Showing papers on "Equivariant map published in 1968"

Equivariant $K$-theory

Abstract: The purpose of this thesis is to present a fairly complete account of equivariant K-theory on compact spaces. Equivariant K-theory is a generalisation of K-theory, a rather well-known cohomology theory arising from consideration of the vector-bundles on a space. Equivariant K-theory, or KG-theory, is defined not on a space but on G-spaces, i.e. pairs (X,α), where X is a space and α is an action of a fixed group G on X, and it arises from consideration of G-vector-bundles on X, i.e. vector-bundles on whose total space G acts in a suitable way (of 3.1). In this thesis G will always be a compact group. But KG-theory does not appear in the first three chapters, which are introductory. Chapter 1 consists of preliminary discussions of little relevance to the sequel, but which permit me to make a few propositions in the later chapters shorter or more elegant. It was intended to be amusing, and the reader may prefer to omit it. Chapter 2 is devoted to the representation-theory of compact groups. When X is a point a G-vector-bundle on X is just a representation-module for G, so the representation-ring, or character-ring, R(G) plays a fundamental role in KG-theory. In chapter 2 I investigate its algebraic structure, and in particular when G is a compact Lie group I determine completely its prime ideals. To do this I have to discuss first the space of conjugacy-classes of a compact Lie group, and outline an induced-representation construction for obtaining finite-dimensional modules for G from modules for suitable subgroups not of finite index. Chapter 3 is a rather full collection of technical results concerning G-vector-bundles: they are all essentially well-known, but have not been stated in the equivariant case. Chapter 4 presents basic equivariant K-theory. I show that it can be defined in three ways: by G-vector-bundles, by complexes of G-vector-bundles, and by Fredholm complexes of infinite-dimensional G-vector-bundles. This chapter also treats the continuity of KG with respect to inverse limits of G-spaces, the Thorn homomorphism for a G-vector-bundle and the periodicity-isomorphism, and the question of extending KG to non-compact spaces. In chapter 5 I obtain for KG(X) a filtration and spectral sequence generalising those of [6], but without dissecting the space X. My method is based on a Cech approach: for each open covering of X I construct an auxiliary space homotopy-equivalent to X which has the natural filtration that X lacks. Also in chapter 5 I prove the localisation-theorem (5.3), which, together with the theory of chapter 6, is one of the most important tools in applied KG-theory. KG(X) is a module over the character-ring R(G), so one can localise it at the prime ideals of R(G), which I have determined in 2.5. The simplest and most important case of the localisation-theorem states that, if β is the prime ideal of characters of G vanishing at a conjugacy-class γ, and if Xγ is the part of X where elements in γ have fixed-points, then the natural restriction-map KG(X) r KG(Xγ) induces an isomorphism when localised at β. In chapter 6 I show how to associate to certain maps f : X r Y of (G-spaces a homomorphism f! : KG(X) r KG(Y). It is the analogue of the Gysin homomorphism in ordinary cohomology-theory; but it can also be regarded as a generalisation of the induced-representation construction of 2.4. In the important special case when f is a fibration whose fibre is a rational algebraic variety I prove that f! is left-inverse to the natural map f! : KG(Y) r KG(X); and I apply that to obtain the general Thom isomorphism theorem. Finally in chapter 7 I prove the theorem towards which my thesis was originally directed. Just as a G-module defines a vector-bundle on the classifying-space BG for G (of [1]), so a G~vector-bundle on X defines a vector-bundle on the space XG fibred over BG with fibre X. Thus one gets a homomorphism α : KG(X) r K(XG). I prove that if KG(X) and K(XG) are given suitable topologies then in certain circumstances K(XG)is complete and α induces an isomorphism of the completion of KG(X) with K(XG). This generalises the theorem of Atiyah-Hirsebruch that R(G)^ ≅ K(BG).

Bott periodicity and the index of elliptic operators

Abstract: IN an expository article (1) I have indicated the deep connection between the Bott periodicity theorem (on the homotopy of the unitary groups) and the index of elliptio operators. It ia the purpose of this paper to elaborate on this connection and in particular to show how elliptio operators can be used to give a rather direct proof of the periodicity theorem. As hinted at in (1) the merit of such a proof is that it immediately extends to all the various generalizations of the periodicity theorem. Thus we obtain the "Thorn isomorphism' theorem together with its equivariant and real forms. The equivariant case is particularly noteworthy because for this no proof of the Thom isomorphism theorem is known (even when the base space is a point) which does not use elliptio operators. In fact a main purpose of this paper is to present the proof for the equivariant case. This proof supersedes an earlier (unpublished) proof (7) wbioh, though relying on elliptio operators, was more indirect than our present one. Besides the fnnfJft.Tnpmt.nl use of elliptio operators there is another novel feature of our treatment. This is that we exploit the multiplicative structure of X-theory to produce a short-cut in the formal proof of the periodicity theorem. The situation is briefly as follows. One has the Bott map

Actions of SO(2) on 3-Manifolds

Abstract: The purpose of this report is to give a complete equivariant and topological classification of the effective actions of the circle group, SO(2), on closed, connected 3-manifolds. The equivariant classification was given in [3] together with the topological classification of actions with fixed points. Actions without fixed points and without special exceptional orbits are manifolds admitting singular fiberings in the sense of Seifert [4]. Most of these manifolds were shown topologically distinct in [2]. In the last section of the present report we complete the classification of SO(2) actions by considering those manifolds not treated in [2] and fixed point free actions with special exceptional orbits.

Equivariant Homology Theories

Abstract: The bordism theory, originally invented by Thom, has been generalized to a (generalized) homology theory by Atiyah, and independently by Conner and Floyd. Meanwhile, a similar phenomenon occured to the K-theory, originally invented by Grothendieck, which was responsible for solutions of many delicate problems in differential topology. Since then an (abstract) form of such homology theory was formulated and studied by a number of authors, notably G. W. Whitehead, E. Brown, A. Dold. These theories are defined on a suitable category of topological spaces and continuous maps.

Equivariant maps onto minimal flows

Abstract: Let G be a connected Lie group. (X, G) is aflow i fX is a compact Hausdorff space with a joint ly continuous group action by G. T h e flow (X, G) is minimal if every orbit is dense or, equivalently, if X has no proper closed non-empty invariant set. We wish to capture two ideas about minimal sets. The first is that if (X, G) is a minimal flow, then not only is each orbit dense, but also each orbit actually winds a round X in much the same way that a Kronecker line on the torus winds a round the torus. This fact is made precise and explained fur ther in T h e o r e m 2.1 and the discussion immediately following it. As a consequence of this theorem, we obtain as Theo rem 2.2 a s tatement about equivariant maps of flows onto minimal flows, which generalizes the main result of Chu and Geraghty in [2]. The second idea is that the space X of a minimal flow should have some homogenei ty property. A known result, due to A. A. Markoff [7], is that if X is finite-dimensional, then X has the same dimension at each point. The conjecture that X has a transitive set of homeomorphisms commut ing with G is shown to be false by enlarging the space of Floyd's example [6] and making it into a flow unde r the reals in the usual way. Instead, our result is of a relative ra ther than an absolute nature. Namely, if rr is an equivariant mapping between minimal flows (X, G) and (Y, G), then under suitable conditions X is the bundle space of a fiber bundle with base space Y and with projection zr. Such a result is proved as Theo rem 3.1 under the assumption that everything is differentiable.

Coinciding directions for imbeddings of spaces

Abstract: 1. Equivariant cohomology. Let A be a space with a continuous fixed point free involution a: A --A. Consider the two possible actions of a on the coefficient group Z of integers: the trivial action and the nontrivial one. Let I9+(A) (resp. H7+(A)) denote the equivariant (resp. residual) ith cohomology group of A for the trivial action; and let HL (A) (resp. m (A)) denote the corresponding equivariant (resp. residual) cohomology groups for the nontrivial action of a on Z (the singular homology is considered throughout; see [1]). The groups 0 i ~~~0 H+ (A) and Hi (A) can also be described as the corresponding cohomology groups of the orbit space A/a, with the constant and twisted coefficients { Z}, respectively. There are exact sequences: