We show that the mapping class group of a handlebody is a virtual duality group, in the sense of Bieri and Eckmann. In positive genus we give a description of the dualising module of any torsion-free, finite-index subgroup of the handlebody mapping class group as the homology of the complex of non-simple disc systems.

We give a self-contained and streamlined rendition of Andrea Bianchi's recent proof of the Mumford conjecture using moduli spaces of branched covers. 

We show that a certain locus inside the moduli space ​ of hyperbolic surfaces, given by surfaces with “sufficiently many” short geodesics, is a classifying space of the handlebody mapping class group. A consequence of the construction is that the top weight cohomology of , studied by Chan–Galatius–Payne, maps injectively into the cohomology of the handlebody mapping class group.

Over a field of characteristic zero, we show that two commutative differential graded (dg) algebras are quasi-isomorphic if and only if they are quasi-isomorphic as associative dg algebras. This answers a folklore problem in rational homotopy theory, showing that the rational homotopy type of a space is determined by its associative dg algebra of rational cochains. We also show a Koszul dual statement, under an additional completeness hypothesis: two homotopy complete dg Lie algebras whose universal enveloping algebras are quasi-isomorphic as associative dg algebras must themselves be quasi-isomorphic. The latter result applies in particular to nilpotent Lie algebras (not differential graded), in which case it says that two nilpotent Lie algebras whose universal enveloping algebras are isomorphic as associative algebras must be isomorphic.

We prove a comparison isomorphism between singular cohomology and sheaf cohomology.

We correct some oversights in the paper A spectral sequence for stratified spaces and configuration spaces of points by the second author. In particular, we explain that an additional hypothesis should be added to Theorem 4.15 in said paper.

A recent theorem of Hyde proves that the factorization statistics of a random polynomial over a finite field are governed by the action of the symmetric group on the configuration space of n distinct ordered points in R-3. Hyde asked whether this result could be explained geometrically. We give a geometric proof of Hyde's theorem as an instance of the Grothendieck-Lefschetz trace formula applied to an interesting, highly nonseparated algebraic space. An advantage of our method is that it generalizes uniformly to any Weyl group. In the process we study certain non-Hausdorff models for complements of hyperplane arrangements, first introduced by Proudfoot.

We give a proof of the Homotopy Transfer Theorem following Kadeishvili's original strategy. Although Kadeishvili originally restricted himself to transferring a dg algebra structure to an A-infinity-structure on homology, we will see that a small modification of his argument proves the general case of transferring any kind of infinity-algebra structure along a quasi-isomorphism, under weaker hypotheses than existing proofs of this result.

Let X be a topological space. We consider certain generalized configuration spaces of points on X, obtained from the cartesian product X-n by removing some intersections of diagonals. We give a systematic framework for studying the cohomology of such spaces using what we call 'twisted commutative dg algebra models' for the cochains on X. Suppose that X is a 'nice' topological space, R is any commutative ring, H-c(circle) (X, R) -> H-circle(X, R) is the zero map, and that H-c(circle) (X, R) is a projective R-module. We prove that the compact support cohomology of any generalized configuration space of points on X depends only on the graded R-module H-c(circle) (X, R). This generalizes a theorem of Arabia.