Synopsis: We define and study a notion of coarse complement of a subspace in a given metric space. We define a notion of manifold-like object in the coarse category and prove a version of coarse Alexander duality for such spaces: coarse cohomology of the complement of a subspace is determined by the coarse homology of the subspace.
2. Coarse cohomology of configuration space and coarse embedding, arXiv, To appear in Algebraic & Geometric Topology.
Synopsis: We define a notion of coarse cohomology of the configuration space of a metric space X. We identify a class of degree n in that cohomology which obstructs coarse embedding of X into any uniformly contractible (n-1)-manifold.
Synopsis: We relate the coarse cohomology of a space to a more computable cohomology, called boundedly supported cohomology. As a consequence, we show that coarse cohomology of the complement (introduced in [B-Okun]) can be computed in terms of Alexander-Spanier cohomology for many spaces.
4. On the girth of groups acting on CAT(0) cube complexes (with D. Gulbrandsen , P. Mishra , and P. Parija) , arXiv, To appear in Algebraic & Geometric Topology.
Synopsis: It is known that every Cayley graph of an abelian group has a `small' cycle. We prove a partial converse for CAT(0) cubulated groups: Either the group is virtually abelian, or some Cayley graphs of the group only have `large' cycles.
Synopsis: A metric space having small Uryson 1-width roughly means that the space is close to being a 1-dimensional complex. We investigate which spaces inherit this property from their universal covers: that is, we study which spaces have small Uryson 1-width when their universal covers do.
6. Cyclic orders and actions of Leary-Minasyan groups on coarse PD(n) spaces (with K. Schreve) , arXiv , submitted
Synopsis: Cannon conjecture states that any torsion free hyperbolic group with sphere boundary is a fundamental group of a closed 3-manifold. We use tools from [B-Okun] to show that a high dimensional generalization of this conjecture for CAT(0) groups is false.