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We study diffeomorphisms of compact, oriented surfaces, developing methods of distinguishing those which have positive factorizations into Dehn twists from those which satisfy the weaker condition of being right-veering. We use these to construct open book decompositions of Stein-fillable –manifolds whose monodromies have no positive factorization.
A Morse –function is a generic smooth map from a smooth manifold to a surface. In the absence of definite folds (in which case we say that the Morse –function is indefinite), these are natural generalizations of broken (Lefschetz) fibrations. We prove existence and uniqueness results for indefinite Morse –functions mapping to arbitrary compact, oriented surfaces. “Uniqueness” means there is a set of moves which are sufficient to go between two homotopic indefinite Morse –functions while remaining indefinite throughout. We extend the existence and uniqueness results to indefinite, Morse –functions with connected fibers.
We compute the motivic Donaldson–Thomas invariants of the one-loop quiver, with an arbitrary potential. This is the first computation of motivic Donaldson–Thomas invariants to use in an essential way the full machinery of –equivariant motives, for which we prove a dimensional reduction result similar to that of Behrend, Bryan and Szendrői in their study of degree-zero motivic Donaldson–Thomas invariants. Our result differs from theirs in that it involves nontrivial monodromy.
We embed triangulated categories defined by quivers with potential arising from ideal triangulations of marked bordered surfaces into Fukaya categories of quasiprojective –folds associated to meromorphic quadratic differentials. Together with previous results, this yields nontrivial computations of spaces of stability conditions on Fukaya categories of symplectic six-manifolds.
In this paper we will promote the 3D index of an ideal triangulation of an oriented cusped –manifold (a collection of –series with integer coefficients, introduced by Dimofte, Gaiotto and Gukov) to a topological invariant of oriented cusped hyperbolic –manifolds. To achieve our goal we show that (a) admits an index structure if and only if is –efficient and (b) if is hyperbolic, it has a canonical set of –efficient ideal triangulations related by – and – moves which preserve the 3D index. We illustrate our results with several examples.
We show that the spectral sequence induced by the Betti realization of the slice tower for the motivic sphere spectrum agrees with the Adams–Novikov spectral sequence, after a suitable reindexing. The proof relies on a partial extension of Deligne’s décalage construction to the Tot–tower of a cosimplicial spectrum.
We show that cellular approximations of nilpotent Postnikov stages are always nilpotent Postnikov stages, in particular classifying spaces of nilpotent groups are turned into classifying spaces of nilpotent groups. We use a modified Bousfield–Kan homology completion tower whose terms we prove are all –cellular for any . As straightforward consequences, we show that if is –acyclic and nilpotent for a given homology theory , then so are all its Postnikov sections , and that any nilpotent space for which the space of pointed self-maps is “canonically” discrete must be aspherical.
In this paper, we study lower bounds on the –theory of the maximal –algebra of a discrete group based on the amount of torsion it contains. We call this the finite part of the operator –theory and give a lower bound that is valid for a large class of groups, called the finitely embeddable groups. The class of finitely embeddable groups includes all residually finite groups, amenable groups, Gromov’s monster groups, virtually torsion-free groups (eg ), and any group of analytic diffeomorphisms of an analytic connected manifold fixing a given point. We apply this result to measure the degree of nonrigidity for any compact oriented manifold with dimension . In this case, we derive a lower bound on the rank of the structure group , which is roughly defined to be the abelian group of all pairs , where is a compact manifold and is a homotopy equivalence. In many interesting cases, we obtain a lower bound on the reduced structure group , which measures the size of the collections of compact manifolds homotopic equivalent to but not homeomorphic to by any homeomorphism at all (not necessary homeomorphism in the homotopy equivalence class). For a compact Riemannian manifold with dimension greater than or equal to and positive scalar curvature metric, there is an abelian group that measures the size of the space of all positive scalar curvature metrics on . We obtain a lower bound on the rank of the abelian group when the compact smooth spin manifold has dimension and the fundamental group of is finitely embeddable.
Given a free-by-cyclic group determined by any outer automorphism which is represented by an expanding irreducible train-track map , we construct a –complex called the folded mappingtorus of , and equip it with a semiflow. We show that enjoys many similar properties to those proven by Thurston and Fried for the mapping torus of a pseudo-Anosov homeomorphism. In particular, we construct an open, convex cone containing the homomorphism having , a homology class , and a continuous, convex, homogeneous of degree function with the following properties. Given any primitive integral class there is a graph such that:
The inclusion is –injective and .
is a section of the semiflow and the first return map to is an expanding irreducible train track map representing such that .
The logarithm of the stretch factor of is precisely .
If was further assumed to be hyperbolic and fully irreducible then for every primitive integral the automorphism of is also hyperbolic and fully irreducible.
In this paper we give the first example of a surface bundle over a surface with at least three fiberings. In fact, for each we construct –manifolds admitting at least distinct fiberings as a surface bundle over a surface with base and fiber both closed surfaces of negative Euler characteristic. We give examples of surface bundles admitting multiple fiberings for which the monodromy representation has image in the Torelli group, showing the necessity of all of the assumptions made in the main theorem of a recent paper of ours. Our examples show that the number of surface bundle structures that can be realized on a –manifold with Euler characteristic grows exponentially with .
We study the long-time behavior of the Kähler–Ricci flow on compact Kähler manifolds. We give an almost complete classification of the singularity type of the flow at infinity, depending only on the underlying complex structure. If the manifold is of intermediate Kodaira dimension and has semiample canonical bundle, so it is fibered by Calabi–Yau varieties, we show that parabolic rescalings around any point on a smooth fiber converge smoothly to a unique limit, which is the product of a Ricci-flat metric on the fiber and a flat metric on Euclidean space. An analogous result holds for collapsing limits of Ricci-flat Kähler metrics.
We define a –valued homotopy invariant of a –structure on the tangent bundle of a closed –manifold in terms of the signature and Euler characteristic of a coboundary with a –structure. For manifolds of holonomy obtained by the twisted connected sum construction, the associated torsion-free –structure always has . Some holonomy examples constructed by Joyce by desingularising orbifolds have odd .
We define a further homotopy invariant such that if is –connected then the pair determines a –structure up to homotopy and diffeomorphism. The class of a –structure is determined by on its own when the greatest divisor of modulo torsion divides 224; this sufficient condition holds for many twisted connected sum –manifolds.
We also prove that the parametric –principle holds for coclosed –structures.
We continue our study of contact structures on manifolds of dimension at least five using surgery-theoretic methods. Particular applications include the existence of “maximal” almost contact manifolds with respect to the Stein cobordism relation as well as the existence of weakly fillable contact structures on the product . We also study the connection between Stein fillability and connected sums: we give examples of almost contact manifolds for which the connected sum is Stein fillable, while the components are not.
Concerning obstructions to Stein fillability, we show for all that there are almost contact structures on the –sphere which are not Stein fillable. This implies the same result for all highly connected –manifolds which admit almost contact structures. The proofs rely on a new number-theoretic result about Bernoulli numbers.