For each discriminant $D>1$, McMullen constructed the Prym–Teichmüller curves $W_{D}(4)$ and $W_{D}(6)$ in ${\mathcal{M}}_{3}$ and ${\mathcal{M}}_{4}$, which constitute one of the few known infinite families of geometrically primitive Teichmüller curves. In the present paper, we determine for each $D$ the number and type of orbifold points on $W_{D}(6)$. These results, together with a previous result of the two authors in the genus $3$ case and with results of Lanneau–Nguyen and Möller, complete the topological characterisation of all Prym–Teichmüller curves and determine their genus. The study of orbifold points relies on the analysis of intersections of $W_{D}(6)$ with certain families of genus $4$ curves with extra automorphisms. As a side product of this study, we give an explicit construction of such families and describe their Prym–Torelli images, which turn out to be isomorphic to certain products of elliptic curves. We also give a geometric description of the flat surfaces associated to these families and describe the asymptotics of the genus of $W_{D}(6)$ for large $D$.

A foliation $(M,{\mathcal{F}})$ is said to be $2$-calibrated if it admits a closed 2-form $\unicode[STIX]{x1D714}$ making each leaf symplectic. By using approximately holomorphic techniques, a sequence $W_{k}$ of $2$-calibrated submanifolds of codimension-$2$ can be found for $(M,{\mathcal{F}},\unicode[STIX]{x1D714})$. Our main result says that the Lefschetz hyperplane theorem holds for the pairs $(F,F\cap W_{k})$, with $F$ any leaf of ${\mathcal{F}}$. This is applied to draw important consequences on the transverse geometry of such foliations.

We study tautological rings for high-dimensional manifolds, that is, for each smooth manifold $M$ the ring $R^{\ast }(M)$ of those characteristic classes of smooth fibre bundles with fibre $M$ which is generated by generalised Miller–Morita–Mumford classes. We completely describe these rings modulo nilpotent elements, when $M$ is a connected sum of copies of $S^{n}\times S^{n}$ for $n$ odd.

We introduce a new affinely invariant structure on smooth surfaces in ℝ3 by defining a family of reflections in all points of the surface. We show that the bifurcation set of this family has a special structure at ‘ points’, which are not detected by the flat geometry of the surface. These points (without an associated structure on the surface) have also arisen in the study of the centre symmetry set; using our technique we are able to explain how the points are created and annihilated in a generic family of surfaces. We also present the bifurcation set in a global setting.

In this paper, we determine the group of contact transformations modulo contact isotopies for Legendrian circle bundles over closed surfaces of non-positive Euler characteristic. These results extend and correct those presented by the first author in a former work. The main ingredient we use is connectedness of certain spaces of embeddings of surfaces into contact 3-manifolds. This connectedness question is also studied for itself with a number of (hopefully instructive) examples.

We consider a developable surface normal to a surface along a curve on the surface. We call it a normal developable surface along the curve on the surface. We investigate the uniqueness and the singularities of such developable surfaces. We discover two new invariants of curves on a surface that characterize these singularities.

We exhibit a knot $P$ in the solid torus, representing a generator of first homology, such that for any knot $K$ in the 3-sphere, the satellite knot with pattern $P$ and companion $K$ is not smoothly slice in any homology 4-ball. As a consequence, we obtain a knot in a homology 3-sphere that does not bound a piecewise-linear disk in any homology 4-ball.

This paper investigates the space of codimension zero embeddings of a Poincaré duality space in a disk. One of our main results exhibits a tower that interpolates from the space of Poincaré immersions to a certain space of “unlinked” Poincaré embeddings. The layers of this tower are described in terms of the coefficient spectra of the identity appearing in Goodwillie’s homotopy functor calculus. We also answer a question posed to us by Sylvain Cappell. The appendix proposes a conjectural relationship between our tower and the manifold calculus tower for the smooth embedding space.

We give a method for constructing a Legendrian representative of a knot in $S^{3}$ which realizes its maximal Thurston–Bennequin number under a certain condition. The method utilizes Stein handle decompositions of $D^{4}$ , and the resulting Legendrian representative is often very complicated (relative to the complexity of the topological knot type). As an application, we construct infinitely many knots in $S^{3}$ each of which yields a reducible 3-manifold by a Legendrian surgery in the standard tight contact structure. This disproves a conjecture of Lidman and Sivek.

We define an invariant of contact 3-manifolds with convex boundary using Kronheimer and Mrowka’s sutured monopole Floer homology theory ( $SHM$ ). Our invariant can be viewed as a generalization of Kronheimer and Mrowka’s contact invariant for closed contact 3-manifolds and as the monopole Floer analogue of Honda, Kazez, and Matić’s contact invariant in sutured Heegaard Floer homology ( $SFH$ ). In the process of defining our invariant, we construct maps on $SHM$ associated to contact handle attachments, analogous to those defined by Honda, Kazez, and Matić in $SFH$ . We use these maps to establish a bypass exact triangle in $SHM$ analogous to Honda’s in $SFH$ . This paper also provides the topological basis for the construction of similar gluing maps in sutured instanton Floer homology, which are used in Baldwin and Sivek [Selecta Math. (N.S.), 22(2) (2016), 939–978] to define a contact invariant in the instanton Floer setting.

An invertible polynomial in n variables is a quasi-homogeneous polynomial consisting of n monomials so that the weights of the variables and the quasi-degree are well defined. In the framework of the construction of mirror symmetric orbifold Landau–Ginzburg models, Berglund, Hübsch and Henningson considered a pair (f, G) consisting of an invertible polynomial f and an abelian group G of its symmetries together with a dual pair . Here we study the reduced orbifold zeta functions of dual pairs (f, G) and and show that they either coincide or are inverse to each other depending on the number n of variables.

We study the Minkowski symmetry set of a closed smooth curve γ in the Minkowski plane. We answer the following question, which is analogous to one concerning curves in the Euclidean plane that was treated by Giblin and O’Shea (1990): given a point p on γ, does there exist a bi-tangent pseudo-circle that is tangent to γ both at p and at some other point q on γ? The answer is yes, but as pseudo-circles with non-zero radii have two branches (connected components) it is possible to refine the above question to the following one: given a point p on γ, does there exist a branch of a pseudo-circle that is tangent to γ both at p and at some other point q on γ? This question is motivated by the earlier quest of Reeve and Tari (2014) to define the Minkowski Blum medial axis, a counterpart of the Blum medial axis of curves in the Euclidean plane.

We rectify two omissions in the list of generators and include a brief discussion of the localization theorem of Kosniowski and Stong.

In this article we construct symmetric operations for all primes (previously known only for $p=2$). These unstable operations are more subtle than the Landweber–Novikov operations, and encode all $p$-primary divisibilities of characteristic numbers. Thus, taken together (for all primes) they plug the gap left by the Hurewitz map $\mathbb{L}{\hookrightarrow}\mathbb{Z}[b_{1},b_{2},\ldots ]$, providing an important structure on algebraic cobordism. Applications include questions of rationality of Chow group elements, and the structure of the algebraic cobordism. We also construct Steenrod operations of tom Dieck style in algebraic cobordism. These unstable multiplicative operations are more canonical and subtle than Quillen-style operations, and complement the latter.

We apply results from both contact topology and exceptional surgery theory to study when Legendrian surgery on a knot yields a reducible manifold. As an application, we show that a reducible surgery on a non-cabled positive knot of genus $g$ must have slope $2g-1$, leading to a proof of the cabling conjecture for positive knots of genus 2. Our techniques also produce bounds on the maximum Thurston–Bennequin numbers of cables.

We show that the Gromov-Lawson-Rosenberg conjecture for the Semi-Dihedral group of order 16 is true.

We apply the methods of Heegaard Floer homology to identify topological properties of complex curves in $\mathbb{C}P^{2}$. As one application, we resolve an open conjecture that constrains the Alexander polynomial of the link of the singular point of the curve in the case that there is exactly one singular point, having connected link, and the curve is of genus zero. Generalizations apply in the case of multiple singular points.

In this paper we give explicit formulas for differential characteristic classes of principal $G$-bundles with connections and prove their expected properties. In particular, we obtain explicit formulas for differential Chern classes, differential Pontryagin classes and the differential Euler class. Furthermore, we show that the differential Chern class is the unique natural transformation from (Simons–Sullivan) differential $K$-theory to (Cheeger–Simons) differential characters that is compatible with curvature and characteristic class. We also give the explicit formula for the differential Chern class on Freed–Lott differential $K$-theory. Finally, we discuss the odd differential Chern classes.

Groebner bases for the ideals determining mod $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}2$ cohomology of the real flag manifolds $F(1,1,n)$ and $F(1,2,n)$ are obtained. These are used to compute appropriate Stiefel–Whitney classes in order to establish some new nonembedding and nonimmersion results for the manifolds $F(1,2,n)$.