Showing papers on "Equivariant map published in 1987"

The decorated Teichmüller space of punctured surfaces

Abstract: A principal ℝ + 5 -bundle over the usual Teichmuller space of ans times punctured surface is introduced. The bundle is mapping class group equivariant and admits an invariant foliation. Several coordinatizations of the total space of the bundle are developed. There is furthermore a natural cell-decomposition of the bundle. Finally, we compute the coordinate action of the mapping class group on the total space; the total space is found to have a rich (equivariant) geometric structure. We sketch some connections with arithmetic groups, diophantine approximations, and certain problems in plane euclidean geometry. Furthermore, these investigations lead to an explicit scheme of integration over the moduli spaces, and to the construction of a “universal Teichmuller space,” which we hope will provide a formalism for understanding some connections between the Teichmuller theory, the KP hierarchy and the Virasoro algebra. These latter applications are pursued elsewhere.

Equivariant K: Theory and Freeness of Group Actions on C Algebras

Abstract: Introduction: The commutative case.- Equivariant K-theory of C*-algebras.- to equivariant KK-theory.- Basic properties of K-freeness.- Subgroups.- Tensor products.- K-freeness, saturation, and the strong connes spectrum.- Type I algebras.- AF algebras.

The Takens-Bogdanov bifurcation with O(2)-symmetry

Abstract: The versal deformation of a vector field of co-dimension two that is equivariant under a representation of the symmetry group O(2) and has a nilpotent linearization at the origin is studied. An appropriate scaling allows us to formulate the problem in terms of a central-force problem with a small dissipative perturbation. We derive and analyse averaged equations for the angular momentum and the energy of the classical motion. The unfolded system possesses four different types of non-trivial solutions: a steady-state and three others, which are referred to in a wave context as travelling waves, standing waves and modulated waves. The plane of unfolding parameters is divided into a number of regions by (approximately) straight lines corresponding to primary and secondary bifurcations. Crossing one of these lines leads to the appearance or disappearance of a particular solution. We locate secondary saddle-node, Hopf and pitchfork bifurcations as well as three different global, i.e. homoclinic and heteroclinic, bifurcations.

Equivariant Maps for the Symmetric Group

Abstract: M. Ozaydin, Equivariant maps for the symmetric group, unpublished preprint, University of Wisconsin-Madison, 1987, 17

Central configurations of the N-body problem via equivariant Morse theory

Abstract: In this paper we use the equivariant Morse theory to give an estimate of the minimal number of central configurations in the N-body problem in ℝ3. In the case of equal masses we prove that the planar central configurations are saddle points for the potential energy. From this we deduce the presence of non-planar central configurations, for every N ≧ 4. The principal difficulty in applying Morse theory is that the potential function is defined on a manifold on which the group O(3) does not act freely. To avoid this problem the equivariant cohomology functor is applied in order to obtain the Morse inequalities.

Cyclic homology and equivariant theories

Abstract: © Annales de l’institut Fourier, 1987, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Construction of group actions on four-manifolds

Abstract: It is shown that any cyclic group of odd prime order acts on any closed, simply connected topological 4-manifold, inducing the identity on integral homology. The action is locally linear except perhaps at one isolated fixed point. In the case of primes greater than three a more careful argument is used to show that the action can be constructed to be locally linear. Introduction. In this paper we shall show that every closed, simply connected topological 4-manifold M4 admits an action of any cyclic group Zp of odd prime order p. The action will be homologically trivial (that is, induce the identity on integral homology), be pseudofree (that is, have only isolated fixed points) except in certain cases when p = 3, and be locally linear except perhaps at one isolated fixed point. When p = 2 and the intersection form of M4 has even type, then the same conclusion holds. This is also due to S. Kwasik and P. Vogel [1985], by a somewhat different proof. If p = 2 and the intersection form of M has odd type, however, then an action of Z2 which is homologically trivial must have a fixed point set containing two-dimensional components (see ?7). Further it is harder to control the Kirby-Siebenmann triangulation obstruction, which is then not determined by the intersection form. It is interesting to ask whether these actions can be locally linear or smooth (when M4 is smooth). The existence of a locally linear Z2 action implies the vanishing of the Kirby-Siebenmann obstruction (see Kwasik and Vogel [1984]). We shall show, however, that (for p > 3 in general) these actions can be constructed to be locally linear. In a recent preprint Kwasik has shown that this is the case for the fake Cp2 when p is odd by a rather different proof. These actions are homologically trivial and have only isolated fixed points (except sometimes when p = 3). (The G-Signature Theorem shows that not every simply connected 4-manifold admits pseudofree, locally linear actions when p = 3. We shall see, for example, that neither the E8 manifold, the Kummer surface, nor a nontrivial connected sum of copies of Cp2 admits such an action for p = 3.) In broad outline our construction goes as follows. Let a closed, oriented, simply connected 4-manifold M be given. By studying equivariant framed links, we construct a compact smooth 4-manifold with boundary, having the intersection form of M, and admitting a smooth Zp action which is pseudofree and homologically trivial. The action is free on the boundary homology sphere E. In a paper primarily focused on equivariant plumbing diagrams for high dimensional, highly connected manifolds with even intersection forms, Weintraub [1975] carried out most of this. We give an independent development because we need to be able to better control Received by the editors October 16, 1985 and, in revised form, January 7, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 57S17, 57S25, 57N15. Research supported in part by a grant from the National Science Foundation. (?)1987 American Mathematical Society 0002-9947/87 $1.00 + $.25 per page

Best Equivariant Estimators of a Cholesky Decomposition

Abstract: Every positive definite matrix $\Sigma$ has a unique Cholesky decomposition $\Sigma = \theta\theta'$, where $\theta$ is lower triangular with positive diagonal elements. Suppose that $S$ has a Wishart distribution with mean $n\Sigma$ and that $S$ has the Cholesky decomposition $S = XX'$. We show, for a variety of loss functions, that $XD$, where $D$ is diagonal, is a best equivariant estimator of $\theta$. Explicit expressions for $D$ are provided.

Some equivariant k-theory of affine algebraic group actions

Abstract: (1987). Some equivariant k-theory of affine algebraic group actions. Communications in Algebra: Vol. 15, No. 1-2, pp. 181-217.

Group actions and equivariant Lipschitz analysis

Abstract: The idea that one could prove by analytic methods the topological invariance of characteristic classes was suggested in an influential lecture by Singer. The concrete realization of this, in the case of rational Pontrjagin classes, was the result of works by Sullivan, who showed that topological manifolds have essentially unique Lipschitz structures, and Teleman, who showed that Lipschitz manifolds are the type of object for which one can analyze the signature operator. (A refinement of the same method proves the KO[l/2] orientability of topological bundles.) Much of this can be done for manifolds with group actions. The main implication is that for certain sorts of topological group actions (including all smooth and PL actions) one can construct an "equivariant signature operator" whose class in equivariant if-homology is a topological invariant. This class satisfies the G-signature theorem. The topological method has implications for various precise classification problems (e.g. semifree topological actions on the sphere), but here we shall present only the consequences of these Lipschitzian ideas: (A) Topologically locally linear G-manifolds for odd-order G have KOG[1/2] orientations (Madsen-Rothenberg). For all finite G there are Lipschitz structures, but the signature class is a zero divisor for even-order groups. The same method produces, upon application of the localization theorem, "Atiyah-Singer" classes for some actions with nonmanifold fixed set (Weinberger, extending earlier joint work with Cappell). (B) For odd-order groups, topological and linear conjugacy of linear representations are equivalent (Hsiang-Pardon, Madsen-Rothenberg). This is a consequence of (A). Alternatively one can return to the original argument of Atiyah-Bott-Milnor for semifree representations. A stronger result is true: the Atiyah-Bott local numbers associated to representations agree for elements that have discrete fixed sets in topologically equivalent representations. This distinguishes many pairs of representations of even-order groups. (C) R Top(G) = R Lip(G) for all finite G, where R Cat is the Grothendieck group of representations up to Cat conjugacy (Weinberger, after inverting 2). This implies that there is no analytic definition of Reidemeister torsion that works well in the Lipschitz setting. The reason is that, following

The Geometric Finiteness Obstruction

Abstract: The purpose of this paper is to develop a geometric approach to Wall's finiteness obstruction. We will do this for equivariant CW-complexes. The main advantage will be that we can derive all the formal properties of the equivariant finiteness obstruction easily from this geometric description. Namely, the obstruction property, homotopy invariance, the sum and product formulas, and the restriction formula can be stated and proved in a simple manner. Also a characterization of the finiteness obstruction by a universal property is quickly available. This geometric approach is similar to the treatment of Whitehead torsion by Cohen in [3]. In the first section we define a functor Wa from the category of G-spaces to the category of abelian groups. We assign to a finitely dominated G-CW-complex X an element w(X) e Wa(X) called its finiteness obstruction. The finiteness obstruction vanishes if and only if X is G-homotopic to a finite G-CW-complex and satisfies a sum formula and is homotopy invariant. The notion of a universal functorial additive invariant is introduced in the second section where its existence and uniqueness are proved. Product and restriction formulas for the universal additive invariant are obtained by abstract nonsense. We define equivariant Euler characteristics in the third section generalizing the notion of the Euler characteristic of a finite CW-complex. The goal of the fourth section is to prove that the equivariant Euler characteristic and finiteness obstruction determine the universal functorial additive invariant for finite, respectively finitely dominated, G-CW-complexes. The fifth section contains some algebraic computations of Wa in terms of reduced projective class groups of certain integral group rings. In the nonequivariant case Wall's algebraic approach and our geometric one agree. Finally, in the sixth section, the results of the second and fourth sections are used to state an abstract product formula, a restriction formula, and a diagonal product formula. We make some remarks about the simple-homotopy approach to the finiteness obstruction due to Ferry. The treatment by Ferry in [7] is extended by Kwasik in [14] to the equivariant case. In § 6 we construct geometrically an injection I(Y): W(Y)^>Wh(YxS) into the equivariant Whitehead group of Y x S sending our geometric finiteness obstruction to that of Kwasik. A compact Lie group is denoted by G.

Equivariant eilenberg maclane spaces K ( G , μ, 1) for possibly non-connected or empty fixed point sets

Abstract: Equivariant Eilenberg-MacLane spaces are constructed in [1, p. II.13], [3, p. 277], [8, p. 45], however, only for nonempty connected H-fixed point sets for all H⊂G and in the pointed category. This is a reasonable assumption in equivariant homotopy theory (equivariant Posnikov-systems, homology, obstruction theory) but too restrictive for the study of equivariant manifolds. Therefore we develope a treatment of equivariant Eilenberg-MacLane spaces of type one in full generality. They are used, for example, in equivariant L-theory as reference spaces (see [5]) or in [4].

Pulling back fixed points

Abstract: If X, Y are G-spaces (spaces on which a compact Lie group G acts), and f: X ~ Y is a G-map, then the image of a fixed point in X is fixed in I1. When does the inverse image of a fixed point contain a fixed point, in particular if X, Y are G-manifolds and f has degree prime to ]GI (=order of G). In (unpublished work of the author1), this is shown for elementary abelian p-groups, with X, Y more general spaces ("n-near manifolds"). In this paper we show that this is true for abelian p-groups G acting smoothly on X, with p odd or with p=2 with an additional complex hypothesis. We give a counterexample for nonabelian G. The method is to prove a generalization of a theorem of (Bredon, 1973) for 7lip actions, which says that maps of degree prime to p induce injections on the mod p cohomology of the fixed set. We generalize this theorem to abelian p groups, where the action on X is smooth, and p 4= 2 (or other hypothesis). This seems to be new even in the case of elementary abelian p-groups of rank > 1. We actually prove a stronger theorem for a generalization of the notion of degree, which may be non-zero for maps between manifolds of different dimension. The proof involves complex linear equivariant K-theory in an essential way, which requires smoothness and p + 2, or a complexity hypothesis on X in order to produce the appropriate complex linear bundles. Among the corollaries, we show that an abelian p-group (p odd) cannot act smoothly on a closed manifold with exactly one isolated fixed point, generalizing an old result of (Atiyah and Bott, 1964) for G=Z/p. This was found independently by (Ewing and Stong, 1986) by a different argument. I am indebted to Ib Madsen for his comments, in particular for discussions on equivariant K-theory. Thanks are due also to Michael Davis for carefully reading the manuscript, and pointing out several difficulties. I also wish to thank the referee for his comments.

Approximation by equivariant homeomorphisms. I

Abstract: Locally linear (= locally smoothable) actions of finite groups on finite dimensional manifolds are considered in which two incident components of fixed point sets of subgroups either coincide or one has codimension at least three in the other. For these actions, an equivariant o-approximation theorem is proved using engulfing techniques. As corollaries are obtained equivariant ';fibrations are bundles" and "controlled h-cobordism" theorems, as well as an equivariant version of Edwards' cell-like mapping theorem and the vanishing of the set of transfer-invariant G-homotopy topological structures, rel bound- ary, on Tn x Dp (when Tn is the n-torus with trivial G action and Dp is a representation disc). Here we consider locally linear (locally smooth (Br)) PL and topological actions of a finite group G on n-manifolds. In addition, we require that a G-manifold, M, have gaps of codimension > 3 (i.e., for H c G, if MH is a component of the H-fixed point set, MH, and M>H c MH is the subspace of points x for which the isotropy subgroup, Gx, strictly contains H, then either M>H = MH or M>H has codimension > 3 in MH). We shall use the term "not necessarily locally linear G-manifold" when we wish to drop the local linearity hypothesis but retain the property that if MH c M: but MH + M: then MH is a locally flat (inequivariantly) submanifold of MK Of codimension at least three. This, while self-contained, is the second in a series of papers (SW2,...,8,S) in which we analyze the extent of failure of the topological invariance of equivari- ant Whitehead torsion and the consequent failure of subgroups of the equivariant Whitehead group WhGL(M) of Illman (Ill) (cf. Rothenberg (R)) to classify PL or smooth G-h-cobordisms up to topological equivalence (under our gap hypotheses the arguments of Browder and Quinn (BQ) and Rothenberg (R) show that G-h- cobordisms are classified up to PL or smooth equivalence by such subgroups). We eventually conclude in (Sw7 S) (cf. (SW2)) that the topological equivalence classes of G-h-cobordisms on a G-manifold M that are products over the union of fixed point components of M having dimension < 5 are classiSed by a group WhG°P(M) to which WhGL(M) maps homomorphically. The image in WhG°P(M) of the per- tinent subgroup of WhGL(M) classifies up to equivariant homeomorphism, rel M, those G-h-cobordisms on M that admit equivariant handlebody decompositions relative to M. (This homomorphism is not generally surjective and there are G-h- cobordisms that do not admit such handle decompositions (cf. (SW5, S)). In order

The equivariant conner-floyd isomorphism

Abstract: This paper proves two equivariant generalizations of the Conner- Floyd isomorphism relating unitary cobordism and K-theory. It extends a previous result of Okonek for abelian groups to all compact Lie groups. We also show that the result for finite groups is true using either the geometric or homotopical versions of cobordism. 1. Introduction. In (6) Conner and Floyd established a relation between cobordism and .rf-theory. They proved that MU*(X)®MU.K* = k*(X), where MU is unitary cobordism and K is complex A"-theory. A generalization of this result to the equivariant context was proved by Okonek (15). This paper improves upon that generalization in two ways. First, it expands the class of groups considered, from abelian groups to all compact Lie groups. Second, it shows that both of the usual generalizations of cobordism to the equivariant context can be used. The .rf-theory that we use is KG(—), the equivariant K-theory of Segal (17), which is the obvious generalization of K-theory to the equivariant world. Atiyah (2) proved that this theory has Bott periodicity; it is periodic with respect to Spinc-representations of the group. There are two common ways of generalizing cobordism; we will define these carefully in §2. One gives geometric cobordism, which we denote by UG( — ), and the other homotopical cobordism, MUG(—). Both of these are multiplicative cohomology theories. There are multiplicative maps p: U*G(-) -> K*G(-) and p: MUG(-) -* KG(-). Writing U*G for the ring UG(*), and similarly for the other theories, the two main results we will show are THEOREM A. p: UG(X) ®w KG —► KG(X) is an isomorphism when G is a finite group. THEOREM B. p: MUG(X) ®mu* Kg ~* KG(X) is an isomorphism for every compact Lie group G. It should be pointed out that these results use integer grading and not RO(G) grading. We will use integer grading throughout this paper, except on those occa- sions when it is necessary or particularly convenient to use RO(G); we will make it clear at these points that we are using the larger grading. Theorem B has already been shown for G abelian in (15), and we will use that result as a starting point. Theorem A follows from Theorem B, as shown in §3, by

Characteristic classes, singular embeddings, and intersection homology

Abstract: This note announces some results on the relationship between global invariants and local topological structure. The first section gives a local-global formula for Pontrjagin classes or L-classes. The second section describes a corresponding decomposition theorem on the level of complexes of sheaves. A final section mentions some related aspects of “singular knot theory” and the study of nonisolated singularities. Analogous equivariant analogues, with local-global formulas for Atiyah-Singer classes and their relations to G-signatures, will be presented in a future paper.

The equivariant index of c*-elliptic operators

Abstract: Let G be a compact Lie group, and A a C*-algebra with identity. A K-theory of G-equivariant A-vector bundles is developed along with a corresponding theory of Fredholm operators, and the analytic and topological indices of an elliptic equivariant pseudodifferential operator over a C*-algebra A are defined. An index theorem generalizing the Mishchenko–Fomenko theorem is proved. Bibliography: 19 titles.