We prove that there are infinitely many pairs of homeomorphic non-diffeomorphic smooth 4–manifolds, such that in each pair one manifold admits an Einstein metric and the other does not. We also show that there are closed 4–manifolds with two smooth structures which admit Einstein metrics with opposite signs of the scalar curvature.

For suitable subgroups of a finitely generated group, we define the intersection number of one subgroup with another subgroup and show that this number is symmetric. We also give an interpretation of this number.

Given a knot $K$ in the 3–sphere, consider a singular disk bounded by $K$ and the intersections of $K$ with the interior of the disk. The absolute number of intersections, minimised over all choices of singular disk with a given algebraic number of intersections, defines the framing function of the knot. We show that the framing function is symmetric except at a finite number of points. The symmetry axis is a new knot invariant, called the natural framing of the knot. We calculate the natural framing of torus knots and some other knots, and discuss some of its properties and its relations to the signature and other well-known knot invariants.

We show that if a closed atoroidal 3–manifold $M$ contains a genuine lamination, then it is group negatively curved in the sense of Gromov. Specifically, we exploit the structure of the non-product complementary regions of the genuine lamination and then apply the first author’s Ubiquity Theorem to show that $M$ satisfies a linear isoperimetric inequality.

We investigate geometrical interpretations of various structure maps associated with the Landweber–Novikov algebra $S^{\ast }$ and its integral dual $S_{\ast }$. In particular, we study the coproduct and antipode in $S_{\ast }$, together with the left and right actions of $S^{\ast }$ on $S_{\ast }$ which underly the construction of the quantum (or Drinfeld) double $\mathcal{D}\left(S^{\ast }\right)$. We set our realizations in the context of double complex cobordism, utilizing certain manifolds of bounded flags which generalize complex projective space and may be canonically expressed as toric varieties. We discuss their cell structure by analogy with the classical Schubert decomposition, and detail the implications for Poincaré duality with respect to double cobordism theory; these lead directly to our main results for the Landweber–Novikov algebra.

Let $X$ be a 4–manifold with contact boundary. We prove that the monopole invariants of $X$ introduced by Kronheimer and Mrowka vanish under the following assumptions: (i) a connected component of the boundary of $X$ carries a metric with positive scalar curvature and (ii) either $b_2^{+}\left(X\right)>0$ or the boundary of $X$ is disconnected. As an application we show that the Poincaré homology 3–sphere, oriented as the boundary of the positive $E_8$ plumbing, does not carry symplectically semi-fillable contact structures. This proves, in particular, a conjecture of Gompf, and provides the first example of a 3–manifold which is not symplectically semi-fillable. Using work of Frøyshov, we also prove a result constraining the topology of symplectic fillings of rational homology 3–spheres having positive scalar curvature metrics.

We obtain some restrictions on the topology of infinite volume hyperbolic manifolds. In particular, for any $n$ and any closed negatively curved manifold $M$ of dimension $\ge 3$, only finitely many hyperbolic $n$–manifolds are total spaces of orientable vector bundles over $M$.

We construct splittings of some completions of the $\mathbb{Z}∕\left(p\right)$–Tate cohomology of $E\left(n\right)$ and some related spectra. In particular, we split (a completion of) $tE\left(n\right)$ as a (completion of) a wedge of $E\left(n-1\right)$s as a spectrum, where $t$ is shorthand for the fixed points of the $Z∕\left(p\right)$–Tate cohomology spectrum (ie the Mahowald inverse limit ${invlim}_k\left(\left(P_{-k}\wedge \Sigma E\left(n\right)\right)\right)$). We also give a multiplicative splitting of $tE\left(n\right)$ after a suitable base extension.

The topological underpinnings are presented for a new algorithm which answers the question: “Is a given knot the unknot?” The algorithm uses the braid foliation technology of Bennequin and of Birman and Menasco. The approach is to consider the knot as a closed braid, and to use the fact that a knot is unknotted if and only if it is the boundary of a disc with a combinatorial foliation. The main problems which are solved in this paper are: how to systematically enumerate combinatorial braid foliations of a disc; how to verify whether a combinatorial foliation can be realized by an embedded disc; how to find a word in the the braid group whose conjugacy class represents the boundary of the embedded disc; how to check whether the given knot is isotopic to one of the enumerated examples; and finally, how to know when we can stop checking and be sure that our example is not the unknot.

A self-dual harmonic 2–form on a 4–dimensional Riemannian manifold is symplectic where it does not vanish. Furthermore, away from the form’s zero set, the metric and the 2–form give a compatible almost complex structure and thus pseudo-holomorphic subvarieties. Such a subvariety is said to have finite energy when the integral over the variety of the given self-dual 2–form is finite. This article proves a regularity theorem for such finite energy subvarieties when the metric is particularly simple near the form’s zero set. To be more precise, this article’s main result asserts the following: Assume that the zero set of the form is non-degenerate and that the metric near the zero set has a certain canonical form. Then, except possibly for a finite set of points on the zero set, each point on the zero set has a ball neighborhood which intersects the subvariety as a finite set of components, and the closure of each component is a real analytically embedded half disk whose boundary coincides with the zero set of the form.

Theorem 3.1 of “The symmetry of intersection numbers in group theory” is false: the error occurs in the proof of Lemma 3.6. A counterexample is given.