Papers

Working Paper
Ben Knudsen. Working Paper. “Configuration spaces in algebraic topology”.Abstract
These expository notes are dedicated to the study of the topology of configuration spaces of manifolds. We give detailed computations of many invariants, including the fundamental group of the configuration spaces of $\mathbb{R}^2$, the integral cohomology of the ordered---and the mod $p$ cohomology of the unordered---configuration spaces of $\mathbb{R}^n$, and the rational cohomology of the unordered configuration spaces of an arbitrary manifold of odd dimension. We also discuss models for mapping spaces in terms of labeled configuration spaces, and we show that these models split stably. Some classical results are given modern proofs premised on hypercover techniques, which we discuss in detail. 
Submitted
Kathryn Hess and Ben Knudsen. Submitted. “A Kuenneth theorem for configuration spaces”.Abstract
We construct a spectral sequence converging to the homology of the ordered configuration spaces of a product of parallelizable manifolds. To identify the second page of this spectral sequence, we introduce a version of the Boardman–Vogt tensor product for linear operadic modules, a purely algebraic operation. Using the rational formality of the little cubes operads, we show that our spectral sequence collapses in characteristic zero.
Forthcoming
William Dwyer, Kathryn Hess, and Ben Knudsen. Forthcoming. “Configuration spaces of products.” Trans. Amer. Math. Soc.Abstract
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.
Byung Hee An, Gabriel C. Drummond-Cole, and Ben Knudsen. Forthcoming. “Edge stabilization in the homology of graph braid groups.” Geom. Topol.Abstract
We introduce a novel type of stabilization map on the configuration spaces of a graph, which increases the number of particles occupying an edge. There is an induced action on homology by the polynomial ring generated by the set of edges, and we show that this homology module is finitely generated. An analogue of classical homological and representation stability for manifolds, this result implies eventual polynomial growth of Betti numbers. We calculate the exact degree of this polynomial, in particular verifying an upper bound conjectured by Ramos. Because the action arises from a family of continuous maps, it lifts to an action at the level of singular chains, which contains strictly more information than the homology level action. We show that the resulting differential graded module is almost never formal over the ring of edges.
Byung Hee An, Gabriel C. Drummond-Cole, and Ben Knudsen. Forthcoming. “Subdivisional spaces and graph braid groups.” Doc. Math.Abstract

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. 

2018
Ben Knudsen. 2018. “Higher enveloping algebras.” Geom. Topol., 22, 7, Pp. 4013-4066.Abstract

We 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.

2017
Ben Knudsen. 2017. “Betti numbers and stability for configuration spaces via factorization homology.” Algebr. Geom. Topol., 17, 5, Pp. 3137-3187.Abstract

Using 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.

Gabriel C. Drummond-Cole and Ben Knudsen. 2017. “Betti numbers of configuration spaces of surfaces.” J. London Math. Soc., 96, 2, Pp. 367-393.Abstract

We give explicit formulas for the Betti numbers, both stable and unstable, of the unordered configuration spaces of an arbitrary surface of finite type.