Behavior of Quillen (co)homology with respect to adjunctions

This paper aims to answer the following question: Given an adjunction between two categories, how is Quillen (co)homology in one category related to that in the other? We identify the induced comparison diagram, giving necessary and sufficient conditions for it to arise, and describe the various comparison maps. Examples are given. Along the way, we clarify some categorical assumptions underlying Quillen (co)homology: cocomplete categories with a set of small projective generators provide a convenient setup.Comment: Minor corrections. To appear in Homology, Homotopy and Application

Completed power operations for Morava E-theory

We construct and study an algebraic theory which closely approximates the theory of power operations for Morava E-theory, extending previous work of Charles Rezk in a way that takes completions into account. These algebraic structures are made explicit in the case of K-theory. Methodologically, we emphasize the utility of flat modules in this context, and prove a general version of Lazard's flatness criterion for module spectra over associative ring spectra.Comment: Version 3: Minor corrections. Journal version, up to small cosmetic change

2-track algebras and the Adams spectral sequence

In previous work of the first author and Jibladze, the $E_3$-term of the Adams spectral sequence was described as a secondary derived functor, defined via secondary chain complexes in a groupoid-enriched category. This led to computations of the $E_3$-term using the algebra of secondary cohomology operations. In work with Blanc, an analogous description was provided for all higher terms $E_m$. In this paper, we introduce $2$-track algebras and tertiary chain complexes, and we show that the $E_4$-term of the Adams spectral sequence is a tertiary Ext group in this sense. This extends the work with Jibladze, while specializing the work with Blanc in a way that should be more amenable to computations.Comment: v2: Added Appendix A on models for homotopy 2-types. To appear in the Journal of Homotopy and Related Structure

Higher Toda brackets and the Adams spectral sequence in triangulated categories

The Adams spectral sequence is available in any triangulated category equipped with a projective or injective class. Higher Toda brackets can also be defined in a triangulated category, as observed by B. Shipley based on J. Cohen's approach for spectra. We provide a family of definitions of higher Toda brackets, show that they are equivalent to Shipley's, and show that they are self-dual. Our main result is that the Adams differential $d_r$ in any Adams spectral sequence can be expressed as an $(r+1)$-fold Toda bracket and as an $r^{\text{th}}$ order cohomology operation. We also show how the result simplifies under a sparseness assumption, discuss several examples, and give an elementary proof of a result of Heller, which implies that the three-fold Toda brackets in principle determine the higher Toda brackets.Comment: v2: Added Section 7, about an application to computing maps between modules over certain ring spectra. Minor improvements elsewhere. v3: Minor updates throughout; closely matches published versio

The DG-category of secondary cohomology operations

We study track categories (i.e., groupoid-enriched categories) endowed with additive structure similar to that of a 1-truncated DG-category, except that composition is not assumed right linear. We show that if such a track category is right linear up to suitably coherent correction tracks, then it is weakly equivalent to a 1-truncated DG-category. This generalizes work of the first author on the strictification of secondary cohomology operations. As an application, we show that the secondary integral Steenrod algebra is strictifiable.Comment: v3: Minor revision

Moduli spaces of 2-stage Postnikov systems

Using the obstruction theory of Blanc-Dwyer-Goerss, we compute the moduli space of realizations of 2-stage Pi-algebras concentrated in dimensions 1 and n or in dimensions n and n+1. The main technical tools are Postnikov truncation and connected covers of Pi-algebras, and their effect on Quillen cohomology.Comment: Version 3: Added conventions in section 1.3. Minor change

Multiparameter persistence modules in the large scale

A persistence module with $m$ discrete parameters is a diagram of vector spaces indexed by the poset $\mathbb{N}^m$. If we are only interested in the large scale behavior of such a diagram, then we can consider two diagrams equivalent if they agree outside of a ``negligeable'' region. In the $2$-dimensional case, we classify the indecomposable diagrams up to finitely supported diagrams. In higher dimension, we partially classify the indecomposable diagrams up to suitably finite diagrams, and show that the full classification problem is wild.Comment: v2: Added Theorem 9.16 thanks to an argument provided by Steffen Oppermann. Minor changes elsewher

Théorème de Kunneth en homologie de Morse

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal

Quillen cohomology of pi-algebras and application to their realization by Martin Frankland.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (p. 161-162).We use the obstruction theory of Blanc-Dwyer-Goerss to study the realization space of certain - algebras with 2 non-trivial groups. The main technical tool is a result on the Quillen cohomology of truncated -algebras, which is an instance of comparison map induced by an adjunction. We study in more generality the behavior of Quillen (co)homology with respect to adjunctions. As a first step toward applying the obstruction theory to 3-types, we develop methods to compute Quillen cohomology of 2-truncated -algebras via a generalization of group cohomology.Ph.D