Strong Shape Theory of Continuous Maps

映射的形状维数

Shape dimension of maps

该研究定义了推广拓扑空间形状维数的映射形状维数这一数值不变量，给出其性质与应用，解决了形状有限维映射提升形状维数的问题，并构造了相关反例。

到$\boldsymbol{\rm{P}}^m$的代数映射与连续映射族

Families of algebraic and continuous maps to $\mathbb{P}^m$

该研究阐述了如何利用比较到射影空间的代数映射与连续映射空间同调的结果，实现对代数映射族与连续映射族模叠的比较。

Strong Homology Theory of Continuous Maps

The current work is motivated by the papers $[B_3]$, $[B_6]$, $[Be]$, $[Be-Tu]$. In particular, using Theorem 3.7 of $[B_3]$ and methods developed in this paper, the spectral and strong homology groups of continuous maps were defined and studied $[B_6]$, $[Be]$, $[Be-Tu]$. In this paper we will show that strong homology groups of continuous maps are a homology type functor, which is a strong shape invariant and has the semi-continuous property. We will formulate the new axioms and the conjunction on the uniqueness of the constructed functor.

连续映射的“条形码”与Alternative Morse Theory简介

"Barcodes" for continuous maps and a brief introduction to Alternative Morse Theory

该文属于代数拓扑领域，综述连续实值映射的条形码描述，说明如何从中恢复Morse函数的Morse复形，指出条形码是两类向量空间值映射的支集。

Spaces of algebraic and continuous maps between real algebraic varieties

We consider the inclusion of the space of algebraic (regular) maps between real algebraic varieties in the space of all continuous maps. For a certain class of real algebraic varieties, which include real projective spaces, it is well known that the space of real algebraic maps is a dense subset of the space of all continuous maps. Our first result shows that, for this class of varieties, the inclusion is also a homotopy equivalence. After proving this, we restrict the class of varieties to real projective spaces. In this case, the space of algebraic maps has a ` minimum degree\rq filtration b…

Estimates of the topological entropy from below for continuous self-maps on some compact manifolds

Extending our results in "Entropy conjecture for continuous maps of nilmanifolds", to appear in Israel Jour. of Math., we confirm that Entropy Conjecture holds for every continuous self-map of a compact $K(π,1)$ manifold with the fundamental group $π$ torsion free and virtually nilpotent, in particular for every continuous map of an infra-nilmanifold. In fact we prove a stronger version, a lower estimate of the topological entropy of a map by logarithm of the spectral radius of an associated "linearization matrix" with integer entries. From this, referring to known estimates of Mahler measure …