2. Moduli spaces 7 3. Postnikov systems for spaces 10 4. Π-algebras and their modules 14

Every homology or cohomology theory on a category of E∞ ring spectra is Topological André–Quillen homology or cohomology with appropriate coefficients. Analogous results hold more generally for categories of algebras over operads.

In this document we make good on all the assertions we made in the previous paper “Moduli spaces of commutative ring spectra ” [20] wherein we laid out a theory a moduli spaces and problems for the existence and uniqueness of E∞-ring spectra. In that paper, we discussed the the Hopkins-Miller theorem on the Lubin-Tate or Morava spectra En; in particular, we showed how to prove that the moduli space of all E ∞ ring spectra X so that (En)∗X ∼ = (En)∗En as commutative (En) ∗ algebras had the homotopy type of BG, where G was an appropriate variant of the Morava stabilizer group. This is but one point of view on these results, and the reader should also consult [3], [38], and [41], among others. A point worth reiterating is that the moduli problems here begin with algebra: we have a homology theory E ∗ and a commutative ring A in E∗E comodules and we wish to discuss the homotopy type of the space T M(A) of all E∞-ring spectra so that E∗X ∼ = A. We do not, a priori, assume that T M(A) is non-empty, or even that there is a spectrum X so that E∗X ∼ = A as comodules.

Abstract. We show that any closed model category of simplicial algebras over an algebraic theory

The aim of this paper is to give an overview of some of the existing homology theories for commutative (S-)algebras. We do not claim any originality; nor do we pretend to give a complete account. But the results in that field are widely spread in the literature, so for someone who does not actually work in that subject, it can be difficult to trace all the relationships between the different homology theories. The theories we aim to compare are • topological André-Quillen homology • Gamma homology • stable homotopy of Γ-modules • stable homotopy of algebraic theories • the André-Quillen cohomology groups which arise as obstruction groups in the Goerss-Hopkins approach As a comparison between stable homotopy of Γ-modules and stable homotopy of algebraic theories is not explicitly given in the literature, we will give a proof of Theorem 2.1 which says that both homotopy theories are isomorphic

Abstract. We construct and examine the universal Toda bracket of a highly structured ring spectrum R. This invariant of R is a cohomology class in the Mac Lane cohomology of the graded ring of homotopy groups of R which carries information about R and the category of R-module spectra. It determines for example all triple Toda brackets of R and the first obstruction to realizing a module over the homotopy groups of R by an R-module spectrum. For periodic ring spectra, we study the corresponding theory of higher universal Toda brackets. The real and complex K-theory spectra serve as our main examples. 1.

this paper we advertise the category of #-spaces as a convenient framework for doing `algebra' over `rings' in stable homotopy theory. #-spaces were introduced by Segal [Se] who showed that they give rise to a homotopy category equivalent to the usual homotopy category of connective (i.e. (-1)-connected) spectra. Bousfield and Friedlander [BF] later provided model category structures for #-spaces. The study of `rings, modules and algebras' based on #-spaces became possible when Lydakis [Ly] introduced a symmetric monoidal smash product with good homotopical properties. Here we develop model category structures for modules and algebras, set up (derived) smash products and associated spectral sequences and compare simplicial modules and algebras to their Eilenberg--MacLane spectra counterparts. There are other settings for ring spectra, most notably the S-modules and S-algebras of [EKMM] and the symmetric spectra of [HSS], each of these with its own advantages and disadvantages. We believe that one advantage of the #- space approach is its simplicity. The definitions of the stable equivalences, the smash product and the `rings' (which we call Gamma-rings) are given on a few pages. Another feature is that #-spaces nicely reflect the idea that spectra are a homotopical generalization of abelian groups, that the smash product generalizes the tensor product and that algebras over the sphere spectrum generalize classical rings. There is an Eilenberg--MacLane functor H which embeds the category of simplicial abelian groups as a full subcategory of the category of #-spaces. The embedding has a left adjoint, left inverse which on cofibrant objects models spectrum homology. Similarly, simplicial rings embed fully faithfully into Gamma-rings. We give a quick proof (see Section 4) ...

Abstract. We study the connection between the Goodwillie tower of the identity and the lower central series of the loop group on connected spaces. We define the simplicial theory of homotopy n-nilpotent groups. This notion interpolates between infinite loop spaces and loop spaces. We prove that the set-valued algebraic theory obtained by applying π0 is the theory of ordinary n-nilpotent groups and that the Goodwillie tower of a connected space is determined by a certain homotopy left Kan extension. We prove that n-excisive functors of the form ΩF have values in homotopy n-nilpotent groups. 1.

Abstract. We prove that Hi(A, Φ(A)) = 0, i> 0. Here A is a commutative algebra over the prime field Fp of characteristic p> 0 and Φ(A) is A considered as a bimodule, where the left multiplication is the usual one, while the right multiplication is given via Frobenius endomorphism and H • denotes the Hochschild homology over Fp. This result has implications in MacLane homology theory. Among other results, we prove that HML•(A, T) = 0, provided A is an algebra over a field K of characteristic p> 0 and T is a strict homogeneous polynomial functor of degree d with 1 < d < Card(K). 1.

There are many situations in algebraic topology, homotopy theory, and homological algebra in which operads parametrize interesting algebraic structures [10, 16, 27, 30, 35]. In many of these, there is a notion of abelianization or stabilization which provides a notion of homology [1, 2, 14, 42, 44]. In these contexts,