Simplifying 3-manifolds in R^4

Piecewise linear generalized Alexander's theorem in dimension at most 5

We study piecewise linear co-dimension two embeddings of closed oriented manifolds in Euclidean space, and show that any such embedding can always be isotoped to be a closed braid as long as the ambient dimension is at most five, extending results of Alexander (in ambient dimension three), and Viro and independently Kamada (in ambient dimension four). We also show an analogous result for higher co-dimension embeddings.

Symplectic 4-manifolds admit Weinstein trisections

We prove that every symplectic 4-manifold admits a trisection that is compatible with the symplectic structure in the sense that the symplectic form induces a Weinstein structure on each of the three sectors of the trisection. Along the way, we show that a (potentially singular) symplectic braided surface in $\mathbb{CP}^2$ can be symplectically isotoped into bridge position.

Fox reimbedding and Bing submanifolds

Let M be an orientable closed connected 3-manifold. We introduce the notion of amalgamated Heegaard genus of M with respect to a closed separating 2-manifold F, and use it to show that the following two statements are equivalent: (i) a compact connected 3-manifold Y can be embedded in M so that the exterior of the image of Y is a union of handlebodies; and (ii) a compact connected 3-manifold Y can be embedded in M so that every knot in M can be isotoped to lie within the image of Y . Our result can be regarded as a common generalization of the reimbedding theorem by Fox [Fox48] and the charact…

Simplifying indefinite fibrations on 4-manifolds

We present explicit algorithms for simplifying the topology of indefinite fibrations on 4-manifolds, which include broken Lefschetz fibrations and indefinite Morse 2-functions. The algorithms consist of sequences of moves, which modify indefinite fibrations in smooth 1-parameter families. In particular, given an arbitrary broken Lefschetz fibration, we show how to turn it to one with directed and embedded round (indefinite fold) image, and to one with all the fibers and the round locus connected. We also show how to realize any given null-homologous 1-dimensional submanifold with prescribed lo…

用于三角剖分和简化4-流形的实用软件

Practical Software for Triangulating and Simplifying 4-Manifolds

本文开发了从Kirby图构建4-流形三角剖分的软件工具和新的简化启发式方法，并据此给出了若干有界奇异对、cork、plug及K3曲面的更小三角剖分，揭示了4-流形三角剖分的精细结构特征。

每个实3-流形都是实切触的

Every real 3-manifold is real contact

该研究通过证明实3-流形可由标准实$S^3$沿不变纽结手术得到且操作可在切触框架下完成，得出任意实3-流形都容许实切触结构的结论，并给出相关推论与透镜空间等的实例验证。