Showing papers on "Equivariant map published in 1981"

The equivariant Dehn's lemma and loop theorem

Abstract: In [4] the authors observed that the topological methods in the theory of three-dimensional manifolds can be modified to settle some old problems in the classical theory of minimal surfaces in euclidean space (see also [1], [12]). In [4] and [5] we found that we could use the theory of minimal surfaces to extend the theorems of Papakriakopoulous, Whitehead and Shapiro, Stalling and Epstein on the Dehn's lemma, loop theorem and sphere theorem. The key point to our approach to these topological theorems is the following: Given a certain family of maps of the disk or sphere into our three-dimensional manifold M, we minimize the area of the maps (with respect to the pulled back metric) in this family and prove the existence of the minimal map. Then by using the area minimizing property of the map and the tower construction in topology, we prove that any area minimizing map in the family is an embedding. In this way, we realize the solutions to the above topological theorems by minimal surfaces. In [4] and [5] we used the above area minimizing solutions to prove equivariant versions of the loop and the sphere theorem, and we applied these new theorems to the classification of compact group actions on R 3 in [11]. In this paper we generalize some of the theorems in [4] and [5] to compact planar domains by proving the existence of embedded planar domains of least area of a given genus and by proving a certain disjointness property for planar domains of least area. We then use this disjointness property to prove the equivariant Dehn's lemma for planar domains. On the other hand, we use a different variation approach to get a geodesic version of the loop theorem. More precisely, we prove the following: suppose that the induced map i.:~rl(OM)--* \"rr~(M) of the inclusion of the boundary has nontrivial kernel K. Then for any metric on OM, any nontrivial geodesic of least length in K is embedded and any two such geodesics are equal or disjoint. This geodesic loop theorem coupled with the above equivariant Dehn's lemma yields a new version of the equivariant loop theorem in [5]. As the placement of curves on a surface is easier to understand this new equivariant loop theorem is easier to

An equivariant extension theorem and G-retracts with a finite structure

Abstract: Let G be a compact Lie group. An equivariant extension theorem for G-spaces with a finite structure is proved. This theorem is then used to give a characterization of G-ANR's and G-AR's for G-spaces with a finite structure.

Equivariant embeddings of low dimensional symmetric planes

Abstract: We determine the class of all locally compact stable planesM of positive dimensiond≤4 which admit a reflection at each point of some open setU\(U \subseteq M\)M. Apart from the expected possibilities (planes defined by real and complex hermitian forms, and almost projective translation planes), one obtains (subplanes of)H. Salzmann's modified real hyperbolic planes [14; 5.3] and one exceptional plane which was not known before. The caseU=M has been treated [9] and is reproved here in a simpler way. The solution to the problem indicated in the title constitutes the main step in the proof of our results.

On the equivariant homotopy type of -ANRs

Abstract: I show that every metric G-ANR has the G-homotopy type of a G-CW complex. Therefore I. James and G. Segal's results concerning equivariant homotopy type are special cases of the Whitehead theorem for G-CW complexes. In this note G is assumed to be a compact Lie group. A metric G-space X is said to be a G-ANR if for any G-embedding i: X -> Y in a metric G-space Y such that iX is closed in Y, the image iX is a G-retract of some open invariant neighborhood of Y. THEOREM. Every metric G-ANR has the G-homotopy type of a G-CW complex. As a consequence of this theorem and Theorem 5.3 in [3] we obtain the following. COROLLARY (JAMES, SEGAL [1]). Let f: X -> Y be a G map between G-ANR's. Then f is a G-homotopy equivalence iff fH: XH _ yH is an ordinary homotopy equivalence for every closed subgroup H C G. PROOF OF THEOREM. Let X be a metric G-ANR. By the equivariant version of a standard argument (Lemma 4.7 of [4]), it suffices to prove that X is G-dominated by a G-CW complex. Observe that every metric G-space X may be G-embedded as a closed G-subset of a convex G-set in a Banach space of bounded real-valued functions on X with G-action given by g(h)(x) = h(g-'(x)) for g E G, h: X -R, and x E X. Therefore we may assume that X is a closed G-subset of a convex G-set C in a Banach G-space. Being a G-ANR, X is a G-retract of some open neighborhood U of X in C; in particular it is G-dominated by some G-CW complex. This is seen by an easy modification of the proof of Theorem 3.B in [2]. We obtain that U is G-dominated by a G-nerve induced by a locally finite refined slice covering and this G-nerve has the G-homotopy type of a G-CW complex (which is a direct limit of barycentric manifolds in the notation of [2]). REMARK. Matumoto's proof that a barycentric manifold is a G-CW complex is incorrect because it relies on a result of Yang which relies on an incorrect result of Cairns. A correct proof that smooth manifolds are G-CW complexes is in a preprint Triangulation of stratified fibre bundles by Andrei Verona. The weaker Received by the editors August 8, 1980. AMS (MQS) subject classifications (1970). Primary 57E10. i 1981 American Mathematical Society 0002-9939/81/0000-0443/$01.50 193 This content downloaded from 157.55.39.58 on Fri, 26 Aug 2016 05:06:43 UTC All use subject to http://about.jstor.org/terms

Robust Projection Pursuit Estimator for Dispersion Matrices and Principal Components.

Abstract: : This paper proposes and discusses the ROBUST PROJECTION PURSUIT ESTIMATOR for dispersion matrices and their principal components. This estimator finds robust principal components by searching, successively, for directions which maximize (minimize) a robust estimate of scale; the estimate of the dispersion matrix is constructed from the estimated principal components. These estimators are shown below (under mild conditions) to have a number of desirable properties. They are orthogonally equivariant and, within any elliptic underlying density family, asymptotically affinely equivariant. Furthermore, at elliptic densities, they are consistent and weakly continuous (i.e., qualitatively robust). Finally, they have good quantitative robustness - their breakdown point can be as high as 1/2. The robust projection pursuit approach is a promising alternatives to other estimators of dispersion matrices. (Author)

A split action associated with a compact transformation group

Abstract: ABsTRAcr. We associate with an effective action of a compact connected Lie group as a pathwise connected space X a split action of a quotient group G/K on the quotient space X/K. One application of the main theorem states that if X is a compact oriented manifold whose principal cohomology class is a cup product of one-dimensional classes then the action of G on X splits. We prove this in the differentiable case; the topological case has since been dealt with by Schultz.

Equivariant cofibrations and nilpotency

Abstract: Let /: B -» Y be a cofibration whose cofiber is a Moore space. We give necessary and sufficient conditions for/ to be induced by a map of the desuspension of the cofiber into B. These conditions are especially simple if B and Y are nilpotent. We obtain some results on the existence of equivariant Moore spaces, and use them to construct examples of noninduced cofibrations between nilpotent spaces. Our machinery also leads to a cell structure proof of the characterization of pre-nilpotent spaces due to Dror and Dwyer [7], and to a simple proof, for finite fundamental group, of the result of Brown and Kahn [4] that homotopy dimension equals simple cohomological dimension in nilpotent spaces. 0. Introduction. Much work has recently been done on the structure of nonsimply connected spaces, particularly nilpotent spaces. It has been shown that in many ways they are just as tractible as simply connected spaces. For example, Brown and Kahn [4] have shown that for nilpotent spaces homotopy dimension equals simple cohomological dimension. In one significant respect, at least, nilpotent spaces remain as obscure as other nonsimply connected spaces: one cannot visualize how the cell structure relates to the homology. The reason is that nilpotency has so far been analyzed using fibrations, not cofibrations. By \"visualize\" I take as paradigm the homology decomposition of [10], whereby it is possible to picture a simply connected space as arising by attaching Moore spaces together, one for each homology group. (A Moore space is a simply connected CW complex having a single nonvanishing homology group.) Dually, it is possible to kill the homology dimension by dimension by successively attaching Moore spaces. This paper is a step toward producing analogous ideas for nilpotent spaces. The first section derives many algebraic lemmas, several of interest in their own right. They all concern finitely generated modules over finitely generated nilpotent groups tt. In Proposition 1.5 we show that if the trivializing map M —> M /IM splits over Z (the integers) and if 77,(77, M) = 0, then H¡(tt, M) = 0 V i > 1. As a corollary, we have the theorem that if 3/i > 1 such that 77n(77, M) is a free Abelian group and H„+x(tt, M) = 0 then H ¡(it, M) = 0 V (' > n + 1. Additional corollaries relating to nilpotent modules are derived. The main technique here is the idea of resolving a module by a free chain complex of length one that is not acyclic, but whose homology modules are perfect. Received by the editors January 7, 1980 and, in revised form, June 24, 1980. AMS (MOS) subject classifications (1970). Primary 55B25, 55D05, 55E35; Secondary 55D30.

The existence of a coadjoint equivariant momentum mapping for a semidirect product

Abstract: Ansrmacr. We consider a symplectic action of a group G on a symplectic manifold P, which admits a momentum mapping. Assume that G is a semidirect product of G1 by G2. We prove that if the symplectic action of GI has a coadjoint equivariant momentum mapping, and if HI(%,: R) = H2(52: R) 0, then the symplectic action of G has a coadjoint equivariant momentum mapping, where 1 and 52 are the Lie algebras of G1 and G2 respectively.

On lattices acting on boundaries of semi-simple groups

Abstract: It is shown that for a lattice Γ in a semi-simple group of real rank 1 the action on the boundary always admits an equivariant topological Γ-factor. We also show that there are no such factors for SL ( n , ℤ) acting on ℙ n −1 , n ≥3.