Unstable Adams Spectral Sequence Research Articles

Higher structure in the unstable Adams spectral sequence

We describe a variant construction of the unstable Adams spectral the sequence for a space $Y$, associated to any free simplicial resolution of $H^*(Y;R)$ for $R=\mathbb{F}_p$ or $\mathbb{Q}$. We use this construction to describe the differentials and filtration in the spectral sequence in terms of appropriate systems of higher cohomology operations.

A simplicial James–Hopf map and decompositions of the unstable Adams spectral sequence for suspensions

We use combinatorial group theory methods to extend the definition of a classical James-Hopf invariant to a simplicial group setting. This allow us to realize certain coalgebra idempotents at sSet -level and obtain a functorial decomposition of the spectral sequence, associated with the lower p-central series filtration on the free simplicial group.

Vanishing theorems for representation homology and the derived cotangent complex

Let $G$ be a reductive affine algebraic group defined over a field $k$ of characteristic zero. In this paper, we study the cotangent complex of the derived $G$-representation scheme $ {\rm DRep}_G(X)$ of a pointed connected topological space $X$. We use an (algebraic version of) unstable Adams spectral sequence relating the cotangent homology of $ {\rm DRep}_G(X) $ to the representation homology $ {\rm HR}_*(X,G) := \pi_*{\mathcal O}[{\rm DRep}_G(X)] $ to prove some vanishing theorems for groups and geometrically interesting spaces. Our examples include virtually free groups, Riemann surfaces, link complements in $ {\mathbb R}^3 $ and generalized lens spaces. In particular, for any f.g. virtually free group $ \Gamma $, we show that $\, {\rm HR}_i({\rm B}\Gamma, G) = 0 \,$ for all $ i > 0 $. For a closed Riemann surface $\Sigma_g $ of genus $ g \ge 1 $, we have $\, {\rm HR}_i(\Sigma_g, G) = 0 \,$ for all $ i > \dim G $. The sharp vanishing bounds for $ \Sigma_g $ depend actually on the genus: we conjecture that if $ g = 1 $, then $\, {\rm HR}_i(\Sigma_g, G) = 0 \,$ for $ i > {\rm rank}\,G $, and if $ g \ge 2 $, then $\, {\rm HR}_i(\Sigma_g, G) = 0 \,$ for $ i > \dim\,{\mathcal Z}(G) \,$, where $ {\mathcal Z}(G) $ is the center of $G$. We prove these bounds locally on the smooth locus of the representation scheme $ {\rm Rep}_G[\pi_1(\Sigma_g)]\,$ in the case of complex connected reductive groups. One important consequence of our results is the existence of a well-defined $K$-theoretic virtual fundamental class for $ {\rm DRep}_G(X)$ in the sense of Ciocan-Fontanine and Kapranov. We give a new `Tor formula' for this class in terms of functor homology.

Derived Koszul duality and [formula omitted]-homology completion of structured ring spectra

Working in the context of symmetric spectra, we consider algebraic structures that can be described as algebras over an operad O. Topological Quillen homology, or TQ-homology, is the spectral algebra analog of Quillen homology and stabilization. Working in a framework for homotopical descent, the associated TQ-homology completion map appears as the unit map of an adjunction comparing the homotopy categories of O-algebra spectra with coalgebra spectra over the associated comonad via TQ-homology. We prove that this adjunction between homotopy categories can be turned into an equivalence by replacing O-algebras with the full subcategory of 0-connected O-algebras. We also construct the spectral algebra analog of the unstable Adams spectral sequence that starts from the TQ-homology groups TQ⁎(X) of an O-algebra X, and prove that it converges strongly to π⁎(X) when X is 0-connected.

TheT–algebra spectral sequence: Comparisons and applications

In previous work with Niles Johnson the author constructed a spectral sequence for computing homotopy groups of spaces of maps between structured objects such as G-spaces and E_n-ring spectra. In this paper we study special cases of this spectral sequence in detail. Under certain assumptions, we show that the Goerss-Hopkins spectral sequence and the T-algebra spectral sequence agree. Under further assumptions, we can apply a variation of an argument due to Jennifer French and show that these spectral sequences agree with the unstable Adams spectral sequence. From these equivalences we obtain information about filtration and differentials. Using these equivalences we construct the homological and cohomological Bockstein spectral sequences topologically. We apply these spectral sequences to show that Hirzebruch genera can be lifted to E_\infty-ring maps and that the forgetful functor from E_\infty-algebras in H\overline{F}_p-modules to H_\infty-algebras is neither full nor faithful.

ON THE COALGEBRAIC RING AND BOUSFIELD–KAN SPECTRAL SEQUENCE FOR A LANDWEBER EXACT SPECTRUM

AbstractWe construct a Bousfield–Kan (unstable Adams) spectral sequence based on an arbitrary (and not necessarily connective) ring spectrum $E$ with unit and which is related to the homotopy groups of a certain unstable $E$ completion $X_E^{\wedge}$ of a space $X$. For $E$ an $\mathbb{S}$-algebra this completion agrees with that of the first author and Thompson. We also establish in detail the Hopf algebra structure of the unstable cooperations (the coalgebraic module) $E_*(\underline{E}_*)$ for an arbitrary Landweber exact spectrum $E$, extending work of the second author with Hopkins and with Turner and giving basis-free descriptions of the modules of primitives and indecomposables. Taken together, these results enable us to give a simple description of the $E_2$-page of the $E$-theory Bousfield–Kan spectral sequence when $E$ is any Landweber exact ring spectrum with unit. This extends work of the first author and others and gives a tractable unstable Adams spectral sequence based on a $v_n$-periodic theory for all $n$.AMS 2000 Mathematics subject classification: Primary 55P60; 55Q51; 55S25; 55T15. Secondary 55P47

The Bousfield-Kan spectral sequence for periodic homology theories

We construct the Bousfield-Kan (unstable Adams) spectral sequence based on certain nonconnective periodic homology theories E such as complex periodic K -theory, and define an E -completion of a space X . For X = S 2 n +1 and E = K we calculate the E 2 -term and show that the spectral sequence converges to the homotopy groups of the K -completion of the sphere. This also determines all of the homotopy groups of the (unstable) K -theory localization of S 2 n +1 including three divisible groups in negative stems.

The unstable Adams spectral sequence for two-stage towers

Let KP denote a generalized mod 2 Eilenberg–MacLane space and let Y be the fiber of a map X→ KP to which the Massey–Peterson theorem applies. We study the relationship of the mod 2 unstable Adams spectral sequence (UASS) for X and for Y. Given conditions on X, we split the E 2-term for Y, and we use a primary level calculation to compute d 2 for Y up to an error term. If the UASS for X collapses at E 2 (for example, if X is an Eilenberg–MacLane space), the UASS for Y collapses at E 3, and we have the entire UASS for Y. We also give examples and address a conjecture of Bousfield on the UASS for the Lie group SO.

Spherical classes and the Dickson algebra

We attack the conjecture that the only spherical classes in the homology of Q0S0 are Hopf invariant one and Kervaire invariant one elements. We do this by computing products in the E2-term of the unstable Adams spectral sequence converging to π∗(Q0S0), using results about the Dickson algebra and by studying the Lannes–Zarati homomorphism.

A Π-algebra spectral sequence for function spaces

We construct a new kind of spectral sequence which is in some ways analogous to the unstable Adams spectral sequence. This new spectral sequence converts primary information about the homotopy groups of two spaces K K and L L into information about the homotopy groups of the function space of maps from K K to L L .

Homotopy spectral sequences and obstructions

For a pointed cosimplicial spaceX •, the author and Kan developed a spectral sequence abutting to the homotopy of the total space TotX •. In this paper,X • is allowed to be unpointed and the spectral sequence is extended to include terms of negative total dimension. Improved convergence results are obtained, and a very general homotopy obstruction theory is developed with higher order obstructions belonging to spectral sequence terms. This applies, for example, to the classical homotopy spectral sequence and obstruction theory for an unpointed mapping space, as well as to the corresponding unstable Adams spectral sequence and associated obstruction theory, which are presented here.

The unstable Adams spectral sequence for free iterated loop spaces

The unstable Adams spectral sequence for free iterated loop spaces

Some Calculations In the Unstable Adams-Novikov Spectral Sequence

The unstable Adams-Novikov spectral sequence for a space X is a sequence of groups {Er(X)}9 r = 2, 3,..., which converge to the homotopy groups of X, and whose £2term depends on the complex cobordism groups of X. We investigate this spectral sequence when X is the infinite special unitary group SU, or one of the finite groups SU(n), or when X is an odd sphere S2+1. The reader is referred to [2] for the construction and properties of the unstable Adams-Novikov spectral sequence. For some purposes, it is convenient to localize at a prime p, in which case the complex cobordism homology theory, based on the spectrum MU, is replaced by Brown-Peterson homology, based on the spectrum BP. We then have a useful spectral sequence with many of the properties of the stable Adams-Novikov spectral sequence. Namely, the nitrations are less than or equal to the filtrations in the unstable Adams spectral sequence based on mod-p homology. When X is a space for which H*(X; BP) is free over the coefficient ring n*(BP) and cofree as a coalgebra, then the E2-term is isomorphic to an Ext group in an abelian category (see §2; also [2, § 7]). Furthermore, this Ext group may be computed as the homology of an unstable cobar complex which we describe explicitly in Section 2. In particular, these considerations apply to the cases X = SU, X = SU(ri), or X = S2n+1. We first consider the situation where X is a p-local H-space with torsion-free homotopy and torsion-free homology. The results of Wilson [10] and the

The unstable Adams spectral sequence for generalized homology

The unstable Adams spectral sequence for generalized homology

Pairings and Products in the Homotopy Spectral Sequence

Smash and composition pairings, as well as Whitehead products are constructed in the unstable Adams spectral sequence; and these pairings and products are described homologically on the ${E_2}$. level. In the special case of the Massey-Peterson spectral sequence, the composition action is given homologically by the Yoneda product, while the Whitehead product vanishes. It is also shown that the unstable Adams spectral sequence over the rationals, with its Whitehead products, is given by the primitive elements in the rational cobar spectral sequence.

Pairings and products in the homotopy spectral sequence

Smash and composition pairings, as well as Whitehead products are constructed in the unstable Adams spectral sequence; and these pairings and products are described homologically on the E 2 {E_2} . level. In the special case of the Massey-Peterson spectral sequence, the composition action is given homologically by the Yoneda product, while the Whitehead product vanishes. It is also shown that the unstable Adams spectral sequence over the rationals, with its Whitehead products, is given by the primitive elements in the rational cobar spectral sequence.

A vanishing theorem for the unstable Adams spectral sequence

A vanishing theorem for the unstable Adams spectral sequence

An unstable Adams spectral sequence

An unstable Adams spectral sequence