Showing papers on "Equivariant map published in 1983"

Systems of fixed point sets

Abstract: Let G be a compact Lie group. A canonical method is given for constructing a C-space from homotopy theoretic information about its fixed point sets. The construction is a special case of the categorical bar construction. Applications include easy constructions of certain classifying spaces, as well as C-EilenbergMac Lane spaces and Postnikov towers. 0. Introduction. Let G be a compact Lie group and X a G-space. The equivariant homotopy theory of X is reflected to a remarkable extent in its system of fixed point sets, defined as a functor from a certain category 0G to Top, the category of topological spaces. (Our spaces will be compactly generated weak Hausdorff; they may or may not be equipped with a basepoint, depending on the context.) These functors, or systems, have considerable technical advantages over G-spaces; it is easy to apply most homotopy theoretic constructions to them, whereas in many cases it is unclear how to proceed for G-spaces. It is the purpose of this paper to present a canonical way of recovering from any system a G-space which preserves all the homotopy theoretic structure of the system. This allows us to give easy equivariant versions of some standard topological constructions such as Eilenberg-Mac Lane spaces and Postnikov towers, and to simplify other equivariant constructions.1 1. Statements of the main theorems. Throughout, G is a fixed compact Lie group. Definitions. The category of canonical orbits, written 0G, is a topological category with discrete object space \0G\ = (G/77: 77 a closed subgroup of G} and morphisms the G-maps, topologized by requiring the natural bijection (•) Hoxn0ciG/H,G/K)^{G/K)H to be a homeomorphism. By an Oc-space we shall mean a continuous contravariant functor from 0G to Top; these functors form the objects of a topological category in the usual manner. We will also consider GG-rings, Oc-groups, etc., defined similarly. Definition. Let A1 be a G-space. The fixed point set system of X, written $X, is an GG-space defined as follows: $ *( G/77) = X", Received by the editors October 30, 1981 and, in revised form, April 26, 1982. 1980 Mathematics Subject Classification. Primary 57S10, 55N25.

The spherical building and regular semisimple elements

Abstract: Let G be a connected reductive algebraic group defined over a finite field k. The finite group G(k) of k-rational points of G acts on the spherical building B(G), a polyhedron which is functorially associated with G. We identify the subspace of points of B(G) fixed by a regular semisimple element s of G(k) topologically as a subspace of a sphere (apartment) in B(G) which depends on an element of the Weyl group which is determined by s. Applications include the derivation of the values of certain characters of G(k) at s by means of Lefschetz theory. The characters considered arise from the action of G(k) on the cohomology of equivariant sheaves over B(G).

Limits of stable homotopy and cohomotopy groups

Abstract: In this paper we formulate and prove generalizations of a theorem of Lin [7]. Let X be a CW complex with base point x0. Define a free involution T on S∞×(X Λ X) by T (w, xΛy) = (−w, yΛx). The quadratic construction on X is the complexThis construction can be applied to spectra. A complete and thorough account will appear in the work on equivariant stable homotopy theory in preparation by L. G. Lewis, J. P. May, J. McLure and M. Steinberger. Some of the results are announced in [8].

The equivariant Dold theorem mod $k$ and the Adams conjecture

Abstract: In this paper, we state and prove a $G$-equivariant version of the Dold Theorem mod $k$ for finite groups $G$. We then use this theorem to prove an equivariant version of the Adams Conjecture for $G$ cyclic, using the Becker-Gottlieb approach. The case for general $G$ and finite structure groups is also obtained by the methods of Quillen. We would like to express our gratitude to Professor J. P. May for his encouragement and many useful suggestions, and to the referee for his critical reading of the manuscript, and for his improvements on several of our proofs.

On equivariant J-homomorphism for involutions

Abstract: G-complex we then have an equivariant /G-maρ / : K0G (X) -*• πos'°(X) [14], which becomes a homomorphism if X is a suspension in the usual sense. Let R be the euclidean space R with non trivial G-action on the firstp coordinates and Σ* be the one point compactification of B q9 with oo as base point. We have the canonical isomorphism π°s°(Σp'q)^7rPιqf the (p, q)-th equivariant stable homotopy group of Landweber [9,3] (which is πp+q,p of Bredon [5]), and therefore we get an induced map

Invariantly sufficient equivariant statistics and characterizations of normality in translation classes

Abstract: It is shown that an equivariant statistic S is invariantly sufficient iff the generated a-algebra and the a-algebra of the invariant Borel sets are independent, and that if S is invariantly sufficient and equivariant, then the Pitman estimator for location parameter y is given by S - Eo(S). For independent Xi, *--, X, the existence of an invariantly sufficient equivariant linear statistic is characterized by the normality of Xi, * * *, Xn. Then, the independence of Xi, * * *, Xn is replaced by a linear framework in which there are established characterizations of the normality of X = (Xi, * * *, X,) by properties (invariant sufficiency, admissibility, optimality) of the minimum variance unbiased linear estimator for y. 1. Invariantly sufficient translation equivariant statistics. For fixed n E N and all -y E R, let T1: - n > Rn be the translation given by T1(xi, * *, Xn) = (X1 + -y, * * * x, + -y), (Xi, ** , X-) E R n. Further let ?n = {B E 4j I TY1(B) = B, -y E RI denote the u-algebra of the Borel subsets in R' being invariant under all translations T1, y E R. If Po is any probability measure on q n and if YF Py = (Po)" I -y E R), where (PO) T is the image measure of P0 under T, is the corresponding n-dimensional translation class, we use the symbols E, and Vy for the expectation and variance w.r.t. Py. S: (Rn, 4n) (R M) being a statistic, we use the symbol Es for (a version of) the conditional expectation under S w.r.t. P, A statistic S: (Rn, gfn) (R, X) is called (translation) equivariant, if SoTy =S+ yholds, yE R. With these notations we can formulate a theorem which will be useful for characterizations of normality in translation classes by sufficiency. Actually, only partial sufficiency for invariant indicators is needed.

Sufficient admissibility conditions for scale parameter estimator

Abstract: Although there are a number of results available for the admissibility of the best translation equivariant estimator of the parameter, there is hardly any stated explicitly for the best scale equivariant estimator of the scale parameter. In this paper, we derive sufficient conditions for the admissibility of the scale parameter estimators and compara them. The derivations use the well known results due to Brown [1], Farrell [2], and Portnoy [3]. The loss function has been taken to be quadratic.

Differentiable group actions on homotopy spheres. III. Invariant subspheres and smooth suspensions

Abstract: A linear action of an abelian group on a sphere generally contains a large family of invariant linear subspheres. In this paper the problem of finding invariant subspheres for more general smooth actions on homotopy spheres is considered. Classification schemes for actions with invariant subspheres are obtained; these are formally parallel to the classifications discussed in the preceding paper of this series. The realizability of a given smooth action as an invariant codimension two sub- sphere is shown to depend only on the ambient differential structure and an isotopy invariant. Applications of these results to specific cases are given; for example, it is shown that every exotic 10-sphere admits a smooth circle action. In our previous papers in this series (45,44), we have considered the theory of semifree actions on homotopy spheres as formulated by W. Browder and T. Petrie (10) and M. Rothenberg and J. Sondow (34). Specifically, in the first paper a method was presented for describing (at least formally) those exotic spheres admitting such semifree actions- a problem first posed explicitly by Browder in (3, Problem 1, p. 7) -and the seconid paper extended the whole theory to handle certain actions that are not semifree. This paper will treat another problem posed in Browder's paper (3, Problem 3) regarding invariant subspheres of homotopy spheres with group actions. One motivation for considering this question is that linear actions on spheres generally admit a great assortment of invariant linear subspheres (e.g., if the group is abelian and the dimension is much larger than the group's order), and from this viewpoint the existence of invariant subspheres reflects the extent to which an arbitrary smooth action resembles some natural linear model. In particular, this idea is central to the work of Browder and Livesay on free involutions (8) (compare also (3)). The existence of such subspheres is directly related to the realizability of actions as equivariant smooth suspensions, providing basic necessary conditions for such a realization. We shall also consider a dual problem in this paper; namely, the description of those group actions that can be smoothly equivariantly suspended. Questions of this sort first arose in the study of free involutions (16), and their close

$G$-spaces with prescribed equivariant cohomology

Abstract: Let G be a finite group. In this note we study the question of realizing a collection of graded commutative algebras over Q as the cohomology algebras with rational coefficients of the fixed point sets X" (H < G) of a G-space X. Let dC be a graded commutative algebra over Q. The question of realizing Cz as the cohomology with rational coefficients of a space X is answered by Quillen [4] and more directly by Sullivan [5]. In particular, Sullivan constructs a space X of finite type, i.e. v,(X) is a finitely generated abelian group for every i, which realizes CT. Now let G be a finite group which acts on C from the left by algebra isomorphisms. Because of the functoriality of the constructions in [4] and [5] one can construct a G-space X such that H*(X; Q) -CT, where the isomorphism is G-equivariant. The space X in both cases is a rational space. In this note we consider a more general question. Let 9G be the category of canonical orbits of a finite group G [1]. The objects of 9G are the quotient spaces G/H, where H is a subgroup of G (H < G), and the morphisms are the G-maps between them, where G acts on G/H by left multiplication. DEFINITION 1. A system of graded commutative algebras (GA's) for G is a covariant functor from ?G into the category of graded commutative connected algebras over Q. We recall that a GA 6f is said to be connected if do = Q and is said to be of finite type if d" is a finite-dimensional vector space over Q. Let X be a G-space such that each fixed point set XH, H < G, is nonempty and connected. Given X, a system of GA's H*( X) is defined by H*( X)(G/H _=-H*( X'; Q) on objects of G'. If f: G/H G/K is a G-map, then there exists an element g E G such that g-'Hg < K, and the map f is determined by H F gK. The map f induces a map f: X" by x i gx and therefore a unique map H*(X)(f )-*: H*( X"; Q) H*( XK; Q). The main result of this paper is the following THEOREM 2. Given a system H of connected GA's of finite type, there exists a G-CW-complex X of finite type such that H*( X) -H. Received by the editors February 18, 1983. 1980 Mathematics Subject Classification. Primary 57S17.

On the existence of equivariant embeddings of principal bundles into vector bundles

Abstract: Let G be a finite group and let X be, say, a connected CW-complex of dimension k > 1. Let r £->Xbea principal G-bundle andp: V -> X an /«-dimensional G-vector-bundle with trivial action of G on X. By an equivariant embedding of it into p we understand an equivariant embedding h: E -» V commuting with projections. We prove a general embedding theorem, a main special case of which is the following Theorem. If k < m and if the action of G on V is free outside the zero section for p, then any principal G-bundle tr: E -» X can be embedded equivarianlly into p: V -» X.

Regular Riemannian $s$-manifolds of noncompact type

Abstract: In this note it is proven that a regular Riemannian s-manifold of noncompact type (see below) cannot be immersed isometrically and equivariantly in R". Our notation, terminology and basic facts will be those of [3]. Let (M, {S,}) be a connected periodic regular s-manifold which is metrizable, i.e. there is a Riemannian metric g on M which is invariant with respect to the symmetries {S,: x E M}. (Periodicity means that (M, {S,)) has finite order [3, p. 4].) We have the group of isometries I(M, g) which is transitive on M [3, p. 2]. Contained in I(M, g) we have the group of transvections G = Tr(M, {S }) [3, p. 57] which is generated by the "elementary transvections", i.e. by the isometries Sx o Sy-', x, y E M. About the group G one knows: (1) G is a connected Lie group [3, II 32, 125]. (2) G is transitive on M [3, II 33]. It is known [3, IV 24] that under the above conditions (M, {S5}) admits two complementary foliations F, 2 such that: (a) IF is invariant and its leaves are regular s-manifolds with solvable group of transvections. (b) 2 is weakly invariant and its leaves are regular s-manifolds with semisimple group of transvections (compare [2, p. 208]). DEFINITION. We shall say that (M, {S,}) is of noncompact type if the foliation S2 has noncompact leaves. The objective of this note is to prove the following. THEOREM. Let (M, {S,1) be a connected periodic, regular s-manifold which is metrizable and of noncompact type. Then (M, {S,}) admits no isometric equivariant immersion into a finite-dimensional real representation of G = Tr(M, {S,}). PROOF. Let us assume the existence of such an isometric immersion (p, f): (G, M) -(I(Rn), Rn), where T is a Lie group monomorphism and f is an isometric Received by the editors January 8, 1982. AMS (MOS) subject classifications (1970). Primary 53C40, 53B25.

The nontriviality of the first rational homology group of some connected invariant subsets of periodic transformations

Abstract: This note was inspired by some results of P. A. Smith (S), One proves that for any periodic map of a manifold M and any codimension two invariant submanifold P of M containing part of the stationary point set, connected invariant subsets of the complement of P must carry nontrivial one-dimensional rational cycles, provided that M satisfies some simple homological conditions (Theorem A). This fact has interesting consequences in transformation group theory. 0. Introduction. If m > 2 is a positive integer let Zm — Zm be the cyclic group of order m and if m = oo let Z", resp. Zm be the infinite cyclic group, respectively, 5' = {z G C||z| = 1}. We also denote by Gm ", m, n — 2,3,4,..., bo, a semidirect product Z", X rZ" for t: Z" -» AutZm. Such a semidirect product has the inclusion zm ■* Gm,"> tne projection Gmn - Z" and the section s: Z" -» Gmn (tt ° s = id) as part of the data. Clearly if n — oo, Gmn = Zm X Sl. The groups Gm " are regarded as compact Lie groups. Given a compact Lie group G, p: G X M -> M a topological action, N an invariant submanifold, and x E M, then the action /x: G X (M, N) -> (M, N) is called locally smooth at x if there exists a smooth action s: G X V -» V, an invariant submanifold W E V, and an invariant neighborhood % of x G M together with an equivariant homeomorphism ^(%, % D JV) -* (F, W7). The main result of this note is the following:

Equivariant isotopies of semifree $G$-manifolds

Abstract: In the previous paper [3] we studied the set of equivariant isotopy classes of equivariant smooth embeddings of a sphere with semifree linear action into a euclidean representation space. In this paper we will study more general case, i.e., the set of equivariant isotopy classes of equivariant smooth embeddings of a manifold into another manifold, where the manifolds in question have a smooth semifree action. Let G be a compact Lie group, and M, N smooth G-manifolds. Two smooth G-embeddings / and g of M into N are called G-isotopic, if there is a smooth G-map

Sufficiency of a sufficient statistic for equivariant estimation

Abstract: Suppose an estimation problem is invariant under a group of transformations and one is interested in finding an optimal equivariant estimator. The usual proactice is to confine attention to non-randomized equivariant estimators based on a minimal sufficient statistic. A justification of this restriction to a smaller clas of estimators is given in this paper under certain conditions.