An area-preserving homeomorphism isotopic to the identity is said to have rational rotation direction if its rotation vector is a real multiple of a rational class. We give a short proof that any area-preserving homeomorphism of a compact surface of genus at least two, which is isotopic to the identity and has rational rotation direction, is either the identity or has periodic points of unbounded minimal period. This answers a question of Ginzburg and Seyfaddini and can be regarded as a Conley conjecture-type result for symplectic homeomorphisms of surfaces beyond the Hamiltonian case. We also discuss several variations, such as maps preserving arbitrary Borel probability measures with full support, maps that are not isotopic to the identity and maps on lower genus surfaces. The proofs of the main results combine topological arguments with periodic Floer homology.

Anosov representations of hyperbolic groups form a rich class of representations that are closely related to geometric structures on closed manifolds. Any Anosov representation $\rho :\Gamma \to G$ admits cocompact domains of discontinuity in flag varieties $G/Q$ [GW12, KLP18] endowing the compact quotient manifolds $M_\rho $ with a $(G,G/Q)$–structure. In general, the topology of $M_\rho $ can be quite complicated.

In this article, we will focus on the special case when $\Gamma $ is a the fundamental group of a closed (real or complex) hyperbolic manifold N and $\rho $ is a deformation of a (twisted) lattice embedding $\Gamma \to \mathrm {Isom}^\circ (\mathbb {H}_{\mathbb {K}}) \to G$ through Anosov representations. In this case, we prove that $M_\rho $ is a smooth fiber bundle over N, and we describe the structure group of this bundle and compute its invariants. This theorem applies in particular to most representations in higher rank Teichmüller spaces, as well as convex divisible representations, AdS-quasi-Fuchsian representations and $\mathbb {H}_{p,q}$–convex cocompact representations.

Even when $M_\rho \to N$ is a fiber bundle, it is often very difficult to determine the fiber. In the second part of the paper, we focus on the special case when N is a surface, $\rho $ a quasi-Hitchin representation into $\mathrm {Sp}(4,{\mathbb C})$, and $M_\rho $ carries a $(\mathrm {Sp}(4,{{\mathbb C}}),\mathrm {Lag}({{\mathbb C}}^4))$–structure. We show that in this case the fiber is homeomorphic to ${{\mathbb C}}\mathbb {P}^2 \# \overline {{{\mathbb C}}\mathbb {P}}^2$.

This fiber bundle $M_\rho \to N$ is of particular interest in the context of possible generalizations of Bers’ double uniformization theorem in the context of higher rank Teichmüller spaces, since for Hitchin-representations it contains two copies of the locally symmetric space associated to $\rho (\Gamma )$. Our result uses the classification of smooth $4$–manifolds, the study of the $\mathrm {SL}(2, {{\mathbb C}})$–orbits of $\mathrm {Lag}({{\mathbb C}}^4)$ and the identification of $\mathrm {Lag}({{\mathbb C}}^4)$ with the space of (unlabelled) regular ideal hyperbolic tetrahedra and their degenerations.

In this paper, we analyse the possible homotopy types of the total space of a principal SU(2)-bundle over a 3-connected 8-dimensional Poincaré duality complex. Along the way, we also classify the 3-connected 11-dimensional complexes E formed from a wedge of S4’s and S7’s by attaching a 11-cell.

We study homotopy groups of spaces of long links in Euclidean space of codimension at least three. With multiple components, they admit split injections from homotopy groups of spheres. We show that, up to knotting, these account for all the homotopy groups in a range which depends on the dimensions of the source manifolds and target manifold and which roughly generalizes the triple-point-free range for isotopy classes. Just beyond this range, joining components sends both a parametrized long Borromean rings class and a Hopf fibration to a generator of the first nontrivial homotopy group of the space of long knots. For spaces of equidimensional long links of most source dimensions, we describe generators for the homotopy group in this degree in terms of these Borromean rings and homotopy groups of spheres. A key ingredient in most of our results is a graphing map which increases source and target dimensions by one.

This paper considers the large N limit of Wilson loops for the two-dimensional Euclidean Yang–Mills measure on all orientable compact surfaces of genus larger or equal to $1$, with a structure group given by a classical compact matrix Lie group. Our main theorem shows the convergence of all Wilson loops in probability, given that it holds true on a restricted class of loops, obtained as a modification of geodesic paths. Combined with the result of [20], a corollary is the convergence of all Wilson loops on the torus. Unlike the sphere case, we show that the limiting object is remarkably expressed thanks to the master field on the plane defined in [3, 39], and we conjecture that this phenomenon is also valid for all surfaces of higher genus. We prove that this conjecture holds true whenever it does for the restricted class of loops of the main theorem. Our result on the torus justifies the introduction of an interpolation between free and classical convolution of probability measures, defined with the free unitary Brownian motion but differing from t-freeness of [5] that was defined in terms of the liberation process of Voiculescu [67]. In contrast to [20], our main tool is a fine use of Makeenko–Migdal equations, proving uniqueness of their solution under suitable assumptions, and generalising the arguments of [21, 33].

We express the total space of a principal circle bundle over a connected sum of two manifolds in terms of the total spaces of circle bundles over each summand, provided certain conditions hold. We then apply this result to provide sufficient conditions for the existence of free circle and torus actions on connected sums of products of spheres and obtain a topological classification of closed, simply connected manifolds with a free cohomogeneity-four torus action. As a corollary, we obtain infinitely many manifolds with Riemannian metrics of positive Ricci curvature and isometric torus actions.

We prove for the first time that knot Floer homology and Khovanov homology can detect non-fibered knots and that HOMFLY homology detects infinitely many knots; these theories were previously known to detect a mere six knots, all fibered. These results rely on our main technical theorem, which gives a complete classification of genus-1 knots in the 3-sphere whose knot Floer homology in the top Alexander grading is 2-dimensional. We discuss applications of this classification to problems in Dehn surgery which are carried out in two sequels. These include a proof that $0$-surgery characterizes infinitely many knots, generalizing results of Gabai from his 1987 resolution of the Property R Conjecture.

Quasigeodesic behavior of flow lines is a very useful property in the study of Anosov flows. Not every Anosov flow in dimension three is quasigeodesic. In fact, until recently, up to orbit equivalence, the only previously known examples of quasigeodesic Anosov flows were suspension flows. In a recent article, the second author proved that an Anosov flow on a hyperbolic 3-manifold is quasigeodesic if and only if it is non-$\mathbb {R}$-covered, and this result completes the classification of quasigeodesic Anosov flows on hyperbolic 3-manifolds. In this article, we prove that a new class of examples of Anosov flows are quasigeodesic. These are the first examples of quasigeodesic Anosov flows on 3-manifolds that are neither Seifert, nor solvable, nor hyperbolic. In general, it is very hard to show that a given flow is quasigeodesic and, in this article, we provide a new method to prove that an Anosov flow is quasigeodesic.

For a simply-connected closed manifold X of $\dim X \neq 4$, the mapping class group $\pi _0(\mathrm {Diff}(X))$ is known to be finitely generated. We prove that analogous finite generation fails in dimension 4. Namely, we show that there exist simply-connected closed smooth 4-manifolds whose mapping class groups are not finitely generated. More generally, for each $k>0$, we prove that there are simply-connected closed smooth 4-manifolds X for which $H_k(B\mathrm {Diff}(X);\mathbb {Z})$ are not finitely generated. The infinitely generated subgroup of $H_k(B\mathrm {Diff}(X);\mathbb {Z})$ which we detect are topologically trivial, and unstable under the connected sum of $S^2 \times S^2$. These results are proven by constructing and computing an infinite family of characteristic classes using Seiberg–Witten theory.

We introduce the notion of the equivariant covering type of a space X on which a finite group G acts and study its properties. The equivariant covering type measures the size of G-equivariant good covers of X and is thus an extension of the covering type of a space, introduced by Karoubi and Weibel. We show that the equivariant covering type is a G-homotopy invariant and describe its relation with other G-invariants, like the equivariant LS-category, G-genus, and the multiplicative structures of equivariant cohomology theories. We also compute the G-covering type of regular G-graphs, give estimates for orientation-preserving actions on surfaces and for the projectivizations of complex representations of G and cohomology spheres. As an application, we derive estimates of sizes of minimal G-triangulations for various G-spaces.

We study the topological structure of the space $\mathcal{X}$ of isomorphism classes of metric measure spaces equipped with the box or concentration topologies. We consider the scale-change action of the multiplicative group ${\mathbb{R}}_+$ of positive real numbers on $\mathcal{X}$, which has a one-point metric measure space, say $*$, as only one fixed-point. We prove that the ${\mathbb{R}}_+$-action on $\mathcal{X}_* := \mathcal{X} \setminus \{*\}$ admits the structure of non-trivial and locally trivial principal ${\mathbb{R}}_+$-bundle over the quotient space. Our bundle ${\mathbb{R}}_+ \to \mathcal{X}_* \to \mathcal{X}_*/{\mathbb{R}}_+$ is a curious example of a non-trivial principal fibre bundle with contractible fibre. A similar statement is obtained for the pyramidal compactification of $\mathcal{X}$, where we completely determine the structure of the fixed-point set of the ${\mathbb{R}}_+$-action on the compactification.

The Haefliger–Thurston conjecture predicts that Haefliger's classifying space for $C^r$-foliations of codimension $n$ whose normal bundles are trivial is $2n$-connected. In this paper, we confirm this conjecture for piecewise linear (PL) foliations of codimension $2$. Using this, we use a version of the Mather–Thurston theorem for PL homeomorphisms due to the author to derive new homological properties for PL surface homeomorphisms. In particular, we answer the question of Epstein in dimension $2$ and prove the simplicity of the identity component of PL surface homeomorphisms.

We recover Reidemeister’s theorem using $\mathcal {C}^{\infty }$ functions and transversality.

Let $\eta $ be [-11pc] [-7pc]a closed real 1-form on a closed Riemannian n-manifold $(M,g)$. Let $d_z$, $\delta _z$ and $\Delta _z$ be the induced Witten’s type perturbations of the de Rham derivative and coderivative and the Laplacian, parametrized by $z=\mu +i\nu \in \mathbb C$ ($\mu ,\nu \in \mathbb {R}$, $i=\sqrt {-1}$). Let $\zeta (s,z)$ be the zeta function of $s\in \mathbb {C}$, defined as the meromorphic extension of the function $\zeta (s,z)=\operatorname {Str}({\eta \wedge }\,\delta _z\Delta _z^{-s})$ for $\Re s\gg 0$. We prove that $\zeta (s,z)$ is smooth at $s=1$ and establish a formula for $\zeta (1,z)$ in terms of the associated heat semigroup. For a class of Morse forms, $\zeta (1,z)$ converges to some $\mathbf {z}\in \mathbb {R}$ as $\mu \to +\infty $, uniformly on $\nu $. We describe $\mathbf {z}$ in terms of the instantons of an auxiliary Smale gradient-like vector field X and the Mathai–Quillen current on $TM$ defined by g. Any real 1-cohomology class has a representative $\eta $ satisfying the hypothesis. If n is even, we can prescribe any real value for $\mathbf {z}$ by perturbing g, $\eta $ and X and achieve the same limit as $\mu \to -\infty $. This is used to define and describe certain tempered distributions induced by g and $\eta $. These distributions appear in another publication as contributions from the preserved leaves in a trace formula for simple foliated flows, giving a solution to a problem stated by C. Deninger.

When restricted to alternating links, both Heegaard Floer and Khovanov homology concentrate along a single diagonal $\delta$-grading. This leads to the broader class of thin links that one would like to characterize without reference to the invariant in question. We provide a relative version of thinness for tangles and use this to characterize thinness via tangle decompositions along Conway spheres. These results bear a strong resemblance to the L-space gluing theorem for three-manifolds with torus boundary. Our results are based on certain immersed curve invariants for Conway tangles, namely the Heegaard Floer invariant $\operatorname {HFT}$ and the Khovanov invariant $\widetilde {\operatorname {Kh}}$ that were developed by the authors in previous works.

In this paper we determine the homotopy types of the reduced suspension space of certain connected orientable closed smooth $five$-manifolds. As applications, we compute the reduced $K$-groups of $M$ and show that the suspension map between the third cohomotopy set $\pi ^3(M)$ and the fourth cohomotopy set $\pi ^4(\Sigma M)$ is a bijection.

We develop two methods for expressing the global index of the gradient of a 2 variable polynomial function $f$: in terms of the atypical fibres of $f$, and in terms of the clusters of Milnor arcs at infinity. These allow us to derive upper bounds for the global index, in particular refining the one that was found by Durfee in terms of the degree of $f$.

We show that if an open set in $\mathbb{R}^d$ can be fibered by unit n-spheres, then $d \geq 2n+1$, and if $d = 2n+1$, then the spheres must be pairwise linked, and $n \in \left\{0, 1, 3, 7 \right\}$. For these values of n, we construct unit n-sphere fibrations in $\mathbb{R}^{2n+1}$.

In this note, we give an alternate proof of the Farrell–Jones isomorphism conjecture for the affine Artin groups of type $\widetilde B_n$.

We introduce Chern classes in $U(m)$-equivariant homotopical bordism that refine the Conner–Floyd–Chern classes in the $\mathbf {MU}$-cohomology of $B U(m)$. For products of unitary groups, our Chern classes form regular sequences that generate the augmentation ideal of the equivariant bordism rings. Consequently, the Greenlees–May local homology spectral sequence collapses for products of unitary groups. We use the Chern classes to reprove the $\mathbf {MU}$-completion theorem of Greenlees–May and La Vecchia.