Bousfield-Kan Spectral Sequence Research Articles

An unstable change of rings for Morava E–theory

The Bousfield-Kan (or unstable Adams) spectral sequence can be constructed for various homology theories such as Brown-Peterson homology theory BP, Johnson-Wilson theory $E(n)$, or Morava $E$-theory $E_n$. For nice spaces the $E_2$-term is given by Ext in a category of unstable comodules. We establish an unstable Morava change of rings isomorphism between $\text{Ext}_{\mathcal{U}_{\Gamma_B}}(B,M)$ and $\text{Ext}_{\mathcal{U}_{E_{n*}E_n}/I_{n}}(E_{n*}/I_{n},E_{n*}\otimes_{BP_*} M)$ where $(B,\Gamma_B)$ denotes the Hopf Algebroid $(v_n^{-1}BP_*/I_{n}, v_n^{-1}BP_*BP/I_{n} )$. We show that the latter groups can be interpreted as $\text{Ext}$ in the category of continuous modules over the profinite monoid of endomorphisms of the Honda formal group law. By comparing this with the cohomology of the Morava stabilizer group we obtain an unstable Morava vanishing theorem when $p-1 \nmid n$.

Enhanced $$A_{\infty }$$-obstruction theory

An $$A_n$$-algebra $$A= (A,m_1, m_2, \ldots , m_n)$$ is a special kind of $$A_\infty $$-algebra satisfying the $$A_\infty $$-relations involving just the $$m_i$$ listed. We consider obstructions to extending an $$A_{n-1}$$ algebra to an $$A_n$$-algebra. We enhance the known techniques by extending the Bousfield–Kan spectral sequence to apply to the homotopy groups of the space of minimal (i.e. $$m_1=0)$$$$A_\infty $$-algebra structures on a given graded projective module. We also consider the Bousfield–Kan spectral sequence for the moduli space of $$A_\infty $$-algebras. We compute up to the $$E_2$$ terms and differentials $$d_2$$ of these spectral sequences in terms of Hochschild cohomology.

A Bousfield–Kan Algorithm for Computing the Effective Homotopy of a Space

This paper is devoted to a constructive version of the Bousfield–Kan spectral sequence (BKSS). The BKSS provides a combinatorial basis for the famous Adams spectral sequence and its descendants. Its systematic description in the book “Homotopy Limits, Completions and Localizations” remains a relatively difficult text, often cited by its sweet nickname, the “Yellow Monster.” The modern constructive point of view gives an opportunity to reread this essential text and to use it to produce a new algorithm computing homotopy groups, more precisely computing the effective homotopy of a given space, a new concept much richer than the ordinary homotopy groups. Without changing the general philosophy of the BKSS, the constructive constraint leads to a significant reorganization of this rich material and, as it is most often the case, finally to a simpler and more explicit description. Combined with our own basic tools, effective homology and effective homotopy, the description of the BKSS given here is finally not so complicated and could also help the topologists interested by this nice subject.

The rational homology of spaces of long links

We provide a complete understanding of the rational homology of the space of long links of m strands in the euclidean space of dimension d > 3. First, we construct explicitly a cosimplicial chain complex whose totalization is quasi-isomorphic to the singular chain complex of the space of long links. Next we show (using the fact that the Bousfield-Kan spectral sequence associated to this cosimplcial chain complex collapses at the page 2) that the homology Bousfield-Kan spectral sequence associated to the Munson-Volic cosimplicial model for the space of long links collapses at the page 2 rationally, and this solves a conjecture of Munson-Volic. Our method enables us also to determine the rational homology of the high dimensional analogues of spaces of long links. The last result of this paper states that the radius of convergence of the Poincar\'e series for the space of long links (modulo immersions) tends to zero as m goes to the infinity.

Lifting homotopy T-algebra maps to strict maps

The settings for homotopical algebra—categories such as simplicial groups, simplicial rings, A∞ spaces, E∞ ring spectra, etc.—are often equivalent to categories of algebras over some monad or triple T. In such cases, T is acting on a nice simplicial model category in such a way that T descends to a monad on the homotopy category and defines a category of homotopy T-algebras. In this setting there is a forgetful functor from the homotopy category of T-algebras to the category of homotopy T-algebras.Under suitable hypotheses we provide an obstruction theory, in the form of a Bousfield–Kan spectral sequence, for lifting a homotopy T-algebra map to a strict map of T-algebras. Once we have a map of T-algebras to serve as a basepoint, the spectral sequence computes the homotopy groups of the space of T-algebra maps and the edge homomorphism on π0 is the aforementioned forgetful functor. We discuss a variety of settings in which the required hypotheses are satisfied, including monads arising from algebraic theories and operads. We also give sufficient conditions for the E2-term to be calculable in terms of Quillen cohomology groups.We provide worked examples in G-spaces, G-spectra, rational E∞ algebras, and A∞ algebras. Explicit calculations, connected to rational unstable homotopy theory, show that the forgetful functor from the homotopy category of E∞ ring spectra to the category of H∞ ring spectra is generally neither full nor faithful. We also apply a result of the second named author and Nick Kuhn to compute the homotopy type of the space E∞(Σ+∞CokerJ,LK(2)R).

Euler series, Stirling numbers and the growth of the homology of the space of long links

We study the Bousfield-Kan spectral sequence associated to the Munson-Volić cosimplicial model for the space of long links with $\ell$ strings in $\mathbb R^N$. We compute explicitly some Euler-Poincaré series associated to the second page of that spectral sequence and deduce exponential growth of their Betti numbers.

Formality of Sinha’s cosimplicial model for long knots spaces and the Gerstenhaber algebra structure of homology

Sinha has constructed a cosimplicial space X for a fixed integer N. One of his main result states that for N > 3, X is a cosimplicial model for the space of long knots (modulo immersions). On the other hand, Lambrechts, Turchin and Volic showed that for N > 3 the homology Bousfield-Kan spectral sequence associated to Sinha's cosimplicial space collapses at the page 2 rationally. Their approach consists first to prove the formality of some other diagrams approximating X and next deduce the collapsing result. In this paper, we prove directly the formality of Sinha's cosimplicial space and this allows us to construct a very short proof of the collapsing result for N > 2. Moreover, we prove that the isomorphism between the page 2 and the rational homology of the space of long knots modulo immersions is of Poisson algebras, when N > 3.

Computing the first stages of the Bousfield-Kan spectral sequence

In this paper, an algorithm computing the terms E 1 and E 2 of the Bousfield-Kan spectral sequence of a 1-reduced simplicial set X is defined. In order to compute the ordinary description of the first level E 1, some elementary operations of Homological Algebra are sufficient. On the contrary, to compute the stage E 2 it is necessary to know more information about the previous groups, in particular with respect to the generators. This additional information can be reached by computing the effective homology of RX, RX being the free simplicial Abelian group generated by X. The algorithm to get the effective homology of RX from the effective homology of X can be considered the main result in our paper. Moreover, we include a combinatorial proof of the convergence of the Bousfield-Kan spectral sequence, based on the tapered nature of the stage E 1.

Homologa efectiva y sucesiones espectrales (abstract only)

Effective homology and spectral sequences are two different techniques of Algebraic Topology which can be used for the computation of homology and homotopy groups. In this work we try to relate both methods, showing that the effective homology method can also be used to produce algorithms computing some spectral sequences. In the thesis we focus our attention on two particular situations: spectral sequences associated with filtered complexes and the Bousfield-Kan spectral sequence. The first part of the memoir is devoted to spectral sequences of filtered complexes, which under some good conditions converge to the homology groups of the initial complex. Making use of the effective homology method, we have developed several algorithms computing the different components of these spectral sequences: groups and differential maps in every stage, convergence level and filtration of the homology groups induced by the initial filtration. These algorithms have been implemented as a new module for the Kenzo system, a program of Symbolic Computation in Algebraic Topology, and can be used to determine two classical examples of spectral sequences, those of Serre and Eilenberg-Moore. Other spectral sequences are not defined by means of filtered complexes. This is the case of the Bousfield-Kan spectral sequence, related with the computation of homotopy groups, which is studied in the second part of the thesis. Our algorithms for the computation of spectral sequences of filtered complexes cannot be applied, but the effective homology method can be useful again to develop a constructive version of the Bousfield-Kan spectral sequence. As a first necessary step, our main result is an algorithm computing the effective homology of the free simplicial Abelian group RX generated by a 1-reduced simplicial set X. This algorithm allows the construction of the levels 1 and 2 of the spectral sequence; for the computation of the higher "pages", we present a sketch of a new algorithm which is not finished yet. Furthermore, we include a proof of the convergence of the spectral sequence.

Coformality and rational homotopy groups of spaces of long knots

We show that the Bousfield-Kan spectral sequence which computes the rational homotopy groups of the space of long knots in R-d for d >= 4 collapses at the E 2 page. The main ingredients in the proof are Sinha's cosimplicial model for the space of long knots and a coformality result for the little balls operad.

The algebraic $K$-theory of a diagram of rings

In this paper, we consider of or functors from a small category to the category of rings, and the corresponding diagrams of groups $K_i.$ Classically, this was initiated by Milnor. The main result of this paper is the direct comparison of the filtration in classical algebraic $K$-theory discussed in J. Duflot, Simplicial groups that are models for algebraic $K$-theory, Manuscripta Math. 113 (2004), no. 4, 423–470 and J. Duflot and C.T. Marak, A filtration in algebraic $K$-theory, J. Pure Applied Algebra 151 (2000), no. 2, 135–162 to a corresponding filtration in the Bousfield-Kan spectral sequence associated to a Tot-tower of simplicial groups attached to the diagram of rings.

THE BOUSFIELD–KAN SPECTRAL SEQUENCE FOR MORAVA $K$-THEORY

AbstractWe construct a spectral sequence converging to the $E_{2}$-term of the Bousfield–Kan spectral sequence (BKSS) for a wide variety of homology theories. Using this, the $E_{2}$-term of the BKSS based on $K(1)$-theory for the odd spheres is computed and the unstable $K(1)$-completion is computed.

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

Finiteness of Total Cofibres

The suspension of the total cofibre (Y ) of a diagram Y of infinite CW complexes, indexed by the face lattice of a polytope, is in general not homotopy equivalent to a finite CW complex. However, if Y satisfies additional conditions (making it into a locally finite non-linear sheaf on a projective toric variety) we can show that (Y ) is a homotopy finite space. The main tools used in the proof are a Bousfield–Kan spectral sequence for total cofibres, and a modification of the Cech construction to calculate sheaf cohomology groups. Mathematics Subject Classifications (2000): Primary 55P99, Secondary 55N30.

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.

Derived functors of graded algebras

A number of spectral sequences arising in homotopy theory have the derived functors of a graded algebraic functor as their E 2-term. We here describe conditions for the vanishing of such derived functors, yielding vanishing lines for the spectral sequences. We also show that under these conditions the nth derived functor, for large n, depends only on low-dimensional information. The applications we have in mind include certain cases of the Bousfield-Kan spectral sequence, the Quillen homology of a graded algebra (with applications to H. Miller's Grothendieck spectral sequence), and the wedge, smash, and homology spectral sequences.

André-Quillen cohomology and the homotopy groups of mapping spaces: understanding the E2-term of the Bousfield-Kan spectral sequence

Let X and Y be two topological spaces. We note that a discussion of the homotopy groups of the space of continuous functions from X to Y leads to a discussion of the algebra derivations from the cohomology algebra of Y to the cohomology algebra of X. This observation, and the Bousfield-Kan spectral sequence, allow us to apply the commutative algebra cohomology of André and Quillen to the study of the homotopy type of spaces of maps.