Thom Isomorphism Research Articles

Bordism groups and shape theory

The THOM isomorphism Theorem (1) allows an immediate extension to the strong shape category of compacta: The shape homology TBG n (X) with coefficient in a THOM spectrum turns out to be isomorphic to the bordism group Ω G n(X) which is defined like Ω G n (X) but with strong shape morphisms replacing continuous mappings (Theorem 1.2).

An algebraic filtration of H*(MO;ℤ2)

Let 2* denote the dual of the mod two Steenrod algebra. In [5] an algebraic filtration B*(n) of H*(BO; ℤ2) was constructed such that each B*(n) is a bipolynomial sub Hopf algebra and sub 2*-comodule of H*(BO; ℤ2). In Lemma 3.1 we prove that the Thom isomorphism determines a corresponding filtration of H*(MO;ℤ2) by polynomial subalgebras and sub 2*-comodules M*(n). Let (n) denote the subalgebra of 2 generated by Sq2k, 0 ≦ k < n, and let *(n) be its dual, a quotient Hopf algebra of 2*. In Section 3 we construct a polynomial algebra and *(n)-comodule R(n) such that M*(n)≃2*□*(n)R(n) as algebras and 2*-comodules. Here □ denotes the cotensor product defined in [9, §2]. Dually it will follow that M*(n) has a sub (n)-module and subcoalgebra T(n) such that M*(n)≃2⊗n)T(n) as coalgebras and 2-modules. We also show that M*(n) can not be realised as the homology of a spectrum for n≧4. Of course M*(0)=H*(MO;ℤ2), M*(1)=H*(MSO;ℤ2), M*(2)=H* (MSpin;ℤ2) and M*(3)=H*(MO<8>;ℤ2). Moreover, it follows from [4; Thm. 2.10, Cor. 2.11] that M*(n)=Images[H*(MO<ϕ(n)>;ℤ2)→H*(MO;ℤ2)] and M*(n) ≃ Image [H*(MO;ℤ2)→ H*(MO<ϕ(n)>;ℤ2)]. Here MO<k> id the Thom spectrum of BO<k>, the (k−1)-connected covering of BO, and ϕ(n)=8s + 2t where n = 4s + t, 0≦t≦3. In Section 4 we sketch the odd primary analogue—a filtration pM*(n) of H*(MUp, 0;ℤp) for p an odd prime. MUp, 0 is the Thom spectrum of the (2p-3)-connected factor of the Adams splitting [2] of BU(p).

H ∗ (MO 〈8 〉; Z/2) is an Extended A ∗ 2 -Coalgebra

We show that H*(MO(8>; Z/2) is an extended A*-coalgebra, where A* is the subalgebra of the Steenrod algebra generated by (Sq', Sq2, Sq4). The method yields an analogous result for H*( M Spin; Z/2). Recently Don Davis conjectured [D] that H*(MO(8); Z/2) is an extended A*-coalgebra and discussed various consequences of such a result. We apply the method introduced in [P] to prove his conjecture. Let A* C A* be the subalgebra generated by (Sq', Sq2, Sq4). THEOREM A. H*MO (8) is an extended A *-coalgebra, i.e., there is an A *-coalgebra N such that H*MO (8) A* ?Al N as an A*-coalgebra. COROLLARY (BAHRI AND MAHOWALD [BMJ). A*//A2 is a direct summand in H*MO (8). THEOREM B. H*M Spin is an extended A,*-coalgebra. PROOF OF THEOREM A. Let p: BO 3], where a(m) is the number of ones in the dyadic expansion of m. H*BO(8) is a sub-Hopf algebra of H* BO, and hence by Borel's theorem is also polynomial [B]. Let p1 be the coalgebra primitive in H2, l BO (and, via the Thom isomorphism, in H21l MO). From the inductive formula for Newton polynomials, pj is indecomposable in H* MO. So we can consider the polynomial subalgebra P2 = Z/2 [ p8, P4 P, p2, * *. C H*MO. P~7/~r84 2 3 l H M p * The map QH'BO n-* QH'BO(8) of indecomposable quotients is an isomorphism if a(n 1) > 3; and thus so is the map PH, BO<(8) ' PH, BO of coalgebra primitives. So the generators of P2 lie in H*MO (8), and thus all of P2 does. The coaction 4 on P2 C H*MO(8) is known [BP]: ipj = p,_. Since A2 (A*2)* = A/(8, 24, ,.24...), the augmentation ideal P2 is clearly a subcomodule of H*MO<(8) over A2 (although not over A). Thus so is the ideal I generated by P2 in H*MO(8). Received by the editors February 3, 1982. 1980 Mlathematicts Subjec-t Classification. Primary 57R90; Secondary 55N22. 55R40. 55S 10. 'Supported in part by a grant from the NSF. (01983 American Mathematical Society 0002-9939/82/0(X)0-0443/$( 1.50

On the mapping torus of an automorphism

Let ρ \rho be an automorphism of a C ∗ {C^ * } -algebra A A . The mapping torus T ρ ( A ) {T_\rho }(A) is the C ∗ {C^ * } -algebra of A A -valued continuous functions x x on [ 0 , 1 ] [0,1] satisfying x ( 1 ) = ρ ( x ( 0 ) ) x(1) = \rho (x(0)) . Using his Thom isomorphism theorem, A. Connes has shown that the K K -groups of T ρ ( A ) {T_\rho }(A) , with indices reversed, are isomorphic to those of the crossed product A × ρ Z A{ \times _\rho }Z . We provide here an alternative proof of this fact which gives an explicit description of the isomorphism.

A Note on the Thom Isomorphism

A Note on the Thom Isomorphism

Connes' analogue of the Thom isomorphism for the Kasparov groups

Connes' analogue of the Thom isomorphism for the Kasparov groups

A note on the Thom isomorphism

The generalized homology version of the Thom isomorphism theorem is exploited to give easy proofs of several recent theorems.

An analogue of the thom isomorphism for crossed products of a [formula omitted] algebra by an action of [formula omitted

In this paper we show the existence and uniqueness of a natural isomorphism ø j α of K j ( A) with K j+1 ( A ⋊ α R ), j ϵ Z /2 where ( A, R , α) is a C∗ dynamical R -system, K is the functor of topological K theory and A ⋊ α R is the crossed product of A by the action of R . The Pimsner-Voiculescu exact sequence is obtained as a corollary. We show that given an α-invariant trace τ on A, with dual trace \\ ̂ gt, one has \\ ̂ gtø 1 α[u] = ( 1 2 iπ) τ(δ(u)u∗) for any unitary u in the domain of the derivation δ of A associated to the action α. Finally, we show that the crossed product of C( S 3) (continuous functions on the 3 sphere) by a minimal diffeomorphism is a simple C∗ algebra with no nontrivial idempotent.

A Direct Summand in H ∗ (MO 〈8 〉, Z 2 )

A Direct Summand in H ∗ (MO 〈8 〉, Z 2 )

Equivariant K-theory

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).

Cohomology and homology theories for categories of principal 𝐺-bundles

Introduction. In [4] G. W. Whitehead gives a method by which all extraordinary homology and cohomology theories may be generated by fundamental homotopy constructions. Unfortunately the definition of the extraordinary theories used is not sufficiently general so as to include all theories that would normally be considered as (co) homology theories. An example of a theory not included is (co) homology with local coefficients. In this paper, we will show how to extend the methods of Whitehead so as to include such theories. Let G be a fixed continuous group. If we consider the category of principal G-bundles over C.W. complexes, we may define a connected sequence of functors from this category to the category of abelian groups, satisfying the usual axioms of extraordinary (co) homology theory, with principal G-bundles replacing spaces, and bundle maps and homotopies replacing maps and homotopies. It is the purpose of this paper to show that such theories may be generated by homotopy constructions. In particular, we will indicate how local coefficient theory may be generated in this manner. In order to develop the homology theories spoken of above, it will first be necessary to define a class of spaces closely related to C.W. complexes but with just enough additional generality so as to include spaces for which a useful C.W. decomposition may not be available, but which are structured enough to give us a good deal of the type of information usually associated with a C.W.-decomposition. The definition of cohomology groups used in this paper was proposed by Professor Israel Berstein of Cornell University and will be exploited by him in a forthcoming paper on the Thom Isomorphism. This paper represents part of a doctoral dissertation written under Professor Berstein at Cornell University. 1. Piecewise-C.W. complexes. We will need three lemmas from homotopy theory. The first two will not be proved as they are quite easy. In the following,

On the Thom Isomorphism Theorem

In what follows (E, p, B) will always denote a fibering in the sense of Serre, † with arc-wise connected base B, projection p:E → B, and fibre F = p−1(b), b ɛ B. Coefficients will always be taken in a commutative ring A with a unit element, and the cohomology will be singular (cubical), as in (3).

A Thom isomorphism for infinite rank Euclidean bundles

An equivariant Thom isomorphism theorem in operator Ktheory is formulated and proven for infinite rank Euclidean vector bundles over finite dimensional Riemannian manifolds. The main ingredient in the argument is the construction of a non-commutative C ? -algebra associated to a bundle E ! M, equipped with a compatible connection r, which plays the role of the algebra of functions on the infinite dimensional total space E. If the base M is a point, we obtain the Bott periodicity isomorphism theorem of Higson-Kasparov-Trout [19] for infinite dimensional Euclidean spaces. The construction applied to an even finite rank spin c -bundle over an even-dimensional proper spin c -manifold reduces to the classical Thom isomorphism in topological K-theory. The techniques involve noncommutative geometric functional analysis.