Related papers: A Theorem on Multiplicative Cell Attachments with …

For a 1-connected CW-complex $X$, let $\mathcal{E}(X)$ denote the group of homotopy classes of self-homotopy equivalences of $X$. The aim of this paper is to prove that, for every $n\in\Bbb N$, there exists a 1-connected rational CW-complex…

Let $R$ be a ring spectrum and $ E\to X$ an $R$-module bundle of rank $n$. Our main result is to identify the homotopy type of the group-like monoid of homotopy automorphisms of this bundle, $hAut^R(E)$. This will generalize the result…

Given a diagram of rings, one may consider the category of modules over them. We are interested in the homotopy theory of categories of this type: given a suitable diagram of model categories M(s) (as s runs through the diagram), we…

In the 1980's, Ravenel introduced sequences of spectra $X(n)$ and $T(n)$ which played an important role in the proof of the Nilpotence Theorem of Devinatz--Hopkins--Smith. In the present paper, we solve the homotopy limit problem for…

Let $R$ be a commutative ring with unit. We consider the homotopy theory of the category of spectral sequences of $R$-modules with the class of weak equivalences given by those morphisms inducing a quasi-isomorphism at a certain fixed page.…

We define quasicategories of E_n-structured coalgebras, bialagebras and comodules. We show that: n-fold loop spaces, suspension spectra thereof, descent data for maps of E_n-ring spectra, descent corings of morphisms of E_n-ring spectra and…

The bootstrap category in E-theory for C*-algebras over a finite space X is embedded into the homotopy category of certain diagrams of K-module spectra. Therefore it has infinite n-order for every n. The same holds for the bootstrap…

We show that Ravenel's spectrum $X(2)$ is the versal $E_1$-$S$-algebra of characteristic $\eta$. This implies that every $E_1$-$S$-algebra $R$ of characteristic $\eta$ admits an $E_1$-ring map $X(2)\to R$, i.e. an $\mathbb{A}_\infty$…

We construct a motivic spectral sequence for the relative homotopy invariant K-theory of a closed immersion of schemes $D \subset X$. The $E_2$-terms of this spectral sequence are the cdh-hypercohomology of a complex of equi-dimensional…

The family of Thom spectra $y(n)$ interpolates between the sphere spectrum and the mod two Eilenberg--MacLane spectrum. Computations of Mahowald, Ravenel, Shick, and the authors show that the associative ring spectrum $y(n)$ has type $n$.…

A parametrized spectrum E is a family of spectra E_x continuously parametrized by the points x of a topological space X. We take the point of view that a parametrized spectrum is a bundle-theoretic geometric object. When R is a ring…

We define a notion of a connectivity structure on an $\infty$-category, analogous to a $t$-structure but applicable in unstable contexts -- such as spaces, or algebras over an operad. This allows us to generalize notions of n-skeleta,…

For a connected based space $X$, let $[X,X]$ be the set of all based homotopy classes of base point preserving self map of $X$ and let $\E(X)$ be the group of self-homotopy equivalences of $X$. We denote by $\A_{\sharp}^k(X)$ the set of…

We extend Geisser and Hesselholt's result on ``bi-relative K-theory'' from discrete rings to connective ring spectra. That is, if $\mathcal A$ is a homotopy cartesian $n$-cube of ring spectra (satisfying connectivity hypotheses), then the…

We show that the homotopy category of commutative algebra spectra over the Eilenberg-Mac Lane spectrum of the integers is equivalent to the homotopy category of E-infinity-monoids in unbounded chain complexes. We do this by establishing a…

We construct a new spectrum of units for a commutative symmetric ring spectrum that detects the difference between a periodic ring spectrum and its connective cover. It is augmented over the sphere spectrum. The homotopy cofiber of its…

Let $n \geq 1$, $p$ a prime, and $T(n)$ any representative of the Bousfield class of the telescope $v_n^{-1}F(n)$ of a finite type $n$ complex. Also, let $E_n$ be the Lubin-Tate spectrum, $K(E_n)$ its algebraic $K$-theory spectrum, and…

Let n \geq 1 and let p be any prime. Also, let E_n be the Lubin-Tate spectrum, G_n the extended Morava stabilizer group, and K(n) the nth Morava K-theory spectrum. Then work of Devinatz and Hopkins and some results due to Behrens and the…

Using methods developed by Franke, we obtain algebraic classification results for modules over certain symmetric ring spectra ($S$-algebras). In particular, for any symmetric ring spectrum $R$ whose graded homotopy ring $\pi_*R$ has graded…

For X a simply-connected finite H-space, there is a Bousfield-Kan spectral sequence which converges to the homotopy of its K-completion. When X=Spin(2n+1), we expect that these homotopy groups equal the v1-periodic homotopy groups in…