Configuration spaces of products

We show that the configuration spaces of a product of parallelizable manifolds may be recovered from those of the factors as the Boardman-Vogt tensor product of right modules over the operads of little cubes of the appropriate dimension. We also discuss an analogue of this result for manifolds that are not necessarily parallelizable, which involves a new operad of skew little cubes.Comment: 21 pages, 1 figure. To appear in Transactions of the AMS. May vary slightly from published versio

Projection spaces and twisted Lie algebras

A projection space is a collection of spaces interrelated by the combinatorics of projection onto tensor factors in a symmetric monoidal background category. Examples include classical configuration spaces, orbit configuration spaces, the graphical configuration spaces of Eastwood--Huggett, the simplicial configuration spaces of Cooper--de Silva--Sazdanovic, the generalized configuration spaces of Petersen, and Stiefel manifolds. We show that, under natural assumptions on the background category, the homology of a projection space is calculated by the Chevalley--Eilenberg complex of a certain generalized Lie algebra. We identify conditions on this Lie algebra implying representation stability in the classical setting of finite sets and injections.Comment: 32 pages. To appear in Contemporary Mathematics. May differ slightly from published versio

The topological complexity of pure graph braid groups is stably maximal

We prove Farber's conjecture on the stable topological complexity of configuration spaces of graphs. The conjecture follows from a general lower bound derived from recent insights into the topological complexity of aspherical spaces. Our arguments apply equally to higher topological complexity.Comment: 9 pages. Minor changes. To appear in Forum of Mathematics, Sigma. May differ slightly from published versio

Subdivisional spaces and graph braid groups

We study the problem of computing the homology of the configuration spaces of a finite cell complex $X$. We proceed by viewing $X$, together with its subdivisions, as a subdivisional space--a kind of diagram object in a category of cell complexes. After developing a version of Morse theory for subdivisional spaces, we decompose $X$ and show that the homology of the configuration spaces of $X$ is computed by the derived tensor product of the Morse complexes of the pieces of the decomposition, an analogue of the monoidal excision property of factorization homology. Applying this theory to the configuration spaces of a graph, we recover a cellular chain model due to \'{S}wi\k{a}tkowski. Our method of deriving this model enhances it with various convenient functorialities, exact sequences, and module structures, which we exploit in numerous computations, old and new.Comment: 71 pages, 15 figures. Typo fixed. May differ slightly from version published in Documenta Mathematic

Embedding calculus and smooth structures

We study the dependence of the embedding calculus Taylor tower on the smooth structures of the source and target. We prove that embedding calculus does not distinguish exotic smooth structures in dimension 4, implying a negative answer to a question of Viro. In contrast, we show that embedding calculus does distinguish certain exotic spheres in higher dimensions. As a technical tool of independent interest, we prove an isotopy extension theorem for the limit of the embedding calculus tower, which we use to investigate several further examples.Comment: 34 pages, 1 figur