Bordism of G-manifolds and integrality theorems
TL;DR: In this article, the authors define equivariant cobordism U,*(X) along the lines of G. W. Whitehead [23], using all representations of the compact Lie group G for suspending.
About: This article is published in Topology.The article was published on 1970-11-01 and is currently open access. It has received 126 citations till now. The article focuses on the topics: Cobordism & Equivariant map.
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01 Oct 1986TL;DR: A brief sketch of the basic concepts of space-level equivariant homotopy theory can be found in this paper, where the authors provide an introduction to the basic ideas and constructions of spectrum-level Equivariant Homotopy Theory.
Abstract: The last decade has seen a great deal of activity in this area. The chapter provides a brief sketch of the basic concepts of space-level equivariant homotopy theory. It also provides an introduction to the basic ideas and constructions of spectrum-level equivariant homotopy theory. The chapter also illustrates ideas by explaining the fundamental localization and completion theorems that relate equivariant to nonequivariant homology and cohomology. To retain the homeomorphism between orbits and homogeneous spaces one shall always restrict attention to closed subgroups. The class of compact Lie groups has two big advantages: the subgroup structure is reasonably simple (nearby subgroups are conjugate), and there are enough representations (any sufficiently nice (7-space embeds in one).
745 citations
Cites methods from "Bordism of G-manifolds and integral..."
...8.2. Bordism. The case of bordism is the greatest success of the method outlined in Section 7. The correct equivariant form of bordism to use is tom Dieck’s homotopical equivariant bordism [ 16 ]....
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TL;DR: In this article, it was shown that the reduced complex cobordism o*(X) of a finite complex is generated by its elements of positive degree as a module over the complex ring.
364 citations
01 Sep 2018
TL;DR: In this article, the authors introduce graduate students and researchers to global equivariant homotopy theory based on the new notion of global equivalences for orthogonal spectra, a much finer notion of equivalence than what is traditionally considered.
Abstract: Equivariant homotopy theory started from geometrically motivated questions about symmetries of manifolds. Several important equivariant phenomena occur not just for a particular group, but in a uniform way for all groups. Prominent examples include stable homotopy, K-theory or bordism. Global equivariant homotopy theory studies such uniform phenomena, i.e., universal symmetries encoded by simultaneous and compatible actions of all compact Lie groups. This book introduces graduate students and researchers to global equivariant homotopy theory. The framework is based on the new notion of global equivalences for orthogonal spectra, a much finer notion of equivalence than what is traditionally considered. The treatment is largely self-contained and contains many examples, making it suitable as a textbook for an advanced graduate class. At the same time, the book is a comprehensive research monograph with detailed calculations that reveal the intrinsic beauty of global equivariant phenomena.
162 citations
115 citations
TL;DR: In this article, a sequence of homology theories for finite complex X for a fixed prime p is studied, and the main result states that there is an n such that ⋯, ϱ(n,n + 1), ϱn−1, n−n) are all epimorphisms; each of the remaining homomorphisms fails to be onto; and n is the projective dimension of BP ∗ (X) as a module over the coefficient ring.
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01 Jan 1964
Abstract: 1. The bordism groups. This note presents an outline of the authors' efforts to apply Thorn's cobordism theory [ó] to the study of differentiable periodic maps. First, however, we shall outline our scheme for computing the oriented bordism groups of a space [ l ] . These preliminary remarks bear on a problem raised by Milnor [4]. A finite manifold is the finite disjoint union of compact connected manifolds with boundary each of which carries a O-differential structure. The boundary of a finite manifold, B, is denoted by dB. A closed manifold is a finite manifold with void boundary. We now define the oriented bordism groups of a pair (X, ^4). An oriented singular manifold in (X, A) is a map ƒ: (B} dB ) —»(X, A) of an oriented finite manifold. Such a singular manifold bords in (X, A) if and only if there is a finite oriented manifold W and a map F: W—->X such that BC.dW as a finite regular submanifold whose orientation is induced by that of W and such that F\ jB=/, F(dW— B) C.A. From two such oriented singular manifolds (Bl fx) and (£?, /2) a disjoint union (B\\JB n 2l fxKJf2) is formed with B\C\B% = 0 and / i U / 2 | £?==ƒ,, * = 1 , 2. Obviously ( £ » , ƒ ) = ( J 3 n , ƒ). We £ay that two singular manifold (5J, /i) and (J5J,/2) are bordant in (X, yl) if and only if the disjoint union (JB*U -~B1,f\\Jf<ï) bords in (X, ^4). By the well-known angle straightening device [5] this is shown to form an equivalence relation. The oriented bordism class of (B,f) is written \B, ƒ] and the collection of all such bordism classes is On(X, A). An abelian group structure is imposed on £2n(X, A) by disjoint union, and then following Atiyah we refer to fin(X, A) as an oriented bordism group of (X, ^4). The weak direct sum fi*(X, A) = ^2Q 0n(X, A) is a graded right module over the oriented Thorn cobordism ring £2. For any ƒ: (B, dB)—>(X, ^4) and any closed oriented manifold V the module product is given by [B, / ] [ F W ] = [BX V, g] where g(x9 y) =ƒ(*). For any map : (X, A)-*(Y, B) there is an induced homomorphism <£*: ön(X, A)-JÇln(Y, B) given by <£*([£, ƒ]) = [B, f]. There is also d*: Qn(X, A)-*Qn-i(A) given by d*([5», ƒ ] ) = [3B», f\dB-*A]. Actually 0*: &*(X, i4)-*Q*(F, 5 ) and d*: J2*(X, ^4)~>fts|c(^4) are fl-module homomorphisms of degree 0 and 1 .
751 citations
TL;DR: In this paper, the authors present a generalisation of K-theory to non-compact spaces, namely equivariant Ktheory on G-spaces, which is a generalization of the notion of vector-bundles.
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).
625 citations
502 citations