Morita [Osaka J. Math. 21 (1984), 545–563] showed that for each integer k \geq 1 , there are examples of flat \mathbb{S}^{1} -bundles for which the k -th power of the Euler class does not vanish. Haefliger [Enseign. Math. (2) 24 (1978), 154] asked if the same holds for flat odd-dimensional sphere bundles. In this paper, for a manifold M with a free torus action, we prove that certain M -bundles are cobordant to a flat M -bundle and as a consequence, we answer Haefliger’s question. We show that all monomials in the Euler class and Pontryagin classes p_{i} for i\leq n-1 are non-trivial in H^{*}(\operatorname{BDiff}^{\delta}_{+}(\mathbb{S}^{2n-1});\mathbb{Q}) .

We consider an area-minimizing integral current T of codimension higher than 1 in a smooth Riemannian manifold \Sigma . In a previous paper we have subdivided the set of interior singular points with at least one flat tangent cone according to a real parameter, which we refer to as the ‘singularity degree’. In this paper, we show that the set of points for which the singularity degree is strictly larger than 1 is (m-2) -rectifiable. In a subsequent work, we prove that the remaining flat singular points form a \mathcal{H}^{m-2} -null set, thus concluding that the singular set of T is (m-2) -rectifiable.

Wandering Fatou components were recently constructed by Astorg et al. (2016) for higher-dimensional holomorphic maps on projective spaces. Their examples are polynomial skew products with a parabolic invariant line. In this paper we study this wandering domain problem for polynomial skew product f with an attracting invariant line L (which is the more common case). We show that if f is unicritical (in the sense that the critical curve has a unique transversal intersection with L ), then every Fatou component of f in the basin of L is an extension of a one-dimensional Fatou component of f|_{L} . As a corollary there is no wandering Fatou component. We will also discuss the multicritical case under additional assumptions.

In a setting of a complex manifold with a positive line bundle and a submanifold, we consider the optimal Ohsawa–Takegoshi extension operator, sending a holomorphic section of the line bundle on the submanifold to the holomorphic extension of it on the ambient manifold with the minimal L^{2} -norm. We show that for a tower of submanifolds and large tensor powers of the line bundle, the extension operators act transitively modulo some small defect, which is a Toeplitz type operator. We calculate the first significant term in the asymptotic expansion of this “transitivity defect”. As a byproduct, we deduce composition rules for Toeplitz type operators, the extension and restriction operators and calculate the second term in the asymptotic expansion of the optimal constant in the semi-classical version of the extension theorem.

Let H_{D}(T) denote the Hilbert class polynomial of the imaginary quadratic order of discriminant D . We study the rate of growth of the greatest common divisor of H_{D}(a) and H_{D}(b) as |D| \to \infty for a and b belonging to various Dedekind domains. We also study the modular support problem: if for all but finitely many D every prime ideal dividing H_{D}(a) also divides H_{D}(b) , what can we say about a and b ? If we replace H_{D}(T) by T^{n}-1 and the Dedekind domain is a ring of S -integers in some number field, then these are classical questions that have been investigated by Bugeaud–Corvaja–Zannier, Corvaja–Zannier, and Corrales-Rodrigáñez–Schoof.

For a closed, orientable, irreducible 3 -manifold M that admits a co-orientable taut foliation with one-sided branching, we show that \pi_{1}(M) is left orderable.

Let M be a closed hyperbolic manifold containing a totally geodesic hypersurface S , and let N be a closed Riemannian manifold homotopy equivalent to M with sectional curvature bounded above by -1 . We study the following question: if \pi_{1}(S) can be represented by a totally geodesic hyperbolic hypersurface in N , then must N be isometric to M ? We show that many such S are rigid in the sense that the answer to this question is positive. On the other hand, we construct examples of S for which the answer is negative.

The goal of this paper is to establish a monotonicity formula for perimeter minimizing sets in \operatorname{RCD}(0,N) metric measure cones, together with the associated rigidity statement. The applications include sharp Hausdorff dimension estimates for the singular strata of perimeter minimizing sets in non-collapsed \operatorname{RCD} spaces and the existence of blow-down cones for global perimeter minimizers in Riemannian manifolds with non-negative Ricci curvature and Euclidean volume growth.

In this paper, we describe how to explicitly construct infinitely many finite simple groups as characteristic quotients of the rank 2 free group F_{2} . This shows that a “baby” version of the Wiegold conjecture [in: Geometry, Rigidity, and Group Actions (2011), 609–643] fails for F_{2} and provides counterexamples to two conjectures in the theory of noncongruence subgroups of \mathrm{SL}_{2}(\mathbb{Z}) by Chen [Math. Ann. 371 (2018), 41–126]. Our main result explicitly produces, for every prime power q\ge 7 , the groups \mathrm{SL}_{3}(\mathbb{F}_{q}) and \mathrm{SU}_{3}(\mathbb{F}_{q}) as characteristic quotients of F_{2} . Our strategy is to study specializations of the Burau representation for the braid group B_{4} , exploiting an exceptional relationship between F_{2} and B_{4} first observed by Dyer, Formanek, and Grossman [Arch. Math. (Basel) 38 (1982), 404–409]. Weisfeiler’s strong approximation theorem guarantees that our specializations are surjective for infinitely many primes, but they are not effective. To make our result effective, we give another proof of surjectivity via a careful analysis of the maximal subgroup structures of \mathrm{SL}_{3}(\mathbb{F}_{q}) and \mathrm{SU}_{3}(\mathbb{F}_{q}) . These examples are minimal in the sense that no finite simple group of the form \mathrm{PSL}_{2}(\mathbb{F}_{q}) appears as a characteristic quotient of F_{2} .

In their 2021 and 2022 papers, Cristofaro-Gardiner, Humilière, Mak, Seyfaddini, and Smith defined links spectral invariants on connected compact surfaces and used them to show various results on the algebraic structure of the group of area-preserving homeomorphisms of surfaces, particularly in cases where the surfaces have genus zero. We show that on surfaces with higher genus, for a certain class of links, the invariants will satisfy a local quasimorphism property. Subsequently, we generalize their results to surfaces of any genus. This extension includes the non-simplicity of (i) the group of hameomorphisms of a closed surface, and (ii) the kernel of the Calabi homomorphism inside the group of hameomorphisms of a surface with non-empty boundary. Moreover, we prove that the Calabi homomorphism extends (non-canonically) to the C^{0} -closure of the set of Hamiltonian diffeomorphisms of any surface. The local quasimorphism property is a consequence of a quantitative Künneth formula for a connected sum in Heegaard–Floer homology, inspired by the results of Ozsváth and Szabó.