Algorithmic aspects of homeomorphism problems

一类具有算法可解同胚问题的开曲面

A class of open surfaces with algorithmically solvable homeomorphism problem

本文提出拓扑自动机这一由有限数据定义的非紧流形新类，并证明当维数为2时其同胚问题算法可判定。

Algorithmic homeomorphism of 3-manifolds as a corollary of geometrization

In this paper we prove two results, one semi-historical and the other new. The semi-historical result, which goes back to Thurston and Riley, is that the geometrization theorem implies that there is an algorithm for the homeomorphism problem for closed, oriented, triangulated 3-manifolds. We give a self-contained proof, with several variations at each stage, that uses only the statement of the geometrization theorem, basic hyperbolic geometry, and old results from combinatorial topology and computer science. For this result, we do not rely on normal surface theory, methods from geometric group…

通过几何 Markov 分区对广义 pseudo-Anosov 同胚进行算法分类

An Algorithmic Classification of Generalized Pseudo-Anosov Homeomorphisms via Geometric Markov Partitions

本文利用几何 Markov 分区的几何类型作为完全不变量，给出实现判据与判定算法，实现广义 pseudo-Anosov 同胚拓扑共轭类的算法分类。

可浸入性与可嵌入性的算法方面

Algorithmic aspects of immersibility and embeddability

本文研究浸入理论的算法可判定性，刻画了何种维数条件下流形可浸入问题可判定，并由此证明一类带边流形的光滑可嵌入性不可判定。

h-cobordism and s-cobordism Theorems: Transfer over Semialgebraic and Nash categories, Uniform bound and Effectiveeness

The h-cobordism theorem is a noted theorem in differential and PL topology. A generalization of the h-cobordism theorem for possibly non simply connected manifolds is the so called s-cobordism theorem. In this paper, we prove semialgebraic and Nash versions of these theorems. That is, starting with semialgebraic or Nash cobordism data, we get a semialgebraic homeomorphism (respectively a Nash diffeomorphism). The main tools used are semialgebraic triangulation and Nash approximation. One aspect of the algebraic nature of semialgebraic or Nash objects is that one can measure their complexities.…

Algorithmic and topological aspects of semi-algebraic sets defined by quadratic polynomial

In this thesis, we consider semi-algebraic sets over a real closed field $R$ defined by quadratic polynomials. Semi-algebraic sets of $R^k$ are defined as the smallest family of sets in $R^k$ that contains the algebraic sets as well as the sets defined by polynomial inequalities, and which is also closed under the boolean operations (complementation, finite unions and finite intersections). We prove new bounds on the Betti numbers as well as on the number of different stable homotopy types of certain fibers of semi-algebraic sets over a real closed field $R$ defined by quadratic polynomials, i…