Related papers: Calculating obstruction groups for E-infinity ring…
Goerss--Hopkins obstruction theory is a powerful tool for constructing structured ring spectra from purely algebraic data. Using the formalism of model $\infty$-categories, we provide a generalization that applies in an arbitrary…
We provide computational tools to calculate the strict units of commutative ring spectra. We describe the Goerss-Hopkins-Miller spectral sequence for computing strict units of $E_\infty$-$H\mathbb{F}_p$-algebras, and use it to compute the…
We extend the Bousfield-Kan spectral sequence for the computation of the homotopy groups of the space of minimal A-infinity algebra structures on a graded projective module. We use the new part to define obstructions to the extension of…
We present an abstract version of Goerss-Hopkins theory in the setting of a prestable $\infty$-category equipped with a suitable periodicity operator. In the case of the $\infty$-category of synthetic spectra, this yields obstructions to…
We develop an obstruction theory for the extension of truncated minimal $A$-infinity bimodule structures over truncated minimal $A$-infinity algebras. Obstructions live in far-away pages of a (truncated) fringed spectral sequence of…
In this survey paper on commutative ring spectra we present some basic features of commutative ring spectra and discuss model category structures. As a first interesting class of examples of such ring spectra we focus on (commutative)…
The space of E-infinity structures on an simplicial operad C is the limit of a tower of fibrations, so its homotopy is the abutment of a Bousfield-Kan fringed spectral sequence. The spectral sequence begins (under mild restrictions) with…
We characterize ring spectra morphisms from the algebraic cobordism spectrum $\QTR{Bbb}{MGL}$ (\QCITE{cite}{}{Vo1}) to an oriented spectrum $\QTR{Bbb}{E}$ (in the sense of Morel \QCITE{cite}{}{Mo}) via formal group laws on the…
Previous work constructed a generalized truncated Brown-Peterson spectrum of chromatic height 2 at the prime 2 as an E_infinity-ring spectrum, based on the study of elliptic curves with level-3 structure. We show that the natural map…
We apply Goerss--Hopkins obstruction theory for motivic spectra to study the motivic Morava $E$-theories. We find that they always admit $\mathbb{E}_\infty$ structures, but that these may admit "exotic" $\mathbb{E}_\infty$ automorphisms not…
An algebraic criterion, in terms of closure under power operations, is determined for the existence and uniqueness of generalized trun- cated Brown-Peterson spectra of height 2 as E_\infty-ring spectra. The criterion is checked for an…
Using the descent spectral sequence for a Galois extension of ring spectra, we compute the Picard group of the higher real $K$-theory spectra of Hopkins and Miller at height $n=p-1$, for $p$ an odd prime. More generally, we determine the…
We develop an obstruction theory for the existence and uniqueness of a solution to the gluing problem for a destriction functor and apply it to some well-known biset functors. The obstruction groups for this theory are reduced cohomology…
This paper contains a complete computation of the homotopy ring of the spectrum of topological modular forms constructed by Hopkins and Miller. The computation is done away from 6, and at the (interesting) primes 2 and 3 separately, and in…
We develop two approaches to obstruction theory for deformations of derived isomorphism classes of complexes $Z^\bullet$ of modules for a profinite group $G$ over a complete local Noetherian ring $A$ of positive residue characteristic…
Let $p\geq 5$ be a prime number, let $n\geq 2$ be a natural number and let $\text{Heis}(p^n)$ denote the Heisenberg group modulo $p^n$. We study the Lyndon-Hochschild-Serre spectral sequence $E(\text{Heis}(p^n))$ associated to…
For each odd prime p, we exhibit p-groups G of p-rank two such that (suitably defined) Chern classes of unitary representations of G fail to generate the following rings: 1. The even degree integral cohomology of G; 2. The final page of the…
In the previous paper, we studied obstructions to the existence of complex sections on almost complex manifolds up to cobordism. We determined the obstruction rationally, in terms of the Chern classes. In this paper, we study the torsion…
The $2$-primary Hopf invariant $1$ elements in the stable homotopy groups of spheres form the most accessible family of elements. In this paper we explore some properties of the $\mathcal{E}_\infty$ ring spectra obtained from certain…
We compute the homotopy groups of spectra associated by a theorem of Lurie to the Shimura curves of discriminants 6, 10, and 14, beginning with a computation of integral rings of automorphic forms on these curves. As an application, we find…