We compare the sheaf-theoretic and singular chain versions of Poincaré duality for intersection homology, showing that they are isomorphic via naturally defined maps. Similarly, we demonstrate the existence of canonical isomorphisms between the singular intersection cohomology cup product, the hypercohomology product induced by the Goresky–MacPherson sheaf pairing, and, for PL pseudomanifolds, the Goresky–MacPherson PL intersection product. We also show that the de Rham isomorphism of Brasselet, Hector and Saralegi preserves product structures.

Kronheimer and Mrowka (Invent. Math. 130 (1997) 209–255) constructed a variant of Seiberg–Witten invariants for a $4$–manifold $X$ with contact boundary. Using Furuta’s finite-dimensional approximation, we refine this invariant in the case $H^1\left(X,\partial X;ℝ\right)=0$.

The bigerbes introduced here give a refinement of the notion of $2$–gerbes, representing degree four integral cohomology classes of a space. Defined in terms of bisimplicial line bundles, bigerbes have a symmetry with respect to which they form “bundle $2$–gerbes” in two ways; this structure replaces higher associativity conditions. We provide natural examples, including a Brylinski–McLaughlin bigerbe associated to a principal $G$–bundle for a simply connected simple Lie group. This represents the first Pontryagin class of the bundle, and is the obstruction to the lifting problem on the associated principal bundle over the loop space to the structure group consisting of a central extension of the loop group; in particular, trivializations of this bigerbe for a spin manifold are in bijection with string structures on the original manifold. Other natural examples represent “decomposable” $4$–classes arising as cup products, a universal bigerbe on $K\left(\mathbb{Z},4\right)$ involving its based double loop space, and the representation of any $4$–class on a space by a bigerbe involving its free double loop space. The generalization to “multigerbes” of arbitrary degree is also described.

We characterise completely when limit sets, as parametrised by Cannon–Thurston maps, move discontinuously for a sequence of algebraically convergent quasi-Fuchsian groups.

We construct examples of fibred three-manifolds with first Betti number at least $2$ and with fibred faces all of whose monodromies extend to a handlebody.

By work of W Thurston, knots and links in the $3$–sphere are known to either be torus links; or to contain an essential sphere or torus in their complement; or to be hyperbolic, in which case a unique hyperbolic volume can be calculated for their complement. We employ a construction of Turaev to associate a family of hyperbolic $3$–manifolds of finite volume to any classical or virtual link, even if nonhyperbolic. These are in turn used to define the Turaev volume of a link, which is the minimal volume among all the hyperbolic $3$–manifolds associated via this Turaev construction. In the case of a classical link, we can also define the classical Turaev volume, which is the minimal volume among all the hyperbolic $3$–manifolds associated via this Turaev construction for the classical projections only. We then investigate these new invariants.

We compute the equivariant $\mathit{\text{KO}}$–homology of the classifying space for proper actions of ${\text{SL}}_3\left(\mathbb{Z}\right)$ and ${\text{GL}}_3\left(\mathbb{Z}\right)$. We also compute the Bredon homology and equivariant $K$–homology of the classifying spaces for proper actions of ${\text{PSL}}_2\left(\mathbb{Z}\left[\frac{1}{p}\right]\right)$ and ${\text{SL}}_2\left(\mathbb{Z}\left[\frac{1}{p}\right]\right)$ for each prime $p$. Finally, we prove the unstable Gromov–Lawson–Rosenberg conjecture for a large class of groups whose maximal finite subgroups are odd order and have periodic cohomology.

We prove that the homotopy theory of $N_{\infty }$ operads is equivalent to a homotopy theory of discrete operads, and we construct free and associative operadic realizations of every indexing system. This resolves a conjecture of Blumberg and Hill in the affirmative.

Prime power–fold cyclic branched covers along smoothly slice knots all bound rational homology balls. This phenomenon, however, does not characterize slice knots. We give a new construction of nonslice knots that have the above property. The sliceness obstruction comes from computing twisted Alexander polynomials, and we introduce new techniques to simplify their calculation.

We introduce the category of locally $k$–standard $T$–manifolds, which includes well-known classes of manifolds such as toric and quasitoric manifolds, good contact toric manifolds and moment–angle manifolds. They are smooth manifolds with well-behaved actions of tori. We study their topological properties, such as fundamental groups and equivariant cohomology algebras. Then we discuss when the torus equivariant cohomology algebra distinguishes them up to weakly equivariant homeomorphism.

We study an abelian category of unstable modules with the top $k$ Steenrod operations at the prime 2. We show that this category has homological dimension at most $k$. We establish forgetful functors, suspension functors, loop functors and Frobenius functors between such modules. The forgetful functors induce an inverse system of $\text{Ext}$ groups, and the inverse system stabilizes when the covariant module is bounded above. We define an analogue of the $\mathrm{\Lambda }$ algebra in this context and verify that its cohomology computes $\text{Ext}$.

Our aim is to provide space level maps between configuration spaces of graphs that are predicted by algebraic manipulations of cellular chains. More explicitly, we consider edge contraction and half-edge deletion, and identify the homotopy cofibers in terms of configuration spaces of simpler graphs. The construction’s main benefit lies in making the operations functorial — in particular, graph minors give rise to compatible maps at the level of fundamental groups as well as generalized (co)homology theories.

As applications we provide a long exact sequence for half-edge deletion in any generalized cohomology theory, compatible with cohomology operations such as the Steenrod and Adams operations, allowing for inductive calculations in this general context. We also show that the generalized homology of unordered configuration spaces is finitely generated as a representation of the opposite graph-minor category.

Let $X$ be a surface, possibly with boundary. Suppose it has infinite genus or infinitely many punctures, or a closed subset which is a disk with a Cantor set removed from its interior. For example, $X$ could be any surface of infinite type with only finitely many boundary components. We prove that the mapping class group of $X$ does not satisfy the Tits alternative. That is, $\text{Map}\left(X\right)$ contains a finitely generated subgroup that is not virtually solvable and contains no nonabelian free group.

Modular functors, ie consistent systems of projective representations of mapping class groups of surfaces, were constructed for nonsemisimple modular categories decades ago. Concepts from homological algebra have not been used in this construction although it is an obvious question how they should enter in the nonsemisimple case. We elucidate the interplay between the structures from topological field theory and from homological algebra by constructing a homotopy coherent projective action of the mapping class group $\text{SL}\left(2,\mathbb{Z}\right)$ of the torus on the Hochschild complex of a modular category. This is a further step towards understanding the Hochschild complex of a modular category as a differential graded conformal block for the torus. Moreover, we describe a differential graded version of the Verlinde algebra.