Central cohomology operations and K-theory

For stable degree zero operations, and also for additive unstable operations of bidegree (0,0), it is known that the centre of the ring of operations for complex cobordism is isomorphic to the corresponding ring of connective complex K-theory operations. Similarly, the centre of the ring of BP operations is the corresponding ring for the Adams summand of p-local connective complex K-theory. Here we show that, in the additive unstable context, this result holds with BP replaced by BP for any n. Thus, for all chromatic heights, the only central operations are those coming from K-theory.Comment: 13 page

Algebras of operations in K-theory

We describe explicitly the algebras of degree zero operations in connective and periodic p-local complex K-theory. Operations are written uniquely in terms of certain infinite linear combinations of Adams operations, and we give formulas for the product and coproduct structure maps. It is shown that these rings of operations are not Noetherian. Versions of the results are provided for the Adams summand and for real K-theory.Comment: 25 page

Music Therapists’ Knowledge of and Attitudes Towards Sustainability: Instruments

Sustainability has become a common point of conversation and concern in today’s society. The purpose of this project was to explore salient issues, attitudes and practices in music therapy sustainability. Information was gathered through an in-depth review of the materials used in the make and manufacturing of commonly used instruments in music therapy practice. In addition, a survey was sent to music therapy professionals with the MT-BC (Music Therapist – Board Certified) credentials to ascertain current knowledge of and attitudes toward sustainability within the profession

Stable and Unstable Operations in mod p Cohomology Theories

We consider operations between two multiplicative, complex orientable cohomology theories. Under suitable hypotheses, we construct a map from unstable to stable operations, left-inverse to the usual map from stable to unstable operations. In the main example, where the target theory is one of the Morava K-theories, this provides a simple and explicit description of a splitting arising from the Bousfield-Kuhn functorComment: 28 pages; corrected proof of proposition 3.2, other minor improvement

Gamma (co)homology of commutative algebras and some related representations of the symmetric group

This thesis covers two related subjects: homology of commutative algebras and certain representations of the symmetric group. There are several different formulations of commutative algebra homology, all of which are known to agree when one works over a field of characteristic zero. During 1991-1992 my supervisor, Dr. Alan Robinson, motivated by homotopy-theoretic ideas, developed a new theory, Γ-homology [Rob, 2]. This is a homology theory for commutative rings, and more generally rings commutative up to homotopy. We consider the algebraic version of the theory. Chapter I covers background material and Chapter II describes Γ-homology. We arrive at a spectral sequence for Γ-homology, involving objects called tree spaces. Chapter III is devoted to consideration of the case where we work over a field of characteristic zero. In this case the spectral sequence collapses. The tree space, Tn, which is used to describe Γ-homology has a natural action of the symmetric group Sn. We identify the representation of Sn on its only non-trivial homology group as that given by the first Eulerian idempotent en(l) in QSn. Using this, we prove that Γ-homology coincides with the existing theories over a field of characteristic zero. In fact, the tree space, Tn, gives a representation of Sn+l. In Chapter IV we calculate the character of this representation. Moreover, we show that each Eulerian representation of Sn is the restriction of a representation of Sn+1. These Eulerian representations are given by idempotents en(j), for j=1, ..., n, in QSn, and occur in the work of Barr [B], Gerstenhaber and Schack [G-S, 1], Loday [L, 1,2,3] and Hanlon [H]. They have been used to give decompositions of the Hochschild and cyclic homology of commutative algebras in characteristic zero. We describe our representations of Sn+1 as virtual representations, and give some partial results on their decompositions into irreducible components. In Chapter V we return to commutative algebra homology, now considered in prime characteristic. We give a corrected version of Gerstenhaber and Schack's [G-S, 2] decomposition of Hochschild homology in this setting, and give the analagous decomposition of cyclic homology. Finally, we give a counterexample to a conjecture of Barr, which states that a certain modification of Harrison cohomology should coincide with André/Quillen cohomology