We use rational formality of configuration spaces and the bar construction to study the cohomology of the space of braids in dimension four or greater. We provide a diagram complex for braids and a quasi-isomorphism to the de Rham cochains on the space of braids. The quasi-isomorphism is given by a configuration space integral followed by Chen’s iterated integrals. This extends results of Kohno and of Cohen and Gitler on the cohomology of the space of braids to a commutative differential graded algebra suitable for integration. We show that this integration is compatible with Bott–Taubes configuration space integrals for long links via a map between two diagram complexes. As a corollary, we get a surjection in cohomology from the space of long links to the space of braids. We also discuss to what extent our results apply to the case of classical braids.

Manifold calculus of functors has in recent years been successfully used in the study of the topology of various spaces of embeddings of one manifold in another. Given a space of embeddings, the theory produces a Taylor tower whose purpose is to approximate this space in a suitable sense. Central to the story are deep theorems about the convergence of this tower. We provide an exposition of the convergence results in the special case of embeddings into $\mathbb R^n$, which has been the case of primary interest in applications. We try to use as little machinery as possible and give several improvements and restatements of existing arguments used in the proofs of the main results.

We study the beginning of the Taylor tower, supplied by manifold calculus of functors, for the space of r-immersions, which are immersions without r-fold self-intersections. We describe the first r layers of the tower and discuss the connectivities of the associated maps. We also prove several results about r-immersions that are of independent interest.

If $D$ is a Reedy category and $M$ is a model category, the category $M^{D}$ of $D$-diagrams in $M$ is a model category under the Reedy model category structure. If $C \to D$ is a Reedy functor between Reedy categories, then there is an induced functor of diagram categories $M^{D} \to M^{C}$. Our main result is a characterization of the Reedy functors $C \to D$ that induce right or left Quillen functors $M^{D} \to M^{C}$ for every model category $M$. We apply these results to various situations, and in particular show that certain important subdiagrams of a fibrant multicosimplicial object are fibrant.