In this paper we study the growth of Casson handles which appear inside smooth four-manifolds. A simply-connected and smooth four-manifold admits decompositions of its intersection form. Casson handles appear around one side of the end of them, when the type is even. They are parameterized by signed infinite trees and their growth measures some of the complexity of the smooth structure near the end. We show that with respect to some decompositions of the forms on the K3 surface, the corresponding Casson handles cannot be of bounded type in our sense. In particular they cannot be periodic. The same holds for all logarithmic transforms which are homotopically equivalent to the K3 surface. We construct Yang–Mills gauge theory over Casson handles of bounded type, and verify that transversality works over them.

We construct a natural, tame action of the monoid of injective self-maps of the set of natural numbers on the homotopy groups of a symmetric spectrum. This extra algebraic structure allows a conceptual and uniform understanding of various phenomena related to ${\pi }_{\ast }$–isomorphisms, semistability and the relationship between naive and true homotopy groups for symmetric spectra.

We describe the connected components of the complement of a natural “diagonal” of real codimension $1$ in a stratum of quadratic differentials on ${ℂ\mathbb{P}}^1$. We establish a natural bijection between the set of these connected components and the set of generic configurations that appear on such “flat spheres”. We also prove that the stratum has only one topological end. Finally, we elaborate a necessary toolkit destined to evaluation of the Siegel–Veech constants.

To a presentation of an oriented link as the closure of a braid we assign a complex of bigraded vector spaces. The Euler characteristic of this complex (and of its triply-graded cohomology groups) is the HOMFLYPT polynomial of the link. We show that the dimension of each cohomology group is a link invariant.

We construct a model for the space of automorphisms of a connected $p$–compact group in terms of the space of automorphisms of its maximal torus normalizer and its root datum. As a consequence we show that any homomorphism to the outer automorphism group of a $p$–compact group can be lifted to a group action, analogous to a classical theorem of de Siebenthal for compact Lie groups. The model of this paper is used in a crucial way in our paper ‘The classification of 2-compact groups’ [arXiv:math.AT/0611437], where we prove the conjectured classification of $2$–compact groups and determine their automorphism spaces.

In the second of a pair of papers, we complete our geometric construction of “Lagrangian matching invariants” for smooth four-manifolds equipped with broken fibrations. We prove an index formula, a vanishing theorem for connected sums and an analogue of the Meng–Taubes formula. These results lend support to the conjecture that the invariants coincide with Seiberg–Witten invariants of the underlying four-manifold, and are in particular independent of the broken fibration.

We prove that the well-rounded retract of ${SO}_n\setminus {SL}_nℝ$ is a minimal ${SL}_n\mathbb{Z}$–invariant spine.

We give formulae for the Ozsváth–Szabó invariants of $4$–manifolds $X$ obtained by fiber sum of two manifolds $M_1$, $M_2$ along surfaces ${\Sigma }_1$, ${\Sigma }_2$ having trivial normal bundle and genus $g\ge 1$. The formulae follow from a general theorem on the Ozsváth–Szabó invariants of the result of gluing two $4$–manifolds along a common boundary, which is phrased in terms of relative invariants of the pieces. These relative invariants take values in a version of Heegaard Floer homology with coefficients in modules over certain Novikov rings; the fiber sum formula follows from the theorem that this “perturbed” version of Heegaard Floer theory recovers the usual Ozsváth–Szabó invariants, when the $4$–manifold in question has $b^{+}\ge 2$. The construction allows an extension of the definition of Ozsváth–Szabó invariants to $4$–manifolds having $b^{+}=1$ depending on certain choices, in close analogy with Seiberg–Witten theory. The product formulae lead quickly to calculations of the Ozsváth–Szabó invariants of various $4$–manifolds; in all cases the results are in accord with the conjectured equivalence between Ozsváth–Szabó and Seiberg–Witten invariants.

We study atomic right-angled Artin groups – those whose defining graph has no cycles of length $\le 4$, and no separating vertices, separating edges, or separating vertex stars. We show that these groups are not quasi-isometrically rigid, but that an intermediate form of rigidity does hold. We deduce from this that two atomic groups are quasi-isometric iff they are isomorphic.

We determine the abstract commensurator $Com\left(F\right)$ of Thompson’s group $F$ and describe it in terms of piecewise linear homeomorphisms of the real line. We show $Com\left(F\right)$ is not finitely generated and determine which subgroups of finite index in $F$ are isomorphic to $F$. We also show that the natural map from the commensurator group to the quasi-isometry group of $F$ is injective.

We prove that manifolds of Lusternik–Schnirelmann category $2$ necessarily have free fundamental group. We thus settle a 1992 conjecture of Gomez-Larrañaga and Gonzalez-Acuña by generalizing their result in dimension $3$ to all higher dimensions. We also obtain some general results on the relations between the fundamental group of a closed manifold $M$, the dimension of $M$ and the Lusternik–Schnirelmann category of $M$, and we relate the latter to the systolic category of $M$.

In this paper, we study and almost completely classify contact structures on closed 3–manifolds which are totally geodesic for some Riemannian metric. Due to previously known results, this amounts to classifying contact structures on Seifert manifolds which are transverse to the fibers. Actually, we obtain the complete classification of contact structures with negative (maximal) twisting number (which includes the transverse ones) on Seifert manifolds whose base is not a sphere, as well as partial results in the spherical case.

We prove a combination theorem for trees of (strongly) relatively hyperbolic spaces and finite graphs of (strongly) relatively hyperbolic groups. This gives a geometric extension of Bestvina and Feighn’s Combination Theorem for hyperbolic groups and answers a question of Swarup. We also prove a converse to the main Combination Theorem.

We prove an equivariant version of the fact that word-hyperbolic groups have finite asymptotic dimension. This is important in connection with our forthcoming proof of the Farrell–Jones conjecture for $K_{\ast }\left(RG\right)$ for every word-hyperbolic group $G$ and every coefficient ring $R$.