Using geodesic length functions, we define a natural family of real codimension 1 subvarieties of Teichmüller space, namely the subsets where the lengths of two distinct simple closed geodesics are of equal length. We investigate the point set topology of the union of all such hypersurfaces using elementary methods. Finally, this analysis is applied to investigate the nature of the Markoff conjecture.

Suppose $f:V\to W$ is an embedding of closed oriented manifolds whose normal bundle has the structure of a complex vector bundle. It is well known in both complex and symplectic geometry that one can then construct a manifold $\tilde{W}$ which is the blow-up of $W$ along $V$. Assume that $dimW\ge 2dimV+3$ and that $H^1\left(f\right)$ is injective. We construct an algebraic model of the rational homotopy type of the blow-up $\tilde{W}$ from an algebraic model of the embedding and the Chern classes of the normal bundle. This implies that if the space $W$ is simply connected then the rational homotopy type of $\tilde{W}$ depends only on the rational homotopy class of $f$ and on the Chern classes of the normal bundle.

Let $G$ be a word-hyperbolic group, obtained as a graph of free groups amalgamated along cyclic subgroups. If $H_2\left(G;\mathbb{Q}\right)$ is nonzero, then $G$ contains a closed hyperbolic surface subgroup. Moreover, the unit ball of the Gromov–Thurston norm on $H_2\left(G;ℝ\right)$ is a finite-sided rational polyhedron.

On any given compact manifold $M^{n+1}$ with boundary $\partial M$, it is proved that the moduli space $\mathcal{ℰ}$ of Einstein metrics on $M$, if non-empty, is a smooth, infinite dimensional Banach manifold, at least when ${\pi }_1\left(M,\partial M\right)=0$. Thus, the Einstein moduli space is unobstructed. The usual Dirichlet and Neumann boundary maps to data on $\partial M$ are smooth, but not Fredholm. Instead, one has natural mixed boundary-value problems which give Fredholm boundary maps.

These results also hold for manifolds with compact boundary which have a finite number of locally asymptotically flat ends, as well as for the Einstein equations coupled to many other fields.

We prove that every closed hyperbolic $3$–manifold has a family of (possibly infinite sheeted) coverings with the property that the Cheeger constants in the family tend to zero. This is used to show that, if in addition the fundamental group of the manifold is LERF, then it satisfies the Lubotzky–Sarnak conjecture.

We continue our study of the monoid of right-veering diffeomorphisms on a compact oriented surface with nonempty boundary, introduced in [Invent. Math. 169 (2007) 427–449]. We conduct a detailed study of the case when the surface is a punctured torus; in particular, we exhibit the difference between the monoid of right-veering diffeomorphisms and the monoid of products of positive Dehn twists, with the help of the Rademacher function. We then generalize to the braid group $B_n$ on $n$ strands by relating the signature and the Maslov index. Finally, we discuss the symplectic fillability in the pseudo-Anosov case by comparing with the work of Roberts [Proc. London Math. Soc. (3) 82/83 (2001) 747–768/443–471].

We show that every hyperbolic knot complement contains a closed quasi-Fuchsian surface.

We give a new geometric obstruction to the iterated Bing double of a knot being a slice link: for $n>1$ the $\left(n+1\right)$–st iterated Bing double of a knot is rationally slice if and only if the $n$–th iterated Bing double of the knot is rationally slice. The main technique of the proof is a covering link construction simplifying a given link. We prove certain similar geometric obstructions for $n\le 1$ as well. Our results are sharp enough to conclude, when combined with algebraic invariants, that if the $n$–th iterated Bing double of a knot is slice for some $n$, then the knot is algebraically slice. Also our geometric arguments applied to the smooth case show that the Ozsváth–Szabó and Manolescu–Owens invariants give obstructions to iterated Bing doubles being slice. These results generalize recent results of Harvey, Teichner, Cimasoni, Cha and Cha–Livingston–Ruberman. As another application, we give explicit examples of algebraically slice knots with nonslice iterated Bing doubles by considering von Neumann $\rho$–invariants and rational knot concordance. Refined versions of such examples are given, that take into account the Cochran–Orr–Teichner filtration.

In this paper we study the string topology (à la Chas–Sullivan) of an orbifold. We define the string homology ring product at the level of the free loop space of the classifying space of an orbifold. We study its properties and do some explicit calculations.

By studying the Heegaard Floer homology of the preimage of a knot $K\subset S^3$ inside its double branched cover, we develop simple obstructions to $K$ having finite order in the classical smooth concordance group. As an application, we prove that all $2$–bridge knots of crossing number at most $12$ for which the smooth concordance order was previously unknown have infinite smooth concordance order.

Given a closed monotone symplectic manifold $M$, we define certain characteristic cohomology classes of the free loop space $LHam\left(M,\omega \right)$ with values in $Q{H}_{\ast }\left(M\right)$, and their $S^1$ equivariant version. These classes generalize the Seidel representation and satisfy versions of the axioms for Chern classes. In particular there is a Whitney sum formula, which gives rise to a graded ring homomorphism from the ring $H_{\ast }\left(\Omega Ham\left(M,\omega \right),\mathbb{Q}\right)$, with its Pontryagin product to $Q{H}_{2n+\ast }\left(M\right)$ with its quantum product. As an application we prove an extension to higher dimensional geometry of the loop space $LHam\left(M,\omega \right)$ of a theorem of McDuff and Slimowitz on minimality in the Hofer metric of a semifree Hamiltonian circle action.

Let $M_1$ and $M_2$ be closed, orientable 3–manifolds. Let $H_i$ denote a Heegaard surface in $M_i$. We prove that if $H_1\#H_2$ comes from stabilizing a lower genus splitting of $M_1\#M_2$ then one of $H_1$ or $H_2$ comes from stabilizing a lower genus splitting. This answers a question of C Gordon (Problem 3.91 from Kirby’s problem list). We also show that every unstabilized Heegaard splitting has a unique expression as the connected sum of Heegaard splittings of prime 3–manifolds.

Throughout this paper we consider smooth maps of positive codimensions, having only stable singularities (see Arnold, Guseĭn-Zade and Varchenko [Monographs in Math. 83, Birkhauser, Boston (1988)]. We prove a conjecture, due to M Kazarian, connecting two classifying spaces in singularity theory for this type of singular maps. These spaces are: 1) Kazarian’s space (generalising Vassiliev’s algebraic complex and) showing which cohomology classes are represented by singularity strata. 2) The space $X_{\tau }$ giving homotopy representation of cobordisms of singular maps with a given list of allowed singularities as in work of Rimányi and the author [Topology 37 (1998) 1177–1191; Mat. Sb. (N.S.) 108 (150) (1979) 433–456, 478; Lecture Notes in Math. 788, Springer, Berlin (1980) 223–244].

We obtain that the ranks of cobordism groups of singular maps with a given list of allowed stable singularities, and also their $p$–torsion parts for big primes $p$ coincide with those of the homology groups of the corresponding Kazarian space. (A prime $p$ is “big” if it is greater than half of the dimension of the source manifold.) For all types of Morin maps (ie when the list of allowed singularities contains only corank $1$ maps) we compute these ranks explicitly.

We give a very transparent homotopical description of the classifying space $X_{\tau }$ as a fibration. Using this fibration we solve the problem of elimination of singularities by cobordisms. (This is a modification of a question posed by Arnold [Itogi Nauki i Tekniki, Moscow (1988) 5–257].)

We analyze the coarse geometry of the Weil–Petersson metric on Teichmuller space, focusing on applications to its synthetic geometry (in particular the behavior of geodesics). We settle the question of the strong relative hyperbolicity of the Weil–Petersson metric via consideration of its coarse quasi-isometric model, the pants graph. We show that in dimension 3 the pants graph is strongly relatively hyperbolic with respect to naturally defined product regions and show any quasi-flat lies a bounded distance from a single product. For all higher dimensions there is no nontrivial collection of subsets with respect to which it strongly relatively hyperbolic; this extends a theorem of Behrstock, Drutu and Mosher [submitted] in dimension 6 and higher into the intermediate range (it is hyperbolic if and only if the dimension is 1 or 2 by Brock and Farb [Amer. J. Math. 128 (2006) 1-22]). Stability and relative stability of quasi-geodesics in dimensions up through 3 provide for a strong understanding of the behavior of geodesics and a complete description of the CAT(0) boundary of the Weil–Petersson metric via curve-hierarchies and their associated boundary laminations.

Recently, Prasad and Yeung classified all possible fundamental groups of fake projective planes. According to their result, many fake projective planes admit a nontrivial group of automorphisms, and in that case it is isomorphic to $\mathbb{Z}∕3\mathbb{Z}$, $\mathbb{Z}∕7\mathbb{Z}$, 7:3 or ${\left(\mathbb{Z}∕3\mathbb{Z}\right)}^2$, where 7:3 is the unique nonabelian group of order 21.

Let $G$ be a group of automorphisms of a fake projective plane $X$. In this paper we classify all possible structures of the quotient surface $X∕G$ and its minimal resolution.