According to a theorem of Eliashberg and Thurston, a $C^2$–foliation on a closed $3$–manifold can be $C^0$–approximated by contact structures unless all leaves of the foliation are spheres. Examples on the $3$–torus show that every neighbourhood of a foliation can contain nondiffeomorphic contact structures.

In this paper we show uniqueness up to isotopy of the contact structure in a small neighbourhood of the foliation when the foliation has no torus leaf and is not a foliation without holonomy on parabolic torus bundles over the circle. This allows us to associate invariants from contact topology to foliations. As an application we show that the space of taut foliations in a given homotopy class of plane fields is not connected in general.

In this paper we discuss and prove $ϵ$–regularity theorems for Einstein manifolds $\left(M^n,g\right)$, and more generally manifolds with just bounded Ricci curvature, in the collapsed setting.

A key tool in the regularity theory of noncollapsed Einstein manifolds is the following. If $x\in M^n$ is such that $V\; ol\left(B_1\left(x\right)\right)>v>0$ and that $B_2\left(x\right)$ is sufficiently Gromov–Hausdorff close to a cone space $B_2\left(0^{n-\ell },y^{\ast }\right)\subset {ℝ}^{n-\ell }\times C\left(Y^{\ell -1}\right)$ for $\ell \le 3$, then in fact $|Rm|\le 1$ on $B_1\left(x\right)$. No such results are known in the collapsed setting, and in fact it is easy to see that without further assumptions such results are false. It turns out that the failure of such an estimate is related to topology. Our main theorem is that for the above setting in the collapsed context, either the curvature is bounded, or there are topological constraints on $B_1\left(x\right)$.

More precisely, using established techniques one can see there exists $ϵ\left(n\right)$ such that if $\left(M^n,g\right)$ is an Einstein manifold and $B_2\left(x\right)$ is $ϵ$–Gromov–Hausdorff close to ball in $B_2\left(0^{k-\ell },z^{\ast }\right)\subset {ℝ}^{k-\ell }\times Z^{\ell }$, then the fibered fundamental group ${\Gamma }_{ϵ}\left(x\right)\equiv Image\left[{\pi }_1\left(B_{ϵ}\left(x\right)\right)\to {\pi }_1\left(B_2\left(x\right)\right)\right]$ is almost nilpotent with $rank\left({\Gamma }_{ϵ}\left(x\right)\right)\le n-k$. The main result of the this paper states that if $rank\left({\Gamma }_{ϵ}\left(x\right)\right)=n-k$ is maximal, then $|Rm|\le C$ on $B_1\left(x\right)$. In the case when the ball is close to Euclidean, this is both a necessary and sufficient condition. There are generalizations of this result to bounded Ricci curvature and even just lower Ricci curvature.

In this paper we prove new classification results for nonnegatively curved gradient expanding and steady Ricci solitons in dimension three and above, under suitable integral assumptions on the scalar curvature of the underlying Riemannian manifold. In particular we show that the only complete expanding solitons with nonnegative sectional curvature and integrable scalar curvature are quotients of the Gaussian soliton, while in the steady case we prove rigidity results under sharp integral scalar curvature decay. As a corollary, we obtain that the only three-dimensional steady solitons with less than quadratic volume growth are quotients of ${ℝ}^3$ or of $ℝ\times {\Sigma }^2$, where ${\Sigma }^2$ is Hamilton’s cigar.

For each finite-dimensional, simple, complex Lie algebra $\mathfrak{g}$ and each root of unity $\xi$ (with some mild restriction on the order) one can define the Witten–Reshetikhin–Turaev (WRT) quantum invariant ${\tau }_M^{\mathfrak{g}}\left(\xi \right)\in ℂ$ of oriented $3$–manifolds $M$. We construct an invariant $J_M$ of integral homology spheres $M$, with values in $\widehat{\mathbb{Z}\left[q\right]}$, the cyclotomic completion of the polynomial ring $\mathbb{Z}\left[q\right]$, such that the evaluation of $J_M$ at each root of unity gives the WRT quantum invariant of $M$ at that root of unity. This result generalizes the case $\mathfrak{g}={sl}_2$ proved by Habiro. It follows that $J_M$ unifies all the quantum invariants of $M$ associated with $\mathfrak{g}$ and represents the quantum invariants as a kind of “analytic function” defined on the set of roots of unity. For example, ${\tau }_M\left(\xi \right)$ for all roots of unity are determined by a “Taylor expansion” at any root of unity, and also by the values at infinitely many roots of unity of prime power orders. It follows that WRT quantum invariants ${\tau }_M\left(\xi \right)$ for all roots of unity are determined by the Ohtsuki series, which can be regarded as the Taylor expansion at $q=1$, and hence by the Lê–Murakami–Ohtsuki invariant. Another consequence is that the WRT quantum invariants ${\tau }_M^{\mathfrak{g}}\left(\xi \right)$ are algebraic integers. The construction of the invariant $J_M$ is done on the level of quantum group, and does not involve any finite-dimensional representation, unlike the definition of the WRT quantum invariant. Thus, our construction gives a unified, “representation-free” definition of the quantum invariants of integral homology spheres.

We classify all rank two affine manifolds in strata in genus three with two zeros. This confirms a conjecture of Maryam Mirzakhani in these cases. Several technical results are proven for all strata in genus three, with the hope that they may shed light on a complete classification of rank two manifolds in genus three.

We prove that the space of smooth Riemannian metrics on the three-ball with nonnegative Ricci curvature and strictly convex boundary is path-connected, and, moreover, that the associated moduli space (ie modulo orientation-preserving diffeomorphisms of the three-ball) is contractible. As an application, using results of Maximo, Nunes and Smith (to appear in J. Differential Geom.), we show the existence of a properly embedded free boundary minimal annulus on any three-ball with nonnegative Ricci curvature and strictly convex boundary.

We show how every $p$–local compact group can be described as a telescope of $p$–local finite groups. As a consequence, we deduce several corollaries, such as a stable elements theorem for the mod $p$ cohomology of their classifying spaces, and a generalized Dwyer–Zabrodsky description of certain related mapping spaces.

Given a closed Riemannian manifold of dimension $n$ and a Morse–Smale function, there are finitely many $n$–part broken trajectories of the negative gradient flow. We show that if the manifold admits a hyperbolic metric, then the number of $n$–part broken trajectories is always at least the hyperbolic volume. The proof combines known theorems in Morse theory with lemmas of Gromov about simplicial volumes of stratified spaces.

We show that for every nonelementary representation of a surface group into $SL\left(2,ℂ\right)$ there is a Riemann surface structure such that the Higgs bundle associated to the representation lies outside the discriminant locus of the Hitchin fibration.

Let $M$ be a compact manifold, possibly with boundary. We show that the group of homeomorphisms of $M$ has the automatic continuity property: any homomorphism from $Homeo\left(M\right)$ to any separable group is necessarily continuous. This answers a question of C Rosendal. If $N\subset M$ is a submanifold, the group of homeomorphisms of $M$ that preserve $N$ also has this property.

Various applications of automatic continuity are discussed, including applications to the topology and structure of groups of germs of homeomorphisms. In an appendix with Frédéric Le Roux we also show, using related techniques, that the group of germs at a point of homeomorphisms of ${ℝ}^n$ is strongly uniformly simple.