Sarah Whitehouse

Operads and Γ-homology of commutative rings

We introduce Γ-homology, the natural homology theory for E[infty infinity]-algebras, and a cyclic version of it. Γ-homology specializes to a new homology theory for discrete commutative rings, very different in general from Andre–Quillen homology. We prove its general properties, including at base change and transitivity theorems. We give an explicit bicomplex for the Γ-homology of a discrete algebra, and elucidate connections with stable homotopy theory.

BASES FOR COOPERATIONS IN K-THEORY

Gaussian polynomials are used to dene bases with good multiplicative properties for the algebra K(K) of coopera- tions in K-theory and for the invariants under conjugation.

On conjugation invariants in the dual Steenrod algebra

We investigate the canonical conjugation, , of the mod 2 dual Steenrod algebra, A, with a view to determining the subspace, A ,o f ele- ments invariant under . We give bounds on the dimension of this subspace for each degree and show that, after inverting 1, it becomes polynomial on a natural set of generators. Finally we note that, without inverting 1,A is far from being polynomial.

The Integral Tree Representation of the Symmetric Group

AbstractLet Tn be the space of fully-grown n-trees and let Vn and Vn′ be the representations of the symmetric groups Σn and Σn+1 respectively on the unique non-vanishing reduced integral homology group of this space. Starting from combinatorial descriptions of Vn and Vn′, we establish a short exact sequence of

Uniqueness of A-infinity structures and Hochschild cohomology

\mathbb{Z}\Sigma _{n + 1}

SYMMETRIC GROUP ACTIONS ON TENSOR PRODUCTS OF HOPF ALGEBROIDS

-modules, giving a description of Vn′ in terms of Vn and Vn+1. This short exact sequence may also be deduced from work of Sundaram.Modulo a twist by the sign representation, Vn is shown to be dual to the Lie representation of Σn, Lien. Therefore we have an explicit combinatorial description of the integral representation of Σn+1 on Lien and this representation fits into a short exact sequence involving Lien and Lien+1.

Stable and unstable operations in mod p cohomology theories

Derived A1 -algebras were developed recently by Sagave. Their advantage over classical A1 -algebras is that no projectivity assumptions are needed to study minimal models of differential graded algebras. We explain how derived A1 -algebras can be viewed as algebras over an operad. More specifically, we describe how this operad arises as a resolution of the operad dAs encoding bidgas, ie bicomplexes with an associative multiplication. This generalises the established result describing the operad A1 as a resolution of the operad As encoding associative algebras. We further show that Sagaves definition of morphisms agrees with the infinity- morphisms of dA1 -algebras arising from operadic machinery. We also study the operadic homology of derived A1 -algebras. 16E45, 18D50; 18G55, 18G10

CENTRAL COHOMOLOGY OPERATIONS AND K-THEORY

Working over a commutative ground ring, we establish a Hochschild cohomology criterion for uniqueness of derived A-infinity algebra structures in the sense of Sagave. We deduce a Hochschild cohomology criterion for intrinsic formality of a differential graded algebra. This generalizes a classical result of Kadeishvili for the case of a graded algebra over a field.