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 CDGA 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.

This volume contains the proceedings of the conference on Manifolds, K-Theory, and Related Topics, held from June 23-27, 2014, in Dubrovnik, Croatia. The articles contained in this volume are a col ...

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.

Preface Part I. Cubical Diagrams: 1. Preliminaries 2. 1-cubes: homotopy fibers and cofibers 3. 2-cubes: homotopy pullbacks and pushouts 4. 2-cubes: the Blakers-Massey Theorems 5. n-cubes: generalized homotopy pullbacks and pushouts 6. The Blakers-Massey Theorems for n-cubes Part II. Generalizations, Related Topics, and Applications: 7. Some category theory 8. Homotopy limits and colimits of diagrams of spaces 9. Cosimplicial spaces 10. Applications Appendix References Index.