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 discrete 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 Swiatkowski. 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.

%B Doc. Math. %G eng %0 Journal Article %J Geom. Topol. %D 2018 %T Higher enveloping algebras %A Ben Knudsen %XWe provide spectral Lie algebras with enveloping algebras over the operad of little G-framed n-dimensional disks for any choice of dimension n and structure group G, and we describe these objects in two complementary ways. The first description is an abstract characterization by a universal mapping property, which witnesses the higher enveloping algebra as the value of a left adjoint in an adjunction. The second, a generalization of the Poincare-Birkhoff-Witt theorem, provides a concrete formula in terms of Lie algebra homology. Our construction pairs the theories of Koszul duality and Day convolution in order to lift to the world of higher algebra the fundamental combinatorics of Beilinson-Drinfeld's theory of chiral algebras. As an application, we show that the stable homotopy types of configuration spaces are homotopy invariants.

%B Geom. Topol. %V 22 %P 4013-4066 %8 2018 %G eng %N 7 %0 Journal Article %J Algebr. Geom. Topol. %D 2017 %T Betti numbers and stability for configuration spaces via factorization homology %A Ben Knudsen %XUsing factorization homology, we realize the rational homology of the unordered configuration spaces of an arbitrary manifold M, possibly with boundary, as the homology of a Lie algebra constructed from the compactly supported cohomology of M. By locating the homology of each configuration space within the Chevalley-Eilenberg complex of this Lie algebra, we extend theorems of Boedigheimer-Cohen-Taylor and Felix-Thomas and give a new, combinatorial proof of the homological stability results of Church and Randal-Williams. Our method lends itself to explicit calculations, examples of which we include.

%B Algebr. Geom. Topol. %V 17 %P 3137-3187 %G eng %N 5 %0 Journal Article %J J. London Math. Soc. %D 2017 %T Betti numbers of configuration spaces of surfaces %A Gabriel C. Drummond-Cole %A Ben Knudsen %XWe give explicit formulas for the Betti numbers, both stable and unstable, of the unordered configuration spaces of an arbitrary surface of finite type.

%B J. London Math. Soc. %V 96 %P 367-393 %G eng %N 2