Showing papers on "Equivariant map published in 1995"

Morse-Bott theory and equivariant cohomology

Abstract: Critical points of functions and gradient lines between them form a cornerstone of physical thinking. In Morse theory the topology of a manifold is investigated in terms of these notions with equally profound success: Smale proved the h-cobordism and generalized Poincare conjectures using surgery cobordisms, see [Mi2].

Spherical functions on affine Lie groups

Abstract: We describe vector valued conjugacy equivariant functions on a group K in two cases -- K is a compact simple Lie group, and K is an affine Lie group. We construct such functions as weighted traces of certain intertwining operators between representations of K. For a compact group $K$, Peter-Weyl theorem implies that all equivariant functions can be written as linear combinations of such traces. Next, we compute the radial parts of the Laplace operators of $K$ acting on conjugacy equivariant functions and obtain a comple- tely integrable quantum system with matrix coefficients, which in a special case coincides with the trigonometric Calogero-Sutherland-Moser multi-particle system. In the affine Lie group case, we prove that the space of equivariant functions having a fixed homogeneity degree with respect to the action of the center of the group is finite-dimensional and spanned by weighted traces of intertwining operators. This space coincides with the space of Wess-Zumino-Witten conformal blocks on an elliptic curve. We compute the radial part of the second order Laplace operator on the affine Lie group acting on equivariant functions, and find that it is a certain parabolic partial differential operator, which degenerates to the elliptic Calogero-Sutherland-Moser hamiltonian as the central charge tends to minus the dual Coxeter number (the critical level). Quantum integrals of this hamiltonian are obtained as radial part of the higher Sugawara operators which are central at the critical level.

Rational Points of Bounded Height on Compactifications of Anisotropic Tori

Abstract: We investigate the analytic properties of the zeta-function associated with heights on equivariant compactifications of anisotropic tori over number fields. This allows to verify conjectures about the distribution of rational points of bounded height.

Generalizing univariate signed rank statistics for testing and estimating a multivariate location parameter

Abstract: We generalize signed rank statistics to dimensions higher than one. This results in a class of orthogonally invariant and distribution free tests that can be used for testing spherical symmetry/location parameter. The corresponding estimator is orthogonally equivariant. Both the test and estimator can be chosen with asymptotic efficiency 1. The breakdown point of the estimator depends only on the scores, not on the dimension of the data. For elliptical distributions, weobtain an affine invariant test with the same asymptotic properties, if the signed rank statisticis applied to standardized data. We also present a method for computing the estimator numerically, and consider a real data example and some simulations. Finally, an application to detection of time-varying signals in spherically symmetric noise is given.

An equivariant version of Grauert's Oka principle

Abstract: For example, a theorem of Oka asserts that every holomorphic line bundle over a domain of holomorphy in (12" is holomorphicaIly trivial if and only if it is topologically trivial. In [G3] Grauer t proves that the category of bundles over a Stein space X with a complex Lie group as a structure group is the same up to isomorphism if considered in the topological or complex analytic category. We would like to underl ine that Grauert ' s method of proof also has far reaching consequences (see e.g. [FR1], [FR2]). In this article we prove the equivariant version of Grauert ' s Oka Principle for a compact Lie group of holomorphic t ransformations on X. The precise statements can be found in w w and w For example, as a consequence of the main results we obtain:

Equivariant sheaves on toric varieties

Abstract: © Foundation Compositio Mathematica, 1995, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). 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.

Deleting-Inserting Theorem for smooth actions of finite nonsolvable groups on spheres

Abstract: The paper presents a method which allows to construct smooth finite nonsolvable group actions on spheres with prescribed fixed point data. The idea is to consider an action on a disk with the required fixed point data, and then to apply equivariant surgery to the equivariant double of the disk to remove the second copy of the fixed point data. In this paper, the method is applied to construct smooth group actions on spheres with exactly one fixed point, and more general actions with fixed point set diffeomorphic to any given closed stably parallelizable smooth manifold. The method is expected to be useful for constructions of smooth group actions on spheres with more complicated fixed point data.

Smooth group actions on definite $4$-manifolds and moduli spaces

Abstract: In this paper we give an application of equivariant moduli spaces to the study of smooth group actions on certain 4-manifolds. A rich source of examples for such actions is the collection of algebraic surfaces (compact and nonsingular) together with their groups of algebraic automorphisms. From this collection, further examples of smooth but generally nonalgebraic actions can be constructed by an equivariant connected sum along an orbit of isolated points. Given a smooth oriented 4-manifold X which is diffeomorphic to a connected sum of algebraic surfaces, we can ask: (i) which (finite) groups can act smoothly on X preserving the orientation, and (ii) how closely does a smooth action on X resemble some equivariant connected sum of algebraic actions on the algebraic surface factors of X? For the purposes of this paper we will restrict our attention to the simplest case, namely X p2(C) #...# p2(C), a connected sum of n copies of the complex projective plane (arranged so thatX is simply connected). Furthermore, ASSUMPTION. All actions will be assumed to induce the identity on H,(X, Z).

Change of universe functors in equivariant stable homotopy theory

Abstract: One striking difference between nonequivariant and equivariant stable homotopy is that, in the equivariant context, one must specify those representations with respect to which spectra are to be stable. One may specify stability with respect only to trivial representations (thereby obtaining what is often called the naive equivariant stable category), with respect to all representations (thereby obtaining the full equivariant stable category), or with respect to any intermediate collection of representations closed under direct sums. The chosen family of representations is usually described by specifying an indexing universe. Change of universe functors transform spectra stable with respect to one set of representations into spectra stable with respect to a second set of representations. This is done either by restriction (that is, by forgetting the stability with respect to some representations) or by induction (that is, by altering the spectra so that they become stable with respect to a larger class of representations). The impact of these transformations on the equivariant homotopy groups of spectra should be viewed as an equivariant generalization of the passage between unstable and stable homotopy groups in the nonequivariant context. Three results concerning this impact are given. One describes when change of universe functors are isomorphisms of categories. The second completely describes the impact of an arbitrary induction functor on the first nonvanishing homotopy groups of a bounded-below spectrum. The third gives a spectral sequence which describes the behavior of an arbitrary induction functor on all the homotopy groups of an arbitrary spectrum. Introduction. This paper continues the study begun in [13] and [12] of the equivariant Hurewicz and suspension maps. In [12], it was shown that direct equivariant generalizations of the Freudenthal suspension theorem necessarily suffer from at least one of two defects—either their hypotheses are unduly restrictive, or they describe the effect of suspension only on the bottom nonvanishing homotopy groups rather than on the homotopy groups in a range of dimensions. One of the purposes of the present paper is to introduce a spectral sequence, promised in [12], which ameliorates this 1991 Mathematics Subject Classification: Primary 55M35, 55P42, 55P91, 55T99, 57S15; Secondary 55N91, 55P20, 55Q10, 55Q91.

Equivariant connected sums of compact self-dual manifolds

Abstract: Due to a recent theorem of Taubes, self-dual conformal structures on compact four-dimensional manifolds exist in abundance [43]. Therefore, finding examples in general is no longer an issue in this subject. To deepen our understanding of self-dual conformal geometry, we take the path of trying to understand the possible symmetry groups of this geometry; i.e. the groups of orientation preserving conformal transformations. The first systematic approach which constructs a large collection of compact self-dual manifolds with a non-trivial group of symmetries is LeBrun's hyperbolic Ansatz [22]. The condition of this Ansatz is that the symmetry group contains an S l := U(1) which acts on the manifold semi-freely, meaning that the isotropy group at any point of the manifold is either trivial or the entire group S I. With this condition on the group action and some favourable topological conditions on the underlying manifold, the geometrical equations of self-duality on the given four-dimensional manifold are reduced to the S 1-monopole equations on hyperbolic 3-space. With this setup, LeBrun explicitly constructed a large number of compact self-dual manifolds with semi-free S l_symmetry including the FubiniStudy metric on the complex projective plane CI? 2 and a one-parameter family of self-dual metrics on C ~ # C ~ first found in [38]. We shall call the self-dual metrics on nCI? 2, n >_ 1, constructed by LeBrun's hyperbolic Ansatz, the LeBrun metrics. After this very successful Ansatz, it is natural to try to find new examples which are beyond the scope of that method.

Determinant Bundles, Quillen Metrics, and Mumford Isomorphisms Over the Universal Commensurability Teichm\"uller Space

Abstract: There exists on each Teichmuller space $T_g$ (comprising compact Riemann surfaces of genus $g$), a natural sequence of determinant (of cohomology) line bundles, $DET_n$, related to each other via certain ``Mumford isomorphisms''. There is a remarkable connection, (Belavin-Knizhnik), between the Mumford isomorphisms and the existence of the Polyakov string measure on the Teichmuller space. This suggests the question of finding a genus-independent formulation of these bundles and their isomorphisms. In this paper we combine a Grothendieck-Riemann-Roch lemma with a new concept of $C^{*} \otimes Q$ bundles to construct such an universal version. Our universal objects exist over the universal space, $T_\infty$, which is the direct limit of the $T_g$ as the genus varies over the tower of all unbranched coverings of any base surface. The bundles and the connecting isomorphisms are equivariant with respect to the natural action of the universal commensurability modular group.

Stably trivial equivariant algebraic vector bundles

Abstract: Let G be a reductive algebraic group over C, let F be a G-module, and let B be an affine G-variety, i.e., an affine variety with an algebraic action of G. Then B x F is in a natural way a G-vector bundle over B, which we denote by F. (All vector bundles here are algebraic.) A G-vector bundle over B is called trivial if it is isomorphic to F for some G-module F. From the endomorphism ring R of the G-vector bundle S, we construct G-vector bundles over B. The bundles constructed this way have the property that when added to S they are isomorphic to F e S for a fixed G-module F. They are called stably trivial. The set of isomorphism classes of G-vector bundles over B which satisfy this condition is denoted by VEC(B, F; S). For such a bundle E we define an invariant p(E) which lies in a quotient of R. This invariant allows us to distinguish non-isomorphic G-vector bundles. When B is a Gmodule, a G-vector bundle over B defines an action of G on affine space. We give criteria which in certain cases allow us to distinguish the underlying actions. The construction and invariants are applied to the following two problems:

Gribov horizon, contact terms and \v{C}ech- De Rham cohomology in 2D topological gravity

Abstract: We point out that averages of equivariant observables of 2D topological gravity are not globally defined forms on moduli space, when one uses the functional measure corresponding to the formulation of the theory as a 2D superconformal model. This is shown to be a consequence of the existence of the Gribov horizon {\it and} of the dependence of the observables on derivatives of the super-ghost field. By requiring the absence of global BRS anomalies, it is nevertheless possible to associate global forms to correlators of observables by resorting to the \v{C}ech-De Rham notion of form cohomology. To this end, we derive and solve the ``descent'' of local Ward identities which characterize the functional measure. We obtain in this way an explicit expression for the \v{C}ech-De Rham cocycles corresponding to arbitrary correlators of observables. This provides the way to compute and understand contact terms in string theory from first principles.

A splitting lemma for equivariant dynamics

Abstract: It is shown that a number of interesting and useful results in equivariant dynamics can be seen as a consequence of a general ‘splitting principle’. Both the results and the principle are actually embodied by the Michel theory of symmetry breaking.

Isotropic geometry and twistors in higher dimensions. II. Odd dimensions, reality conditions, and twistor superspaces

Abstract: This work is a continuation of the study of higher‐dimensional generalizations of the twistor correspondence in conformally compactified Minkowski space begun in Part I [J. Harnad and S. Shnider, J. Math. Phys. 33, 3197–3209 (1992)]. The case of odd‐dimensional spaces is treated, with an equivariant diffeomorphism (Cartan map) defined between Grassmannians (or flag manifolds) of totally isotropic subspaces of a complex vector space with respect to a nondegenerate quadratic form, and minimal orbits in the Grassmannians (or flag manifolds) based upon spinor modules. Then the Cartan maps for the real forms of signature (p,q) are considered. The extensions to super‐Grassmannians are also discussed, with emphasis upon the cases of dimensions 4, 6, and 10. It is found that in dimensions greater than 4, the natural choice for the superdouble flag space correspondence does not result in the expected supernull line foliation arising in the application to the supersymmetric Yang–Mills equations although in dimensio...

A Shapiro lemma for diagrams of spaces with applications to equivariant topology

Abstract: © Foundation Compositio Mathematica, 1995, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). 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.

Cyclic bordism and rack spaces

Abstract: This thesis falls into two parts, the first explores a cyclic version of bordism and the second studies the homotopy groups of rack spaces. In chapters 2-5 we begin by reviewing some theory of cyclic homology but we present it in a topological framework. Then cyclic bordism is introduced as a parallel theory. In particular we prove the equivalence of cyclic and equivariant theories. This enables us to reduce the question of representation of cyclic homology by cyclic bordism to that of representation of ordinary homology by bordism. Finally, we state a fixed point theorem of periodic bordism. In chapters 6-10 we study rack spaces, or the classifying spaces of racks. The homotopy groups of rack spaces are invariants of the rack up to rack isomorphism, and give invariants of semiframednon-split (irreducible) links ill the three-sphere. We describe methods for calculating the second homotopy group in chapter 7 and in the next chapter we apply one of the methods to find generators for the second homotopy of a class of racks, the finite Alexander quotients. Chapter 9 discusses topological racks. The classifying spaces of racks with a non-discrete topology have a cell structure and, although it fails to be a CVV cell structure, it can be used to calculate homotopy groups. The third homotopy group of a rack space is seen to be in one-to-one correspondence with bordism classes of framed labelled immersed surfaces in the three-sphere. We finish in chapter 10 by simplifying such surfaces within bordism to calculate the third homotopy group of the trivial rack and the cyclic racks, [pie]3(B(Cn))=Z2.

Time-periodic spatially periodic planforms in Euclidean equivariant partial differential equations

Abstract: In Rayleigh-Benard convection, the spatially uniform motionless state of a fluid loses stability as the Rayleigh number is increased beyond a critical value. In the simplest case of convection in a pure Boussinesq fluid, the instability is a symmetry-breaking steady-state bifurcation that leads to the formation of spatially periodic patterns. However, in many double-diffusive convection systems the heat-conduction solution actually loses stability via Hopf bifurcation. These hydrodynamic systems provide motivation for the present study of spatiotemporally periodic pattern formation in Euclidean equivariant systems. We call such patterns planforms . We classify, according to spatio-temporal symmetries and spatial periodicity, many of the time-periodic solutions that may be obtained through equivariant Hopf bifurcation from a group-invariant equilibrium. Instead of focusing on plan- forms periodic with respect to a specified planar lattice, as has been done in previous investigations, we consider all planforms that are spatially periodic with respect to some planar lattice. Our classification results rely only on the existence of Hopf bifurcation and planar Euclidean symmetry and not on the particular dif­ferential equation.

The Breden–Löffler Conjecture

Abstract: We give a brief exposition of results of Bredon and others on passage to fixed points from stable C 2 equivariant homotopy (where C 2 is the group of order two) and its relation to Mahowald's root invariant. In particular we give Bredon's easy equivariant proof that the root invariant doubles the stem; the conjecture of the title is equivalent to the Mahowald-Ravenel conjecture that the root invariant never more than triples the stem. Our main result is to verify by computation that the algebraic analogue of this holds in an extensive range: this improves on results of [Mahowald and Shick 1983].

Eqivariant surgery theory : Construction of equivariant normal maps

Abstract: This procedure is one of the important ideas of equivariant surgery theory, and has enabled us to construct various exotic actions (see e.g. [BMol-2], [LaMo], [LaMoPa], [Mo 1-3], [MoU], [Pel-3], [PeR]). A method for (Step I) was presented by T. Petrie in [Pel-3], which we call the equivariant transversality construction. Roughly speaking, it is as follows: Let G be a finite group, and let Y be a compact, smooth G-manifold. If F is a real G-module and a : Y x F > 7 x F i s a proper G-map then a is properly G-homotopic to /?: Y x V -> Y x V such that j8 is transversal to Y x {0}. Then we obtain a G-normal map f:X-+Y, where X = p~(Yx {0}) and /= 0\x: X-+ Y. (Step II) is to convert f\X-*Y to a G-map f':X'^>Y belonging to a prescribed class of maps, e.g. of G-homotopy equivalences, of homotopy equivalences, of Zp-homology equivalences, etc. If some properties of X' are specified before the construction then it is a key to find an adequate real G-module V and an appropriate G-map a. Modified equivariant transversality construction has been employed in [BMol-2], [LaMoPa], [LaMo], [Mol-3], [MoU]. For