David Blanc

NEW MODEL CATEGORIES FROM OLD

Abstract We review Quillens concept of a model category as the proper setting for defining derived functors in non-abelian settings, explain how one can transport a model structure from one category to another by mean of adjoint functors (under suitable assumptions), and define such structures for categories of cosimplicial coalgebras.

A Hurewicz spectral sequence for homology

For any connected space X and ring R, we describe a first-quadrant spectral sequence converging to H* (X; R) , whose E2-term depends only on the homotopy groups of X and the action of the primary homotopy operations on them. We show that (for simply connected X) the E2-term vanishes below a line of slope 1/2; computing part of the E2-term just above this line, we find a certain periodicity, which shows, in particular, that this vanishing line is best possible. We also show how the differentials in this spectral sequence can be used to compute certain Toda brackets.

On realizing diagrams of Π–algebras

Given a diagram ofO‐algebras (graded groups equipped with an action of the primary homotopy operations), we ask whether it can be realized as the homotopy groups of a diagram of spaces. The answer given here is in the form of an obstruction theory, of somewhat wider application, formulated in terms of generalized O‐algebras. This extends a program begun by Dwyer, Kan, Stover, Blanc and Goerss [21; 10] to study the realization of a single O‐algebra. In particular, we explicitly analyze the simple case of a single map, and provide a detailed example, illustrating the connections to higher homotopy operations. 18G55; 55Q05, 55P65

Mapping spaces and M-CW complexes

We describe a procedure for recovering X from the space of maps from M into X, when M is constructed by cofibers of self-maps. This can be used to define an M-CW approximation functor. The case when M is a Moore space is discussed in greater detail.

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 n-th 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 of [3], the Quillen homology of a graded algebra (with applications to the “Grothendieck spectral sequence” of [14]), and the wedge, smash, and homology spectral sequences of [St] and [1].

Stems and spectral sequences

We introduce the category Pstemanc of n‐stems, with a functor Panc from spaces to Pstemanc. This can be thought of as the n‐th order homotopy groups of a space. We show how to associate to each simplicial n‐stem Q an .nC1/‐truncated spectral sequence. Moreover, if Q D PancX is the Postnikov n‐stem of a simplicial space X , the truncated spectral sequence for Q is the truncation of the usual homotopy spectral sequence of X . Similar results are also proven for cosimplicial n‐stems. They are helpful for computations, since n‐stems in low degrees have good algebraic models. 55T05; 18G40, 18G55, 55S45, 55T15, 18G30, 18G10

CW simplicial resolutions of spaces with an application to loop spaces

Abstract We show how a certain type of CW simplicial resolutions of spaces by wedges of spheres may be constructed, and how such resolutions yield an obstruction theory for a given space to be a loop space.

Segal-type algebraic models of n–types

For each n\geq 1 we introduce two new Segal-type models of n-types of topological spaces: weakly globular n-fold groupoids, and a lax version of these. We show that any n-type can be represented up to homotopy by such models via an explicit algebraic fundamental n-fold groupoid functor. We compare these models to Tamsamanis weak n-groupoids, and extract from them a model for (k-1)connected n-types

The configuration space of arachnoid mechanisms

Abstract The configuration spaces of arachnoid mechanisms are analyzed in this paper. These mechanisms consist of k  branches each of which has an arbitrary number of links and a fixed initial point, while all branches end at one common end-point. It is shown that generically, the configuration spaces of such mechanisms are manifolds, and the conditions for the exceptional cases are determined. The configuration space of planar arachnoid mechanisms having k   branches, each with two links is analyzed for both the non-singular and the singular cases.

Realizing coalgebras over the Steenrod algebra

Abstract We describe algebraic obstruction theories for realizing an abstract (co)algebra K ∗ over the mod p Steenrod algebra as the (co)homology of a topological space, and for distinguishing between the p-homotopy types of different realizations. The theories are expressed in terms of the Quillen cohomology of K ∗ .