Almost $\iota$-复形作为浸入曲线

Arnold型不变量的底层Stokes和de Rham结构

Underlying Stokes and de Rham structures for Arnold-type invariants

本文基于局部有限差结构研究浸没曲线和表面的Arnold型不变量，定义局部标准化映射并建立离散Stokes兼容性，推导Shumakovitch型恒等式，统一了全局不变量与局部数据的联系。

Immersions of surfaces in almost complex 4-manifolds

In this note, we investigate the relation between double points and complex points of immersed surfaces in almost-complex 4-manifolds and show how estimates for the minimal genus of embedded surfaces lead to inequalities between the number of double points and the number of complex points of an immersion. We also provide a generalization of a classical genus estimate due to V.A. Rokhlin to the case of immersed surfaces.

Hardness of almost embedding simplicial complexes in $\mathbb R^d$

A map $f\colon K\to \mathbb R^d$ of a simplicial complex is an almost embedding if $f(σ)\cap f(τ)=\emptyset$ whenever $σ,τ$ are disjoint simplices of $K$. Theorem. Fix integers $d,k\ge2$ such that $d=\frac{3k}2+1$. (a) Assume that $P\ne NP$. Then there exists a finite $k$-dimensional complex $K$ that does not admit an almost embedding in $\mathbb R^d$ but for which there exists an equivariant map $\tilde K\to S^{d-1}$. (b) The algorithmic problem of recognition almost embeddability of finite $k$-dimensional complexes in $\mathbb R^d$ is NP hard. The proof is based on the technique from the Mat…

A characterization of virtually embedded subsurfaces in 3-manifolds

The paper introduces the spirality character of the almost fiber part for a closed essentially immersed subsurface of a closed orientable aspherical 3-manifold, which generalizes an invariant due to Rubinstein and Wang. The subsurface is virtually embedded if and only if the almost fiber part is aspiral, and in this case, the subsurface is virtually a leaf of a taut foliation. Besides other consequences, examples are exhibited that non-geometric 3-manifolds with no Seifert fibered pieces may contain essentially immersed but not virtually embedded closed subsurfaces.

Hardness of almost embedding simplicial complexes in $\mathbb{R}^d$, II

A map $f: K \to \mathbb{R}^d$ of a simplicial complex is an almost embedding if $f(σ) \cap f(τ) = \varnothing$ whenever $σ, τ$ are disjoint simplices of $K$. Fix integers $d,k \geqslant 2$ such that $k+2 \leqslant d \leqslant\frac{3k}2+1$. Assuming that the "preimage of a cycle is a cycle" we prove $\mathbf{NP}$-hardness of the algorithmic problem of recognition of almost embeddability of finite $k$-dimensional complexes in $\mathbb{R}^d$. Assuming that $\mathbf{P} \ne \mathbf{NP}$ (and that the "preimage of a cycle is a cycle") we prove that the embedding obstruction is incomplete for $k$-dim…

Eine Bemerkung zu einigen $2$-dimensionalen Komplexen, die im $\mathbb{R}^4$ fast-eingebettet werden können

We observe that many of the 2-complexes constructed by Freedman-Krushkal-Teichner in their paper on the incompleteness of the van Kampen embedding obstruction can actually be PL immersed in $\mathbb{R}^4$ in such a way that the images of the interiors of distinct cells are disjoint. In other words, they PL almost-embed in $\mathbb{R}^4$ with singularities occurring only as self intersections of some 2-cells. This note is written in (not necessarily modern) German.