Characteristic cohomotopy classes for families of 4-manifolds

定号4-流形的同义反复类

Tautological classes of definite 4-manifolds

本文证明了紧致光滑单连通定号4-流形的同义反复类对角化定理，利用Seiberg-Witten理论的族版本方法，完全确定了$\mathbb{CP}^2$等流形的同义反复环并推导出普遍成立的线性关系。

Characteristic classes for families of bundles

The generalized Miller-Morita-Mumford classes of a manifold bundle with fiber $M$ depend only on the underlying $τ_M$-fibration, meaning the family of vector bundles formed by the tangent bundles of the fibers. This motivates a closer study of the classifying space for $τ_M$-fibrations, $Baut(τ_M)$, and its cohomology ring, i.e., the ring of characteristic classes of $τ_M$-fibrations. For a bundle $ξ$ over a simply connected Poincaré duality space, we construct a relative Sullivan model for the universal orientable $ξ$-fibration together with explicit cocycle representatives for the characteri…

由抓取器产生的纽结族

Knotted families from graspers

本文在任意维数$d\geq4$的光滑流形中，构造了嵌入空间同伦群中特定度数的类，并证明其在Taylor塔中被检测到，方法基于$d$维球中的弦链族。

Quotient families of mapping classes

Thurston's fibered face theory allows us to partition the set of pseudo-Anosov mapping classes on different compact oriented surfaces into subclasses with related dynamical behavior. This is done via a correspondence between the rational points on fibered faces in the first cohomology of a hyperbolic 3-manifold and the monodromies of fibrations of the 3-manifold over the circle. In this paper, we generalize examples of Penner, and define quotient families of mapping classes. We show that these mapping classes correspond to open linear sections of fibered faces. The construction gives a simple …

Positivity of divisor classes on the strata of differentials

Three decades ago Cornalba-Harris proved a fundamental positivity result for divisor classes associated to families of stable curves. In this paper we establish an analogous positivity result for divisor classes associated to families of stable differentials.

Characterizing families of graph manifolds via suitable classes of simple fold maps into the plane and embeddability of the Reeb spaces in some 3-dimensional manifolds

Graph manifolds form important classes of $3$-dimensional closed and orientable manifolds. For example, {\it Seifert} manifolds are graph manifolds where hyperbolic manifolds are not. In applying singularity theory of differentiable maps to understanding global topologies of manifolds, graph manifolds have been shown to be characterized as ones admitting so-called simple fold maps into the plane of explicit classes by Saeki and the author. The present paper presents several related new results. Fold maps are higher dimensional variants of Morse functions and simple ones form simple classes, ge…

Adequacy of Link Families

Using computer calculations and working with representatives of pretzel tangles we established general adequacy criteria for different classes of knots and links. Based on adequate graphs obtained from all Kauffman states of an alternating link we defined a new numerical invariant: adequacy number, and computed adequacy polynomial which is the invariant of alternating link families. Adequacy polynomial distinguishes (up to mutation) all families of alternating knots and links whose generating link has at most $n=12$ crossings.

Godbillon-Vey invariants for families of foliations

The classical Godbillon-Vey invariant is an odd degree cohomology class that is a cobordism invariant of a single foliation. Here we investigate cohomology classes of even degree that are cobordism invariants of (germs of) 1-parameter families of foliations.

Characteristic classes via 4-dimensional gauge theory

We construct characteristic classes of 4-manifold bundles using $SO(3)$-Yang-Mills theory and Seiberg-Witten theory for families.

$\mathbb{H}^4$和$\mathbb{H}^5$中紧Coxeter多面体的不可数多个共度量类

Infinitely many commensurability classes of compact Coxeter polyhedra in $\mathbb{H}^4$ and $\mathbb{H}^5$

本文证明了Makarov构造的4维和5维双曲空间中的紧Coxeter多面体族能产生无限多个共度量反射群类。