial action of G on A this is a theorem of Lashof, May and Segal.