Showing papers on "Equivariant map published in 1980"

Equivariant homotopy theory and Milnor’s theorem

Abstract: The foundations of equivariant homotopy and cellular theory are examined; an equivariant Whitehead theorem is proved, and the classical results by Milnor about spaces with the homotopy-type of a CW complex are generalized to the equivariant case. The ambient group G is assumed compact Lie. Further results include equivariant cellular approximation and the procedure for replacement of an arbitrary G-space by a G-CW complex. This is the first of a series of three papers based on the author's thesis [Wal], the object of which is to discuss equivariant homotopy theory in general, and equivariant fibrations in particular, culminating in classification theorems for the various categories of equivariant fibrations and bundles. In the present paper, we discuss the foundations of equivariant homotopy theory and cellular theory and prove an equivariant version of Milnor's theorem on spaces having the homotopy type of CW complexes, where we allow a compact Lie group G to act everywhere. The second paper in this series, Equivariant fibratons and transfer sets up the background for the study of (J-fiber spaces and equivariant stable homotopy theory and contains a description of the equivariant transfer for equivariant fibrations with compact fiber. In the third paper, Equivariant classifying spaces and fibrations, the geometric bar construction is used to construct explicit classifying spaces for equivariant bundles and fibrations, these results depending heavily on the equivariant cellular theory presented here. Also in preparation is a fourth paper which will sequel the present series and will deal with the classificaton of oriented G-spherical fibrations and bundles [Wa2]. The three papers are divided as follows: Equivariant homotopy theory and Milnor's Theorem 1. Notations and definitions 2. Equivariant homotopy groups 3. Equivariant cellular theory 4. Milnor's Theorem 5. Approximation of G-CW complexes by G-simplicial complexes 6. Finite dimensional G-simplicial complexes are G-equilocally convex 7. Reasonable GELC spaces are dominated by G-CW complexes Received by the editors September 7, 1978 and, in revised form, January 15, 1979. AMS (MOS) subject classifications (1970). Primary 54H15.

p-Group actions on homology spheres

Abstract: When an arbitraryp-groupG acts on a ℤn-homologyn-sphereX, it is proved here that the dimension functionn:S(G)→ℤ(S(G) is the set of subgroups ofG), defined byn(H)=dimXH, (dim here is cohomological dimension) is realised by a real representation ofG, and that there is an equivariant map fromX to the sphere of this representation. A converse is also established.

Equivariant $G$-structure on versal deformations

Abstract: Let X0 be an algebraic variety, and (x, 2) its versal deformation. Now let G be an affine algebraic group acting algebraically on Xq. It gives rise to a definite linear G-action on the tangent space of 2. In this paper we establish that if G is linearly reductive then there is an equivariant G-action on (x, 2) which induces given G-action on the special fibre X0 and its linear G-action on the tangent space of the formal moduli 2. Furthermore, such equivariant G-structure is shown to be unique up to noncanonical isomorphism. Let X0 be an algebraic variety together with an action of an algebraic group G, defined over a fixed field k. A question we pose here is to see if there exists an equivariant G-structure on versal deformation of X0. To be more precise, we ask if there exists a versal deformation X0 ^ X i I Spec(k) <^* s where we can provide G-action on X extending the given action on X0, and G-action on the parameter scheme s, such that all the maps entering in the above diagram are compatible with those G-actions. In the case when X0 is an affine cone with the obvious Gm-action, an existence theorem was proven by Pinkham in [4] by an elementary technique, and the question for the general case was left open. The purpose of this paper is to establish an existence theorem and uniqueness for the case of linearly reductive group G, generalizing the case of Gm. Indeed we show that if HX(G, — ) = 0 = H2(G, — ) for a class of G-modules determined by X0, then an equivariant G-structure exists, and is unique up to equivariant isomorphism. Our technique is parallel to the original method of M. Schlessinger in proving the existence theorem for versal deformations [8]. A crucial difference is that we have to stay in the category of deformations and not the isomorphism classes of deformations, since we have to deal with a successive extension problem of G-actions and equivariant isomorphisms. The same Received by the editors November 7, 1977. AMS (MOS) subject classifications (1970). Primary 13D10, 14D15; Secondary 32G05.

Equivariant classifying spaces and fibrations

Abstract: Explicit classifying spaces for equivariant fibrations are constructed using the geometric two-sided bar construction. The constructions are then extended to classify stable equivariant spherical fibrations and equivariant K-theory. The ambient groups is assumed compact Lie. In order to be able to prove an equivariant version of the Adams Conjecture [Wal], it is certainly helpful to have a classifying space for equivariant stable spherical fibrations, and to prove that they lead to a generalized equivariant cohomology theory [MHW]. Stasheff first constructed classifying spaces for various categories of fibrations in [Stl], and these have proved to be an indispensable tool for homotopy theorists. The purpose of this paper is to construct explicit classifying spaces for various categories of stable and unstable equivariant fibrations over suitable base spaces. This will be done using a generalized "classifying space machine" based largely on that of Peter May in Classifying spaces and fibrations [Mal]. As a by-product, we shall also obtain explicit classifying spaces for the various categories of (stable and unstable) equivariant bundles, thereby providing alternate models of spaces constructed by R. Lashof and M. Rothenberg in [Lal], as well as new versions of the classifying spaces for equivariant K-theory. (See, for example, [Mol].) In order to be able to construct universal G-fibrations and to prove a classification theorem, the foundations of G-homotopy theory and G-cellular theory must be put in order. This is done in [Wa2] for G-compact Lie, and will enable us to prove our classification theorem with the full generality of G-compact Lie. The foundational theory of G-fibrations is discussed in [Wa3] and will be referred to here as needed. The technique of our approach to the classification will be to restrain the fibers to lie in an appropriate "category of fibers" which (usually) contains a prototype space F with varying actions of closed subgroups of G. F serves as a homotopy model for the fibers of a given fibraton. The concept of a G-fibration with fiber F [Wa3] may serve as a motivating example. (Without such a prototype, the family of classes of equivariant fibrations over a point are too large to be a set.) When F is compact, we can classify up to strict fiberwise G-homotopy, and when F is not compact, we are still able to classify up to weak fiberwise G-homotopy equivalence. Received by the editors September 7, 1978 and, in revised form, January 15, 1979. AMS (MOS) subject classifications (1970). Primary 54H15.

Equivariant maps which are self homotopy equivalences

Abstract: The aim of this note is (i) to give (in §2) a precise statement and proof of the (to some extent well-known) fact mat the most elementary homotopy theory of "simplicial sets on which a fixed Simplicia! group H acts" is equivalent to the homotopy theory of "simplicial sets over the classifying complex WH", and (ii) to use this (in §1) to prove a classification theorem for simplicial sets with an H-action, which provides classifying complexes for their equivariant maps which are self homotopy equivalences 1 The classification theorem In order to formulate our classification theorem (12) we need some definitions involving 11 Simplicial sets with a simplicial group action, (i) Let V and V be simplicial sets on which a simplicial group 77 acts (from the right) A map V-*V then will be called an equivariant weak homotopy equivalence if it is compatible with the action of 77 and is a weak homotopy equivalence This should not be confused with the stronger notion of weak equivariant-homotopy equivalence, which is a map compatible with the action of 77, which induces weak homotopy equivalences on the fixed point sets of all simplicial subgroups of 77 (and not merely the identity subgroup) (ii) Let F be a simplicial set on which a simplicial group 77 acts Then we denote by haut^ V the simplicial monoid of equivariant weak self homotopy equivalences of V, ie, the simplicial monoid which has as «-simplices the commutative diagrams A[«] XV -> A[«] X V proj\ s proj A[«] in which the top map is an equivariant weak homotopy equivalence 12 Classification theorem Let H be a simplicial group and WH its classifying complex [4, p 87] and let M be a minimal simplicial set [4, p 35] and aut M its simplicial group of automorphisms [1, 13], [4, p 74] 77ie« the function complex (If aut M)WH has the following properties: (i) T«e components of ( W aut M)WH are in natural 1-1 correspondence with the equivariant weak homotopy equivalence classes of simplicial sets with an H-action, which have the weak homotopy type of M Moreover each such class contains Received by the editors October 31, 1979 and, in revised form, January 25, 1980 1980 Mathematics Subject Classification Primary 55P15 'This research was supported in part by the National Science Foundation and the Israeli Academy of Sciences © 1980 American Mathematical Society 0002-9939/80/0000-0630/$017S 670 SELF HOMOTOPY EQUIVALENCES 671 simplicial sets which satisfy the extension condition [4,p 2] and on which the H-action is free (ie principal [4,p 70]) (ii) If V is a simplicial set which satisfies the extension condition and has the homotopy type of M and on which H acts freely, then W haut^ V has the weak homotopy type of the component of (W aut M)WH which (see (i)) corresponds to V Proof This is an immediate consequence of the classification theorem [1, 14] for fibrations over WH and the results of §3 13 Corollary Let H be a simplicial group and let G be a group which has no center and no outer automorphisms (such as, for instance, the symmetric group on n letters, where n ¥= 2 or 6) Then (i) there is only one equivariant weak homotopy equivalence class of simplicial sets with an H-action, which have the weak homotopy type of K(G, 1), and (ii) if V is a simplicial set which satisfies the extension condition and has the homotopy type of K(G, 1) and on which H acts freely, then hautw V is contractible (ie has the weak homotopy type of a point) Proof This follows immediately from the fact that aut K(G, 1) is contractible 14 Corollary If V is a contractible simplicial set which satisfies the extension condition and on which a simplicial group H acts freely, then hautw V is contractible 2 The equivalence of homotopy theories Let H be a simplicial group, let SH denote the category of simplicial sets with a (right) H-action and equivariant maps between them, and let S/ WH be the category of simplicial sets over the classifying complex [4, p 87] WH Then the homotopy theories of SH and S/WH are equivalent in the sense that Theorem 21 and Corollary 25 below hold 21 Theorem The simplicial localizations [3, §3] of SH with respect to the equivariant weak homotopy equivalences (11) and of S/ WH with respect to the weak homotopy equivalences over WH are simplicial categories in the sense of [3, §2], which are homotopically equivalent [3, 25] This follows immediately from [2, 35], [3, 41] and the following propositions 22 Proposition 77te categories S/ WH and SH, with fibrations, cofibrations and weak equivalences as defined below, are closed model categories in the sense of Quillen [5]: (i) The model category structure on S/ WH is the one induced by the usual one on the category of simplicial sets S [5, II, 28]; in particular the weak equivalences are the weak homotopy equivalences over WH (ii) A map in SH is a fibration if the underlying map in S is so, is a cofibration if it is 1-1 and its H-action is relatively free (ie no nonidentity simplex of H fixes a simplex of the range which is not in the image of the domain), and is a weak equivalence if it is an equivariant weak homotopy equivalence (11) The proof is straightforward 672 E DROR, W G DWYER AND D M KAN 23 Proposition There is a pair of adjoint functors A: S/WH ^>SH (the left adjoint) and B: SH -» S/1*77 (the right adjoint) such that, in the terminology of 22 (i) both functors send weak equivalences into weak equivalences, and (ii) for every object U G S/ IP77 and every object V G SH, the adjunction maps U -» BA U and ABV -» V are weak equivalences Proof Given an object U: Y^WH G S/WH, one defines AU by [4, §21] A U = Y XiyH WH, with its 77-action induced by the one on WH and, given an object V G SH, one defines BV as the map (V X WH)/H-+ WH G S induced by the projection V x WH-» 1*77 G SH The adjoint of_a map/: AU^ V G SH is the map T« AU/H ^>(V X WH)/H G S over WH induced by the map A U -> V x WH G SH which is the product of / and the obvious map A U = Y X jyH WH -> 1*77 G SH The rest of the proof is a straightforward verification Also not hard to verify is 24 Proposition Let (S/ WH)m and (SH)+ denote the simplicial categories [3, §2] obtained from the model categories S/WH and SH (22) by giving them the obvious simplicial structure (ie function complexes) Then (S/ WH)m and (SH)^ are closed simplicial model categories in the sense of Quillen [5] Moreover, the functors A and B of 23 induce functors A,: (S/WH),^>(SH) and B,: (SH)m ->(S/if77)„ Let (S/ ifOrY)? c (S/ WH)n and (S^)^ c (S„), denote the subcategories generated by the objects which are cofibrant as well as fibrant (ie the fibrations with WH as base and the fibrant simplicial sets with free H-actions) Then the functors An and B^ (24) send these simplicial subcategories into each other and one has, in view of [3, 48]: 25 Corollary The simplicial categories (S/ W-T?)^ and (S#)^f are homotopically equivalent [3, 25] Moreover, they are weakly equivalent [3, 24] to the simplicial categories of Theorem 21

On equivariant isometric embeddings

Abstract: The representation p can be regarded as a Lie group homomorphism from G into the orthogonal group O(N) which acts on IE N by rotations and reflections; a smooth map X: M ~ I E N is equivariant with respect to p if and only if X(~p) =p(cr) X(p), for all ~r~G, pEM. The main theorem is true in both the C ~ and real analytic categories. We will work in the C ~ category for the time being, and return to the real analytic case in w 4. Moreover, the theorem holds for manifolds with boundary. The main analytic tool used by Nash to prove his isometric embedding theorem is an implicit function theorem based upon the Newton iteration method. The implicit function theorem applies to the equivariant case with virtually no change. In order to apply the implicit function theorem we need to approximate a given G-invariant Riemannian metric on M by a metric induced by an equivariant embedding; we will do this by using the theory of the Laplace operator on compact Riemannian manifolds. According to Gromov and Rokhlin [7], any n-dimensional compact Riemannian manifold can be isometrically embedded in IF, N, where N = (1/2) n(n + 1) + 3 n + 5. No such universal bound is possible in the equivariant case, and in fact, given any positive integer N, it is possible to construct a left invariant

Morse Theoretic Aspects of Yang-Mills Theory

Abstract: Let me start in the manner I have learned of late from all you Physicists, with a modest list of topics to be covered in these two lectures. My topics are: (i) Algebraic topology (ii) Morse theory (iii) Equivariant Morse theory (iv) Pertinence of (i), (ii), and (iii) to the solutions of the classical Yang-Mills Equations.

Weyl group actions and equivariant homotopy equivalence

Abstract: Let G be a compact Lie group and Go its identity component. Then we shall show that the normal representations of the corresponding fixed point components of G-homotopy equivalent manifolds are necessarily isomorphic when G/ Go is a Weyl group of a compact connected Lie group.

On the Homotopy Types of the Groups of Equivariant Diffeomorphisms

Abstract: The purpose of this paper is to study the homotopy type of the group of the equivariant diffeomorphisms of a closed connected smooth G-manifold M, when G is a compact Lie group and the orbit space M/G is homeomorphic to a unit interval [0, 1]. Let Diffg (M)0 denote the group of equivariant C°° diffeomorphisms of the G-manifold M which are G-isotopic to the identity, endowed with C°° topology. If M/G is homeomorphic to [0, 1], then M has two or three orbit types G/H, G/K0 and GjKl. We can choose the isotropy subgroups H, K0, K^ satisfying HdK0nK1. Moreover the G-manifold structure of M is determined by an element r\\ of a factor group N(H)/H9 where N(H) is the normalizer of H in G (see §1). Let Q(N(H)/H; (N(H) fl N(K0))IH, (N(H) n N(rjK1ri~ y)IH)0 denote the connected component of the identity of the space of paths a: [0, l]-+N(H)IH satisfying a(0)e(N(H) n N(X0))/H and a(l)e(N(H) n

Admissible, Minimax and Equivariant Estimators of Life Distributions from Type II Censored Samples

Abstract: In life testing, the unique minimum variance unbiased estimator (MVUE) ? is often used when it exists. However it has been shown for certain distributions that an estimator of the form k? with uniformly smaller mean square error exists. Such extimators are derived here for a class of life distributions and are shown to be admissible, minimax, and (in most cases) equivariant. The underlying distribution from which the samples are drawn follows a generalized life model (GLM) which includes a model proposed by Epstein & Sobel, Weibull, exponential, and Rayleigh distributions as special cases. Results are also given for the Type II asymptotic distribution of largest values, Pareto, and limited distributions. In addition, admissible linear estimators of the form a? + b are obtained and it is shown that they are a form of locally best estimators for some portion of the parameter space. Both k? and a? + b could be used in nonrepetitive estimation problems where bias causes no difficulty.

On the equivariant homotopy of Stiefel manifolds

Abstract: If m=l, then V£(E) is the unit sphere S(E) in E. For any g^G and any orthonormal m-frame (v19 •••,#„) in E, (gvl9 m ',gvm) is also an orthonormal τw-frame in E. This induces a smooth G-action on V%(E). Let £ ' be another representation of G over Λ. We are interested in the set of G-homotopy classes of G-maps from S(E) to V£(E'), [S(E), V£(E')]G. If m=\\> this set is the set of G-homotopy classes of G-maps from sphere to sphere, [5(£), S(E')]G, which was studied in Hauschild [1], Rubinsztein [3] and others. (I am grateful to the referee who informed me that there was a gap in the proof of Rubinsztein's main theorem [3; Theorem 7.2]. This information leads to an improvement of the presentation of this paper.) For any positive integer n, let

Optimality Measures for Monotone Equivariant Cluster Techniques.

Abstract: : Several commonly used optimality measures are compared and classified as to their performance with monotone equivariant cluster techniques A new measure is introduced and compared to the earlier measures The measures are then applied to determining the optimality of several cluster techniques when they are applied to some concrete data that arises from the psychological and social sciences (Author)