On a spectral sequence for the cohomology of infinite loop spaces

  title={On a spectral sequence for the cohomology of infinite loop spaces},
  author={Rune Haugseng and Haynes R. Miller},
  journal={Algebraic \& Geometric Topology},
We study the mod-2 cohomology spectral sequence arising from delooping the Bousfield‐Kan cosimplicial space giving the 2‐nilpotent completion of a connective spectrum X . Under good conditions its E2 ‐term is computable as certain nonabelian derived functors evaluated at H .X/ as a module over the Steenrod algebra, and it converges to the cohomology of  1 X . We provide general methods for computing the E2 ‐term, including the construction of a multiplicative spectral sequence of Serre type… Expand
Algebraic infinite delooping and derived destabilization
Working over the prime field of characteristic two, consequences of the Koszul duality between the Steenrod algebra and the big Dyer-Lashof algebra are studied, with an emphasis on the interplayExpand
On the Derived Functors of Destabilization and of Iterated Loop Functors
These notes explain how to construct small functorial chain complexes which calculate the derived functors of destabilization (respectively iterated loop functors) in the theory of modules over theExpand
Andr\'{e}-Quillen homology of spectral Lie algebras with application to mod p homology of labeled configuration spaces
  • Adela YiYu Zhang
  • Mathematics
  • 2021
We provide a general method to compute the mod 2 André-Quillen homology of spectral Lie algebras. This is the E-page of Knudsen’s spectral sequence, which converges to the mod 2 homology ofExpand


under the suspension homomorphiβms. One also has the unstable homology H*(E0), which with mod p coefficients carries a Pontrjagin product and an action of the mod p Dyer-Lashof algebra R. It isExpand
Operations in the homology spectral sequence of a cosimplicial infinite loop space
Consider the mod 2 homology spectral sequence associated to a cosimplicial space X. We construct external operations whose target is the spectral sequence associated to EΣ2×Σ2(X×X). If X is aExpand
The Mod 2 Homology of Infinite Loop Spaces
We study the spectral sequence that one obtains by applying mod 2 homology to the Goodwillie tower which sends a spectrum X to the suspension spectrum of its 0th space X_0. This converges strongly toExpand
On the André-Quillen cohomology of commutative F[2]-algebras
Quillen and Andre have rigorized and explored a notion of cohomology of commutative algebras or, more generally, simplicial commutative algebras. They were able to do a number of systematicExpand
On simplicial commutative algebras with vanishing André-Quillen homology
Abstract.In this paper, we study the André-Quillen homology of simplicial commutative ℓ-algebras, ℓ a field, having certain vanishing properties. When ℓ has non-zero characteristic, we obtain anExpand
Correction to "The Sullivan Conjecture on Maps from Classifying Space"
On page 49, I assert that given an unstable coalgebra C over the mod p Steenrod algebra A, C E CA, the module of primitives PC is the suspension of an unstable A-module: E 'PC E U. This is true for pExpand
∞-Categories for the Working Mathematician
homotopy theory C.1. Lifting properties, weak factorization systems, and Leibniz closure C.1.1. Lemma. Any class of maps characterized by a right lifting property is closed under composition,Expand
Iterated loop functors and the homology of the Steenrod algebra
Let A be the mod-2 Steenrod algebra. For any unstable A -module M the "unstable homology groups" Hfk(M) = Tor f k(M) are defined by means of unstable projective resolutions of M [2]. We describe hereExpand
The homology of iterated loop spaces
The singular chain complex of the iterated loop space is expressed in terms of the cobar construction. After that we consider the spectral sequence of the cobar construction and calculate its firstExpand
IN [8] WE studied localizations of spaces with respect to homology, and we now develop the analogous stable theory. Let Ho” denote the stable homotopy category of CW-spectra. We show that eachExpand