Showing papers on "Equivariant map published in 1978"

On the lifting of structure groups

Abstract: § i. General principal bundles i. F-structures Let P=(P,~,B,G) be a principal bundle where P and B are topological spaces and @ is a topological group. Let 0: F+G be a continuous homomorphism from a topological group F onto G with kernel K • 0 will be called central, if (i) K is discrete (ii) K is contained in the center of F Thus, in particular, K is abelian. It is easy to check that (ii) follows from (i) if F is connected. Thus every covering projection P: F÷G (F connected) satisfies the conditions above. Throughout this paper we shall assume that p is central. A r-structure on P is a F-principal bundle P=(P,~,B,F) together with a strong bundle map ~: P÷P which is equivariant under the right actions of the structure groups; that is

Finite nilpotent group actions on finite complexes

Abstract: Under certain connectivity and nilpotence hypotheses the group actions of the title are classified (modulo a K 0 ∼ obstruction) up to concordance by the p-localizations of the actions of the p-Sylow subgroups. Smith theory implies that the equivariant homotopy class of these localizations are concordance invariant. Our conclusion is that the original action can be reconstructed, up to concordance, from these localizations.

Equivariant framability of involutions on homotopy spheres

Abstract: Let G be the group Z2. Denote byRn,k theRn+k with non trivial G-action on the first n coordinates. Let ɛn,k be the trivial bundle with fibreRn,k. We say that a G-manifold M is (n,k)-framable if t(M)= =ɛn,k in KOG(M) with t(M) the tangent bundle of M. We show that if G acts on a homotopy sphere ∑n+k such that the fixed point set is a k-dimensional homotopy sphere then ∑ is (n,k)-framable.

The fixed-point construction in equivariant bordism

Abstract: Consider the bordism fl (G) of smooth G-actions. If AT is a subgroup of G, with normalizer NK, there is a standard NK/ Af-action on £2 (ÍQ(A11, Proper). If M has a smooth G-action, a tubular neighborhood of the fixed set of K in M representó an element of fl^XAU, Proper)\"*/*. One thus obtains the \"fixed point homomorphism\" carrying fl,(G) to the sum of the Q (ATXA11, Proper)\"*/*, summed over conjugacy classes of subgroups K. Let P be the collection of primes not dividing the order of G. We show that the ^-localization of is an isomorphism, and give several applications. 1. The fixed-point homomorphism. Let G be a finite group. We shall be concerned with smooth actions of G on compact manifolds M, preserving either an orientation or a unitary structure. The notation is based on that of Stong [8], [9]. The unadorned symbol S2„ denotes either the oriented bordism ring fi^° or the unitary bordism ring fi^. If A\" is a subgroup of G, then AK is the family of all subgroups of G conjugate to subgroups of K, and PK is the family of subgroups conjugate to proper subgroups of K. Thus ß„(G) or fi,(G)(/lG) is the bordism of all G-actions, while ß„(G)(/iG, PG) is the bordism of all G-actions on manifolds with boundary, such that each point x of 9A/ has isotropy subgroup Gx in PG. Beginning with the famous monograph of Conner and Floyd [2], most research in equivariant bordism has involved fixed-point constructions of the following sort. Let K be a subgroup of G. If M is a closed manifold with a smooth Abaction, then the fixed set of K in M has a ^-invariant tubular neighborhood N. Assigning N to M defines a homomorphism fK: fi»(ÄT) -» ti„(K)(AK, PK), which is of interest because the \"relative\" group Q,(K)(AK, PK) is generally easier to compute than is Ü^(K). If r£: ñ»(G) -» &t(K) is the forgetful homomorphism restricting G-actions to AT-actions, there is the composition/*/■£: fi»(G) -» Ü^K^AK, PK). Definition. If AT is a subgroup of G, let (AT) be the set of subgroups conjugate to K in G. If *¥' Q *$ are families of subgroups of G, in the sense of Received by the editors September 19, 1977. AMS (MOS) subject classifications (1970). Primary 57D85.

Characteristic numbers of -manifolds and multiplicative induction

Abstract: It was shown in [6], [7] that suitable global characteristic numbers determine the bordism classes of unoriented G-manifolds if G (ZJk and of unitary G-manifolds if G is cyclic. We show in this paper that there are no other cases in which characteristic numbers determine bordism classes. The proof is based on an explicit computation of the equivariant characteristic numbers of certain manifolds and on the use of a new construction in bordism theory, which we call multiplicative induction, and which should have many more applications. In ? 1 we review some well-known facts about characteristic numbers and explain the notation used in ?2 to state the main results of this paper. ?3 contains calculations. The definition of multiplicative induction and some applications are given in ?4.

Equivariant sheaf cohomdlogy

Abstract: In this paper we study Grothendieck's equivariant sheaf cohomology H(X,G;G) for non-discrete topological groups G and G-sheavesG on a G-Space X. For compact groups and locally compact, totally disconnected groups we obtain detailed results relating H(X,G;-) to H(X;-)G and H(X/G;-). Furthermore we point out the connection between H(X,G;-) and Borel's equivariant cohomology HG(X;-).

Classification of SO(3)-Actions on Five Manifolds

Abstract: P. Orlik and F. Raymond showed that some invariants classify smooth 3-manifolds with smooth ^-action, up to equivariant diffeomorphism (preserving the orientation of the orbit space if it is orientable) [6]. And R. W. Richardson JR. studied SO (3) -actions on S [7]. Also, K. A. Hudson classified smooth SO (3) -actions on connected, simply connected, closed 5-manifolds admitting at least one orbit of dimension three [2]. In this paper, we discuss the equivariant classification of smooth SO (3) -actions on closed, connected, oriented, smooth 5-manifolds such that the orbit space is an orientable surface. We call oriented 50(3)manifolds M and N are equivalent if there is an equivariant homeomorphism between M and N which induces an orientation preserving homeomorphism of the orbit spaces M* to N*. Since there exist various types as the principal orbit, we classify SO (3) -manifolds about each type. It is well known that every subgroups of SO (3) are conjugate to one of the following [4], [5]. 50(2), 0(2), Zn, dihedral group Dn= {x, y ; x =y»= (xyY = \], tetrahedral group T={x,, y ; x= (xyY = y*= 1}, octahedral group O=[x, y\ x= (xyY=y = 1}, and icosahedral group 1= [x, y ; x —

Witt classes of integral representations of an abelian 2-group

Abstract: In this paper the Witt groups of integral representations of an abelian 2-group 7T, WO(7; Z) and W2(7; Z) are calculated. Invariants are listed which completely determine WO(Z4; Z) and W2(Z4; Z) and can be extended to the case 7T = Z2k. If 7T is an elementary abelian 2-group, it is shown that W2(7T; Z) = 0 and Wo(7T; Z[ 2 J) is ring isomorphic to the group ring W(Z [ 2 ])(Hom(r, Z2)). In [1], Alexander, Conner, Hamrick and Vick studied the Witt classes of integral representations of an abelian p-group; however, their results focused on the case where p is an odd prime. In this paper, we study the case where p = 2. Our interest in this algebra stems from an interest in the bordism of manifolds with a differentiable action of an abelian 2-group, say 'r, and the Atiyah-Bott homomorphism ab: 0 (S)-* W*(,r; Z) (cf. [3]) provides a very convenient bordism invariant. The algebra which we will develop here provides the really essential information for a study along the lines of [3]. For reasons of length, bordism related results will appear in a subsequent publication. The groups W*(Z2k; Z) are computed in ? 1, and we show that VW0 has rank 2 1+ 1 and torsion subgroup isomorphic to Z2 while W2 is a free abelian group of rank 2k 1 1. Complete invariants for W*(Z4; Z) are exhibited in ?2; these will be applied to equivariant bordism theory elsewhere. In ?3, we show that for an arbitrary finite abelian 2-group v, the rank of Wk(Qr; Z) is equal to I [Order(r)(1 1/2L) + (1 + (-I)k/2) Order(Hom(,r, Z2))], where L = log2(Order(v)) dim Hom(,r, Z2). Furthermore the torsion subgroup is isomorphic to the group ring Z2(Hom(v, Z2)) if k = 0 and is trivial if k = 2. In the case that v is an elementary abelian 2-group, we establish the fact that Wk(Qr; Z[2 ]) is ring isomorphic to the group ring W(Z[ I ])(Hom(,r, Z2)) if k = 0 and is trivial if k = 2. Although the rank of Wk(v; Z) is in general quite large, the decomposition Received by the editors November 29, 1976 and, in revised form, March 28, 1977. AMS (MOS) subject classifications (1970). Primary IOC05; Secondary 57D85.

On The Equivariant Isotopy Classes of Some Equivariant Imbeddings of Spheres

Abstract: The purpose of this paper is to study the G-isotopy classes of G-imbeddings of spheres into spheres, where the spheres are equipped with semi-free linear G-actions for a finite group G. Let V be an ra-dimensional real G-module. Throughout this paper we shall assume that V is a product module V=JR"0yi of a trivial real G-module R of positive dimension n and an (m—ri)dimensional real G-module Vj on the G-invariant unit sphere S(V^) of which G acts freely. Let W be a real G-module which contains V as a direct summand. Let S and S denote the one-point compactifications of V and W respectively. Then S and S are spheres on which G acts linearly. The direct sum of d copies of V will be denoted by dV.

Equivariant estimators for structural models

Abstract: For invariant statistical models, the use of invariant decision rules in estimation problems with an invariant loss function is prominent in the literature and is fully discussed in the books by Ferguson [2] and Zacks [7] This paper shows that if an invariant statistical model is also a structural model, then the search for minimum risk equivariant estimator may be facilitated by means of a property enjoyed by equivariant estimators for structural models

Equivariant smoothing theory

Abstract: In fact, define smooth local coordinates on M by pulling back the local coordinates on V via the local homeomorphisms. We will denote this smooth structure by Ma. Recall that the differential of a smooth immersion ƒ: V\" -» V2 induces a bundle homomorphism df: TVX -> TV2 of the tangent vector bundles which is an isomorphism on fibres. Call such a bundle homomorphism a representation and let R(TVV TV^) be the space of representations with the C°-topology and I\"*(VX, V£ the space of smooth immersions with the C topology. The Smale-Hirsch theorem for manifolds of the same dimension states: