Adams Operations Research Articles

FREE LIE ALGEBRAS AND ADAMS OPERATIONS

Let G be a group and K a field. For any finite-dimensional KG-module V and any positive integer n, let Ln(V) denote the nth homogeneous component of the free Lie K-algebra generated by (a basis of) V. Then Ln(V) can be considered as a KG-module, called the nth Lie power of V. The paper is concerned with identifying this module up to isomorphism. A simple formula is obtained which expresses Ln(V) in terms of certain linear functions on the Green ring. When n is not divisible by the characteristic of K these linear functions are Adams operations. Some results are also obtained which clarify the relationship between Adams operations defined by means of exterior powers and symmetric powers and operations introduced by Benson. Some of these results are put into a more general setting in an appendix by Stephen Donkin.

The equivariant {$J$}-homomorphism

May’s J-theory diagram is generalized to an equivariant setting. To do this, equivariant orientation theory for equivariant periodic ring spectra (such as KOG) is developed, and classifying spaces are constructed for this theory, thus extending the work of Waner. Moreover, Spin bundles of dimension divisible by 8 are shown to have canonical KOG-orientations, thus generalizing work of Atiyah, Bott, and Shapiro. Fiberwise completions for equivariant spherical fibrations are constructed, also on the level of classifying spaces. When G is an odd order p-group, this allows for a classifying space formulation of the equivariant Adams conjecture. It is also shown that the classifying space for stable fibrations with fibers being sphere representations completed at p is a delooping of the 1-component of QG(S 0 )ˆ p. The “Adams-May square,” relating generalized characteristic classes and Adams operations, is constructed and shown to be a pull-back after completing at p and restricting to G-connected covers. As a corollary, the canonical map from the p-completion of J k G to the G-connected cover of QG(S 0 )ˆ p is shown to split after restricting to G-connected covers.

Test Modules to Calculate Dutta Multiplicities

Using Frobenius maps, the Dutta multiplicity χ∞(F) was first defined by Dutta for a complex F over a local ring of positive characteristic. Nowadays, using localized Chern characters or Adams operations, it is defined for a complex not only over a local ring of positive characteristic but also over a ring that is a homomorphic image of an arbitrary regular local ring. In the paper, we shall give another characterization to Dutta multiplicity using some Galois extension.Furthermore, we shall define the notion of test modules. If a test module for a local ring exists, then the positivity conjecture of Dutta multiplicities is true over the ring. In the paper, it is proved that, if the small Cohen–Macaulay modules conjecture is true, then test modules always exist for complete local domains whose field of fractions are of characteristic 0.

Adams operations, algebras up to homotopy and cyclic homology

We define and study a new A ∞-algebra structure on the cyclic bar complex of a commutative algebra. Unlike the Getzler–Jones’ one, this new structure is compatible with the Adams operations and provides therefore the right setting for studying operations on cyclic homology theories. Its construction relies on the properties of some formal polytopes, which should be thought of as intermediates between Stasheff polytopes and models for cyclic homology operations.

Adams operations, localized Chern characters, and the positivity of Dutta multiplicity in characteristic $0$

The positivity of the Dutta multiplicity of a perfect complex of $A$-modules of length equal to the dimension of $A$ and with homology of finite length is proven for homomorphic images of regular local rings containing a field of characteristic zero. The proof uses relations between localized Chern characters and Adams operations.

Adams operations and the Dennis trace map

We show that the Adams operations on the algebraic K-theory of a commutative unital Q -algebra correspond to Adams operations on Hochschild homology via the Dennis trace map. This establishes a conjecture of Geller and Weibel, Hodge decomposition of Loday symbols in K-theory and cyclic homology, K-theoty 8 (1994) 587–632, that the Dennis trace map preserves the Hodge decomposition.

Operations on locally free classgroups

Let G be a finite group and K a number field. We show that the k-th Adams operation defined by Cassou-Nogues and Taylor on the locally free class group Cl(Z_K G) is a symmetric power operation, if k is coprime to the order of G. Using the equivariant Adams-Riemann-Roch theorem, we furthermore give a geometric interpretation of a formula established by Burns and Chinburg for these operations.

THE K-THEORY LOCALIZATIONS AND [formula omitted]-PERIODIC HOMOTOPY GROUPS OF H-SPACES

We determine the mod p K-theory localizations and v 1 -periodic homotopy groups of finite H-spaces and of other spaces with torsion-free exterior p-adic K-cohomology algebras at an odd prime p. Our localization results generalize those of Mahowald and Thompson (Topology 1992, 31, 133–141) for odd-dimensional spheres. We construct our mod p K-theory localizations as homotopy fibers of unstable maps between infinite loop spaces, and similarly construct a wide array of new spaces having torsion-free exterior p-adic K-cohomology algebras with prescribed Adams operations. This leads, for example, to a classification of the odd mod p K-homology spheres.

An Adams-Riemann-Roch theorem in Arakelov geometry

We prove an analog of the classical Riemann-Roch theorem for Adams operations acting on K-theory, in the context of Arakelov geometry.

A General Theory of Canonical Induction Formulae

So far there exist three independent constructions of two different canonical versions of Brauer's induction theorem for complex characters due to V. Snaith, P. Symonds, and the author. “Canonical” in this context means functorial with respect to restrictions along group homomorphisms. In this article we axiomatize the situation in which the above canonical induction formulae are constructed. Mackey functors and related structures arise in this way naturally as a convenient language. This approach allows us to construct canonical induction formulae for arbitrary Mackey functors. In particular we obtain canonical induction formulae for the Brauer character ring, the group of projective characters, the ring of trivial source modules, and the ring of linear source modules. In most cases, it is not difficult to construct such formulae over the rational numbers. A much more subtle question is whether the constructed formula comes from a canonical induction formula defined over the integers. We give a sufficient condition in the general framework of Mackey functors for a canonical induction formula to be integral. As an application we show how canonical induction formulae allow the construction of functorial maps on representation rings in terms of functorial maps on subrings, as, for example, the span of linear characters in the case of the canonical Brauer induction formula. This will be used in a subsequent article in the case of Adams operations and Chern classes.

Periodic -rings and exponents of finite groups

A -ring is said to be -periodic if its Adams operators satisfy the relation for each . The quotient by the radical of the free periodic -ring generated by one element is described. Using this description, the order of a finite group is shown to divide the group's exponent to the power equal to the dimension of an arbitrary faithful complex representation.

Adams operations for projective modules over group rings

Let R be a commutative ring, Γ a finite group acting on R, and let k∈ℕ be invertible in R. Generalizing a definition of Kervaire, we construct an Adams operation ψk on the Grothendieck group and on the higher K-theory of projective modules over the twisted group ring R#Γ. For this, we generalize Atiyah's cyclic power operations and use shuffle products in higher K-theory. For the Grothendieck group, we show that ψk is multiplicative and that it commutes with base change, with the Cartan homomorphism, and with ψl for any other l which is invertible in R.

Adams operations, iterated integrals and free loop spaces cohomology

We construct by geometrical means a weight-decomposition of Chen's iterated integrals in the case of a free loop fibration pullback.

Adams operations and integral Hermitian-Galois representations

We give an algebraic interpretation of the action of the j -th Adams operator j on Hermitian Galois structure classes for tame covers of schemes of dimension 1 . This generalizes work of Erez and of Erez-Taylor concerning rings of algebraic integers when j = 2 .

Adams and Steenrod operators in dihedral homology

In this article, we define the Adam′s and Steenrod′s operators in the dihedral homology.

Free Lie algebra and lambda-ring structure

Let R be a graded λ-ring. We extend a well-known formula in the universal ring of Witt vectors by replacing the power operations by the Adams operations. Our method provides us an easy way to compute the inverse image by the symmetric power operators of certain elements of R. As a corollary we get identities, found by Klyachko and Hanlon, in the rings 1 + ℤ[[t]]+ and 1 + Rˆ[[t]]+, where R is the representation ring of the symmetric groups.

Hopf rings and unstable operations

We give a geometric interpretation of certain algebraically defined Hopf rings (−) R ∗(−), which will lead to identification of the sets of unstable operations for E(n), BP, and MU with sets of certain functors. As an application, we derive the existence of the BP-unstable Adams operations.

A Hopf-Algebra Approach to Inner Plethysm

We use the Hopf algebra structure of the algebra of symmetric functions to study the Adams operators of the complex representation rings of symmetric groups, and we give new proofs of all of Littlewood′s formulas for inner plethysm. We also study the Adams operations for orthogonal and symplectic group characters.

Adams operations on character restrictions

Adams operations on character restrictions

Adams operations on higherK-theory

We construct Adams operations on higher algebraic K-groups induced by oper- ations such as symmetric powers on any suitable exact category, by constructing an explicit map of spaces, combinatorially dened. The map uses the S-construction of Waldhausen, and deloops (once) earlier constructions of the map.