When the normal bundle $N_{Z∕X}$ is convex with a minor assumption, we prove that genus$-0$ GW–invariants of the blow-up ${Bl}_{Z}X$ of $X$ along a submanifold $Z$, with cohomology insertions from $X$, are identical to GW–invariants of $X$. Under the same hypothesis, a vanishing theorem is also proved. An example to which these two theorems apply is when $N_{Z∕X}$ is generated by its global sections. These two main theorems do not hold for arbitrary blow-ups, and counterexamples are included.

This paper continues our project started in [J. Funct. Anal. 219, 109–133] where Poincaré duality in $K$–theory was studied for singular manifolds with isolated conical singularities. Here, we extend the study and the results to general stratified pseudomanifolds. We review the axiomatic definition of a smooth stratification $S$ of a topological space $X$ and we define a groupoid $T^{S}X$, called the $S$–tangent space. This groupoid is made of different pieces encoding the tangent spaces of strata, and these pieces are glued into the smooth noncommutative groupoid $T^{S}X$ using the familiar procedure introduced by Connes for the tangent groupoid of a manifold. The main result is that $C^{\ast }\left(T^{S}X\right)$ is Poincaré dual to $C\left(X\right)$, in other words, the $S$–tangent space plays the role in $K$–theory of a tangent space for $X$.

We prove a global fixed point theorem for the centralizer of a homeomorphism of the two-dimensional disk $D$ that has attractor–repeller dynamics on the boundary with at least two attractors and two repellers. As one application we give an elementary proof of Morita’s Theorem, that the mapping class group of a closed surface $S$ of genus $g$ does not lift to the group of $C^2$ diffeomorphisms of $S$ and we improve the lower bound for $g$ from $5$ to $3$.

Consider the space of long knots in ${ℝ}^n$, $K_{n,1}$. This is the space of knots as studied by V Vassiliev. Based on previous work [Budney: Topology 46 (2007) 1–27], [Cohen, Lada and May: Springer Lecture Notes 533 (1976)] it follows that the rational homology of $K_{3,1}$ is free Gerstenhaber–Poisson algebra. A partial description of a basis is given here. In addition, the mod–$p$ homology of this space is a free, restricted Gerstenhaber–Poisson algebra. Recursive application of this theorem allows us to deduce that there is $p$–torsion of all orders in the integral homology of $K_{3,1}$.

This leads to some natural questions about the homotopy type of the space of long knots in ${ℝ}^n$ for $n>3$, as well as consequences for the space of smooth embeddings of $S^1$ in $S^3$ and embeddings of $S^1$ in ${ℝ}^3$.

The $k$–dimensional Dehn (or isoperimetric) function of a group bounds the volume of efficient ball-fillings of $k$–spheres mapped into $k$–connected spaces on which the group acts properly and cocompactly; the bound is given as a function of the volume of the sphere. We advance significantly the observed range of behavior for such functions. First, to each nonnegative integer matrix $P$ and positive rational number $r$, we associate a finite, aspherical $2$–complex $X_{r,P}$ and determine the Dehn function of its fundamental group $G_{r,P}$ in terms of $r$ and the Perron–Frobenius eigenvalue of $P$. The range of functions obtained includes $\delta \left(x\right)=x^s$, where $s\in \mathbb{Q}\cap \left[2,\infty \right)$ is arbitrary. Next, special features of the groups $G_{r,P}$ allow us to construct iterated multiple HNN extensions which exhibit similar isoperimetric behavior in higher dimensions. In particular, for each positive integer $k$ and rational $s\ge \left(k+1\right)∕k$, there exists a group with $k$–dimensional Dehn function $x^s$. Similar isoperimetric inequalities are obtained for fillings modeled on arbitrary manifold pairs $\left(M,\partial M\right)$ in addition to $\left(B^{k+1},S^k\right)$.

Let $N^h\in H\left(M,P\right)$ be a hyperbolic structure of bounded geometry on a pared manifold such that each component of ${\partial }_{0}M=\partial M-P$ is incompressible. We show that the limit set of $N^h$ is locally connected by constructing a natural Cannon–Thurston map. This provides a unified treatment, an alternate proof and a generalization of results due to Cannon and Thurston, Minsky, Bowditch, Klarreich and the author.

Let ${\widetilde{\mathcal{ℳ}}}_{g,n}$, for $2g-2+n>0$, be the moduli stack of $n$–pointed, genus $g$, stable complex curves of compact type. Various characterizations and properties are obtained of both the topological and algebraic fundamental groups of the stack ${\widetilde{\mathcal{ℳ}}}_{g,n}$. For instance we show that the topological fundamental groups are linear, extending to all $n\ge 0$ previous results of Morita and Hain for $g\ge 2$ and $n=0,1$.

Let ${\Gamma }_{g,n}$, for $2g-2+n>0$, be the Teichmüller group associated with a compact Riemann surface of genus $g$ with $n$ points removed $S_{g,n}$, ie the group of homotopy classes of diffeomorphisms of $S_{g,n}$ which preserve the orientation of $S_{g,n}$ and a given order of its punctures. Let ${\mathcal{K}}_{g,n}$ be the normal subgroup of ${\Gamma }_{g,n}$ generated by Dehn twists along separating simple closed curves (briefly s.c.c.) on $S_{g,n}$. The above theory yields a characterization of ${\mathcal{K}}_{g,n}$ for all $n\ge 0$, improving Johnson’s classical results for closed and one-punctured surfaces in [Topology 24 (1985) 113-126].

The Torelli group ${\mathcal{T}}_{g,n}$ is the kernel of the natural representation ${\Gamma }_{g,n}\to {Sp}_{2g}\left(\mathbb{Z}\right)$. The abelianization of the Torelli group ${\mathcal{T}}_{g,n}$ is determined for all $g\ge 1$ and $n\ge 1$, thus completing classical results of Johnson [Topology 24 (1985) 127-144] and Mess [Topology 31 (1992) 775-790] for closed and one-punctured surfaces.

We also prove that a connected finite étale cover ${\widetilde{\mathcal{ℳ}}}^{\lambda }$ of ${\widetilde{\mathcal{ℳ}}}_{g,n}$, for $g\ge 2$, has a Deligne–Mumford compactification ${\overline{\mathcal{ℳ}}}^{\lambda }$ with finite fundamental group. This implies that, for $g\ge 3$, any finite index subgroup of ${\Gamma }_g$ containing ${\mathcal{K}}_g$ has vanishing first cohomology group, improving a result of Hain [Math. Sci. Res. Inst. Publ. 28 (1995) 97-143].

Motivated by the programmes initiated by Taubes and Perutz, we study the geometry of near-symplectic $4$–manifolds, ie, manifolds equipped with a closed $2$–form which is symplectic outside a union of embedded $1$–dimensional submanifolds, and broken Lefschetz fibrations on them; see Auroux, Donaldson and Katzarkov adk and Gay and Kirby gkirby. We present a set of four moves which allow us to pass from any given broken fibration to any other which is deformation equivalent to it. Moreover, we study the change of the near-symplectic geometry under each of these moves. The arguments rely on the introduction of a more general class of maps, which we call wrinkled fibrations and which allow us to rely on classical singularity theory. Finally, we illustrate these constructions by showing how one can merge components of the zero-set of the near-symplectic form. We also disprove a conjecture of Gay and Kirby by showing that any achiral broken Lefschetz fibration can be turned into a broken Lefschetz fibration by applying a sequence of our moves.

We characterize the $2$–line of the $p$–local Adams–Novikov spectral sequence in terms of modular forms satisfying a certain explicit congruence condition for primes $p\ge 5$. We give a similar characterization of the $1$–line, reinterpreting some earlier work of A Baker and G Laures. These results are then used to deduce that, for $\ell$ a prime which generates ${\mathbb{Z}}_p^{\times }$, the spectrum $Q\left(\ell \right)$ detects the $\alpha$ and $\beta$ families in the stable stems.

Let $X$ be a smooth irreducible projective complex variety with an ample line bundle $L$, and $D$ be a simple normal crossing hypersurface. We establish the Kobayashi–Hitchin correspondence between tame harmonic bundles on $X-D$ and ${\mu }_L$–stable parabolic $\lambda$–flat bundles with trivial characteristic numbers on $\left(X,D\right)$. In particular, we obtain the quasiprojective version of the Corlette–Simpson correspondence between flat bundles and Higgs bundles.

We prove that a $3$–dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by the metric induced on its boundary. Furthermore, any hyperbolic metric on the torus with cone singularities of positive curvature can be realized as the induced metric on the boundary of a convex polyhedral cusp.

The proof uses the discrete total curvature functional on the space of “cusps with particles”, which are hyperbolic cone-manifolds with the singular locus a union of half-lines. We prove, in addition, that convex polyhedral cusps with particles are rigid with respect to the induced metric on the boundary and the curvatures of the singular locus.

Our main theorem is equivalent to a part of a general statement about isometric immersions of compact surfaces.

Let $M$ be a closed, connected, orientable three-manifold. The purpose of this paper is to study the Seiberg–Witten Floer homology of $M$ given that $S^1\times M$ admits a symplectic form.

We have developed a mathematical theory of the topological vertex—a theory that was originally proposed by M Aganagic, A Klemm, M Mariño and C Vafa on effectively computing Gromov–Witten invariants of smooth toric Calabi–Yau threefolds derived from duality between open string theory of smooth Calabi–Yau threefolds and Chern–Simons theory on three-manifolds.