Volume distortion in homotopy groups

Given a finite metric CW complex $X$ and an element $\alpha \in \pi_n(X)$, what are the properties of a geometrically optimal representative of $\alpha$? We study the optimal volume of $k\alpha$ as a function of $k$. Asymptotically, this function, whose inverse, for reasons of tradition, we call the volume distortion, turns out to be an invariant with respect to the rational homotopy of $X$. We provide a number of examples and techniques for studying this invariant, with a special focus on spaces with few rational homotopy groups. Our main theorem characterizes those $X$ in which all non-torsion homotopy classes are undistorted, that is, their distortion functions are linear.Comment: 49 pages, 4 figures. Accepted for publication in Geometric and Functional Analysis (GAFA

On an extension of the notion of Reedy category

We extend the classical notion of a Reedy category so as to allow non-trivial automorphisms. Our extension includes many important examples occuring in topology such as Segal's category Gamma, or the total category of a crossed simplicial group such as Connes' cyclic category Lambda. For any generalized Reedy category R and any cofibrantly generated model category E, the functor category E^R is shown to carry a canonical model structure of Reedy type

Higher homotopy operations and cohomology

We explain how higher homotopy operations, defined topologically, may be identified under mild assumptions with (the last of) the Dwyer-Kan-Smith cohomological obstructions to rectifying homotopy-commutative diagrams.Comment: 28 page