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We introduce the notion of an operator category and two different models for homotopy theory of –operads over an operator category — one of which extends Lurie’s theory of –operads, the other of which is completely new, even in the commutative setting. We define perfect operator categories, and we describe a category attached to a perfect operator category that provides Segal maps. We define a wreath product of operator categories and a form of the Boardman–Vogt tensor product that lies over it. We then give examples of operator categories that provide universal properties for the operads and () and also a collection of new examples.
We construct bi-Lipschitz embeddings into Euclidean space for bounded-diameter subsets of manifolds and orbifolds of bounded curvature. The distortion and dimension of such embeddings is bounded by diameter, curvature and dimension alone. We also construct global bi-Lipschitz embeddings for spaces of the form , where is a discrete group acting properly discontinuously and by isometries on . This generalizes results of Naor and Khot. Our approach is based on analyzing the structure of a bounded-curvature manifold at various scales by specializing methods from collapsing theory to a certain class of model spaces. In the process, we develop tools to prove collapsing theory results using algebraic techniques.
Let be a closed and oriented –manifold. We define different versions of unfolded Seiberg–Witten Floer spectra for . These invariants generalize Manolescu’s Seiberg–Witten Floer spectrum for rational homology –spheres. We also compute some examples when is a Seifert space.
We construct a family of –dimensional compact manifolds which are simultaneously diffeomorphic to complex Calabi–Yau manifolds and symplectic Calabi–Yau manifolds. They have fundamental groups , their odd-degree Betti numbers are even, they satisfy the hard Lefschetz property, and their real homotopy types are formal. However, is never homotopy equivalent to a compact Kähler manifold for any topological space . The main ingredient to show the non-Kählerness is a structure theorem of cohomology jump loci due to the second author.
We show that if is an annular homeomorphism admitting an attractor which is an irreducible annular continua with two different rotation numbers, then the entropy of is positive. Further, the entropy is shown to be associated to a –robust rotational horseshoe. On the other hand, we construct examples of annular homeomorphisms with such attractors for which the rotation interval is uniformly large but the entropy approaches zero as much as desired.
The developed techniques allow us to obtain similar results in the context of Birkhoff attractors.
Let be a field of characteristic different from . We establish surjectivity of Balmer’s comparison map
from the tensor triangular spectrum of the homotopy category of compact motivic spectra to the homogeneous Zariski spectrum of Milnor–Witt –theory. We also comment on the tensor triangular geometry of compact cellular motivic spectra, producing in particular novel field spectra in this category. We conclude with a list of questions about the structure of the tensor triangular spectrum of the stable motivic homotopy category.
We suggest a method to construct new examples of partially hyperbolic diffeomorphisms. We begin with a partially hyperbolic diffeomorphism which leaves invariant a submanifold . We assume that is an Anosov submanifold for , that is, the restriction is an Anosov diffeomorphism and the center distribution is transverse to . By replacing each point in with the projective space (real or complex) of lines normal to , we obtain the blow-up . Replacing with amounts to a surgery on the neighborhood of which alters the topology of the manifold. The diffeomorphism induces a canonical diffeomorphism . We prove that under certain assumptions on the local dynamics of at the diffeomorphism is also partially hyperbolic. We also present some modifications, such as the connected sum construction, which allows to “paste together” two partially hyperbolic diffeomorphisms to obtain a new one. Finally, we present several examples to which our results apply.
Inspired by the Katz–Mazur theorem on crystalline cohomology and by the numerical experiments of Eskin, Kontsevich and Zorich, we conjecture that the polygon of the Lyapunov spectrum lies above (or on) the Harder–Narasimhan polygon of the Hodge bundle over any Teichmüller curve. We also discuss the connections between the two polygons and the integral of eigenvalues of the curvature of the Hodge bundle by using the works of Atiyah and Bott, Forni, and Möller. We obtain several applications to Teichmüller dynamics conditional on the conjecture.
Consider a flat bundle over a complex curve. We prove a conjecture of Fei Yu that the sum of the top Lyapunov exponents of the flat bundle is always greater than or equal to the degree of any rank- holomorphic subbundle. We generalize the original context from Teichmüller curves to any local system over a curve with nonexpanding cusp monodromies. As an application we obtain the large-genus limits of individual Lyapunov exponents in hyperelliptic strata of abelian differentials, which Fei Yu proved conditionally on his conjecture.
Understanding the case of equality with the degrees of subbundle coming from the Hodge filtration seems challenging, eg for Calabi–Yau-type families. We conjecture that equality of the sum of Lyapunov exponents and the degree is related to the monodromy group being a thin subgroup of its Zariski closure.
We show that there are infinitely many commensurability classes of pseudomodular groups, thus answering a question raised by Long and Reid. These are Fuchsian groups whose cusp set is all of the rationals but which are not commensurable to the modular group. We do this by introducing a general construction for the fundamental domains of Fuchsian groups obtained by gluing together marked ideal triangular tiles, which we call hyperbolic jigsaw groups.
We introduce a class of Weinstein domains which are sublevel sets of flexible Weinstein manifolds but are not themselves flexible. These manifolds exhibit rather subtle behavior with respect to both holomorphic curve invariants and symplectic flexibility. We construct a large class of examples and prove that every flexible Weinstein manifold can be Weinstein homotoped to have a nonflexible sublevel set.
Thanks to a recent result by Jean-Marc Schlenker, we establish an explicit linear inequality between the normalized entropies of pseudo-Anosov automorphisms and the hyperbolic volumes of their mapping tori. As corollaries, we give an improved lower bound for values of entropies of pseudo-Anosovs on a surface with fixed topology, and a proof of a slightly weaker version of the result by Farb, Leininger and Margalit first, and by Agol later, on finiteness of cusped manifolds generating surface automorphisms with small normalized entropies. Also, we present an analogous linear inequality between the Weil–Petersson translation distance of a pseudo-Anosov map (normalized by multiplying by the square root of the area of a surface) and the volume of its mapping torus, which leads to a better bound.
Let be a closed manifold that admits a self-cover of degree . We say is strongly regular if all iterates are regular covers. In this case, we establish an algebraic structure theorem for the fundamental group of : We prove that surjects onto a nontrivial free abelian group , and the self-cover is induced by a linear endomorphism of . Under further hypotheses we show that a finite cover of admits the structure of a principal torus bundle. We show that this applies when is Kähler and is a strongly regular, holomorphic self-cover, and prove that a finite cover of splits as a product with a torus factor.
We classify the minimum-volume smooth complex hyperbolic surfaces that admit smooth toroidal compactifications, and we explicitly construct their compactifications. There are five such surfaces, and they are all arithmetic; ie they are associated with quotients of the ball by an arithmetic lattice. Moreover, the associated lattices are all commensurable. The first compactification, originally discovered by Hirzebruch, is the blowup of an abelian surface at one point. The others are bielliptic surfaces blown up at one point. The bielliptic examples are new and are the first known examples of smooth toroidal compactifications birational to bielliptic surfaces.
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