19 research outputs found
Noncommutative ring spectra
No full textThesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2006.Includes bibliographical references (p. 87-91).Let A be an Ax ring spectrum. We give an explicit construction of topological Hochschild homology and cohomology of A using the Stasheff associahedra and another family of polyhedra called cyclohedra. Using this construction we can then study how THH(A) varies over the moduli space of AO structures on A, a problem which seems largely intractable using strictly associative replacements of A. We study how topological Hochschild cohomology of any 2-periodic Morava K-theory varies over the moduli space of AO structures and show that in the generic case, when a certain matrix describing the multiplication is invertible, the result is the corresponding Morava E-theory. If this matrix is not invertible, the result is some extension of Morava E-theory, and exactly which extension we get depends on the AO structure. To make sense of our constructions, we first set up a general framework for enriching a subcategory of the category of noncommutative sets over a category C using products of the objects of a non-E operad P in C. By viewing the simplicial category as a subcategory of the category of noncommutative sets in two different ways, we obtain two generalizations of simplicial objects.(cont.) For the operad given by the Stasheff associahedra we obtain a model for the 2-sided bar construction in the first case and the cyclic bar and cobar construction in the second case. Using either the associahedra or the cyclohedra in place of the geometric simplices we can define the geometric realization of these objects.by Vigleik Angeltveit.Ph.D
Enriched Reedy categories
No full textWe define the notion of an enriched Reedy category and show
that if A is a C-Reedy category for some symmetric monoidal model category
C and M is a C-model category, the category of C-functors and C-natural
transformations from A to M is again a model category.This research was partially conducted during the period the author was employed by the Clay
Mathematics Institute as a Liftoff Fellow
Graded Tambara functors
No full textWe define the notion of an -graded Tambara functor and prove that any G-spectrum with norm multiplication gives rise to such an -graded Tambara functor.This project was started after Discovery Grant No. DP120101399 from the Australian Research Council paid for the second author to visit the first author. We would also like to thank MSRI for hosting
both of us during the Algebraic Topology program where some of this work was carried out. This material is based on work partially supported by the National Science Foundation under Grant No. 0932078 000 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2014 semester. The second author was also partially supported by National Science Foundation grant number DMS-1710534
RO(S1)-graded TR-groups of Fp, Z and ℓ
No full textAbstractWe give an algorithm for calculating the RO(S1)-graded TR-groups of Fp, completing the calculation started by the second author. We also calculate the RO(S1)-graded TR-groups of Z with mod p coefficients and of the Adams summand ℓ of connective complex K-theory with V(1)-coefficients. These calculations are used elsewhere to compute the algebraic K-theory of certain Z-algebras
Uniqueness of BP⟨n⟩
No full textFix a prime number p and an integer n≥ 0. We prove that if a p-complete spectrum X satisfying a mild finiteness condition has the same mod p cohomology as BP⟨ n⟩ as a module over the Steenrod algebra, then X is weak homotopy equivalent to the p-completion of BP⟨ n⟩.The first author was supported by an ARC Discovery grant. The
second author was partially supported by the DFG through SFB-1085, and thanks the Australian National
University for hosting him while this research was conducted
The cyclic bar construction on A∞H-spaces
No full textWe set up a general framework for enriching a category A over a symmetric monoidal category C using a non-Σ operad P in C. To do this we require A to come with a functor U : A → Δ Σ+ to the category of noncommutative sets. By viewing the simplicial
R(3,10) <= 41
No full textWe improve the upper bound on the Ramsey number R(3,10) from 42 to 41. Hence
R(3,10) is equal to 40 or 41
