Showing papers on "Equivariant map published in 1973"

How to Conjugate C1-Close Group Actions

Abstract: The existence of a map conjugating two Cl-close G-actions has already been proved by Palais [5]. Palais' proof relies essentially on the fact that there exists a representation of G in an orthogonal group 0 (n) and an equivariant embedding of M in IR". The main tool in our approach is to define a "center of mass" for almost constant maps. This enables us to define a specific map conjugating the two group actions if they are Ca-close. Using this notion of center we prove in the last paragraph a differential geometric version of the theorem:

Estimating the Scale Parameter of the Exponential Distribution with Unknown Location

Abstract: Let $X_1,\cdots, X_n$ be i.i.d. random variables each with a density which is $a^{-1} \exp (b - x)$ or 0 according as $x \geqq b$ or $x 0$ are unknown constants. Let $\bar{X} = n^{-1} \sum X_i$ and $M = \min X_i$. The maximum likelihood estimator of $a$ is $\bar{X} - M$ and is, if quadratic loss is assumed, the best affine equivariant estimator of $a$. It is shown that if loss is measured by any member of a large class of "bowl-shaped" functions which includes quadratic loss, the best affine equivariant estimator is inadmissible. The proof entails an examination of the conditional expected loss given the maximal invariant under the scale group. It is carried out by exhibiting a superior alternative. In the case of quadratic loss, for example, the result is as follows. Given any estimator $u = (\bar{X} - M)T\lbrack M(\bar{X} - M)^{-1}\rbrack$, let $T^\ast(y) = T(y)$ if $y 0$. If $T^\ast eq T$ with positive probability, then the estimator, obtained from $u$ by replacing $T$ by $T^\ast$, has uniformly smaller risk than $u$. Using a generalization of the author's conditions for admissibility [Ann. Math. Statist. 41 (1970) 446-457] a class, $B$, of generalized Bayes estimators within $D$, the class of scale equivariant estimators, are obtained with each member of $B$ admissible in $D$. The improper measures determining members of $B$ have densities on the orbit space $R$, created in the parameter space by the action of the group of scale changes. These prior densities, $g$, satisfy $\int^\infty_1 (t^2g(t))^{-1} dt = \int^{-1}_{-\infty} (t^2g(t))^{-1} dt = \infty$.

Equivariant algebraic topology

Abstract: © Annales de l’institut Fourier, 1973, 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.

Hermitian K-theory and geometric applications

Abstract: Hermitian K - Theory in topology.- Some problems in hermitian K - Theory.- Some problems and conjectures in algebraic K-theory.- Unitary algegraic K-theory.- Foundations of algebraic L-theory.- Periodicite de la K - Theorie hermitienne.- Algebraic L-theory III. Twisted laurent extensions.- Surgery and unitary K2.- Surgery groups and inner automorphisms.- Mayer-vietoris sequences in hermitian K-theory.- Groups of singular hermitian forms.- Proper surgery groups and wall-Novikov groups.- Induction in equivariant K - theory and geometric applications.

Equivariant extensions of maps

Abstract: This paper treats extension and retraction properties in the category *$/9 of compact metric spaces with periodic maps of a prime period p; the subspaces and maps in J^p are called equivariant subspaces and maps, respectively. The motivation of the paper is the following question: Let E be a Euclidean space and α: E X E-> E X E be the involution (x, y) -> (y, x), i.e., the symmetry with respect to the diagonal. Suppose that Z is a symmetric (i.e., equivariant) closed subset of ExE which is an absolute retract; that is, Z is a retract of E X E. When does there exist a symmetric (i.e., equivariant) retraction Ex E-+ZΊ This is an extension problem in the category J2/'p. If X and Y are spaces in J£fp, A is a closed equivariant subspace of X and /: A -> Y is an equivariant map, then the existence of an extension of / does not, in general, imply the existence of an equivariant extension. It is shown, however, that if A contains all the fixed points of the periodic map and dim(X— A) < oo, then a condition for the existence of an extension is also sufficient for the existence of an equivariant extension. In particular, it follows that a finite dimensional space X in Sf 'p is an equivariant ANR (i.e., an absolute neighborhood retract in the category Sf v) if and only if it is an ANR and the fixed point set of the periodic map on X is an ANR. Generally speaking, the paper deals with the question of symmetry in extension and retraction problems.

Oriented and weakly complex bordism algebra of free periodic maps

Abstract: Free cyclic actions on a closed oriented (weakly almost complex, respectively) manifold which preserve the orientation (weakly comnplex structure) are considered from the viewpoint of equivariant bordism theory. The author gives an explicit presentation of the oriented bordism module structure and multiplicative structure of all orientation preserving (and reversing) free involutions. The odd period and weakly complex cases are also determined with the aid of the notion of formal group laws. These results are applied to a nonexistence problem for certain equivariant maps. Introduction. As the oriented analogue of the free equivariant unoriented bordism theory Rt*(X, A, r) of Stong [121, K. Komiya and C. M. Wu have respectively defined the free equivariant oriented bordism theories Q4+(X, A, r) and Q*(X, A, r) for involutions (X, A, r) (Komiya [9]), and Q*(X, A, r) for maps of odd prime period (X, A, r) (Wu [17]). The main object of the present paper is to apply Komiya's theories to the geometrical determination of the oriented bordism algebras Q* (Z2) of all orientationpreserving free differentiable involutions and Q(Z2) of all orientation-reversing free differentiable involutions. (Compare with the semi-geometric methods in Stong [11, Chapter VIII]). We also remark in this paper that the equivariant oriented and weakly complex theories of Wu, together with Mis'cenko's theorem [10, Appendix 1], give rise to a simple proof of the structure theorem for Q*(Z p) [2], U*(ZM, U*(Zm) ([3], [5k[6], [7]) and K'NLn(m)) [8]. These results are applied to the nonexistence problem for equivariant maps. In ?1, we define the bordism groups Q4(X, r) and Q(X, r), and then introduce the external product and the Pontrjagin product in these theories. In ??2 and 3, we give two kinds of direct sum decompositions of Q*(S , a) and Q(Sn, a) into isomorphic copies of Q(S, a). Deviating slightly from the Received by the editors July 28, 1971 and, in revised form, February 4, 1972. AMS (MOS) subject classifications (1970). Primary 57D75, 57D85, 57D90; Secondary 55B20, 55C35, 55G37.

Spaces of equivariant self-equivalences of spheres

Abstract: Let F(ST) denote the identity component of the space of homotopy self-equivalences of STM and let F = inj limw FiS" ). This paper studies the homotopy properties of certain equivariant analogs of the infinite loop space F.

The index of a semifree periodic mapping

Abstract: We consider semi-free periodic mappings of homologic spheres and study the structure of the set of fixed points and the properties of the index of a periodic mapping; we establish relations for the degree of an equivariant mapping.

Extending equivariant maps for compact Lie group actions

Abstract: Extensions of maps are studied in the category of spaces with actions of a compact Lie group G. If G acts on a finite dimensional compact metric space X with a finite number of conjugacy classes of isotropy subgroups,Jf Xis a closed equivariant subspace of X such that the action on X X is free and if/: X -• Y is an equivariant map to a compact metric space Y with a (/-action, then an equivariant neighborhood extension of/exists provided that Fis an ANR; if Fis an AR, then ƒ can be equivariantly extended over X.

An equivariant version of Gromov’s theorem

Abstract: In this note we announce an equivariant version of the theorem of Gromov [2], [3], [7] concerning the classification of smooth sections of a differentiable fibre bundle whose r-jets satisfy an \"intrinsic differential inequality\". The development of Gromov's theorem began with the Smale-Hirsch theory of immersions [8], [5], which was clarified and generalized by Phillips [6], Haefliger and Poenaru [4] and Gromov. Phillips' submersion theorem makes clear the essential role played by the assumption that the source manifold is nonclosed (i.e., no compact component meets the boundary); in fact the immersion theorem in positive codimension can be deduced from the submersion theorem using the (nonclosed) normal bundle of the source manifold in the target. I would like to thank my thesis advisor, Richard S. Palais, for his encouragement and generous advice.