A proof of a homeomorphism theorem of Waldhausen

Hyperbolic Geometry and Homotopic Homeomorphisms of Surfaces

The Epstein-Baer theory of curve isotopies is basic to the remarkable theorem that homotopic homeomorphisms of surfaces are isotopic. The groundbreaking work of R. Baer was carried out on closed, orientable surfaces and extended by D. B. A. Epstein to arbitrary surfaces, compact or not, with or without boundary and orientable or not. We give a new method of deducing the theorem about homotopic homeomorphisms from the results about homotopic curves via the hyperbolic geometry of surfaces. This works on all but 13 surfaces where ad hoc proofs are needed.

The boundary of the Milnor fiber for some non-isolated germs of complex surfaces

We study the boundary L_t of the Milnor fiber for the non-isolated singularities in C^3 with equation z^m - g(x,y) = 0 where g(x,y) is a non-reduced plane curve germ. We give a complete proof that L_t is a Waldhausen graph manifold and we provide the tools to construct its plumbing graph. As an example, we give the plumbing graph associated to the germs z^2 - (x^2 - y^3)y^l = 0 with l an interger >1. We prove that the boundary of the Milnor fiber is a Waldhausen manifold new in complex geometry, as it cannot be the boundary of a normal surface singularity.

同痕于伪Anosov同胚的同胚的广义叶状结构：Fathi一个结果的几何实现

Generalized foliations for homeomorphisms isotopic to a pseudo-Anosov homeomorphism: a geometric realization of a result by Fathi

本研究给出Fathi一个结果的新几何证明，证明闭曲面上同痕于伪Anosov同胚的同胚都存在与伪Anosov同胚叶状结构性质相似的稳定、不稳定不变划分。

同胚问题的不可解性

The Unsolvability of the Homeomorphism Problem

本文通过详细证明Markov定理，探讨了高维空间中同胚问题的不可解性及不可识别流形的存在性。

Sous-groupes compacts d'homeomorphismes de la sphere

In this paper, we present a simple proof of the fact that any compact subgroup of homeomorphisms of the 2-sphere is topologically conjugate to a closed subgroup of the orthogonal group O(3).

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…

Lipschitz同胚的形变

Deformations of Lipschitz Homeomorphisms

本文将Perelman证明稳定性定理时所用Siebenmann同胚形变结果推广至Lipschitz情形，建立Lipschitz版粘合定理及带常数追踪的形变理论。

An Additivity theorem for cobordism categories

Using methods inspired from algebraic $K$-theory, we give a new proof of the Genauer fibration sequence, relating the cobordism categories of closed manifolds with cobordism categories of manifolds with boundaries, and of the Bökstedt-Madsen delooping of the cobordism category. Unlike the existing proofs, this approach generalizes to other cobordism-like categories of interest. Indeed we argue that the Genauer fibration sequence is an analogue, in the setting of cobordism categories, of Waldhausen's Additivity theorem in algebraic $K$-theory.

Symmetry of links and classification of lens spaces

We give a concise proof of a classification of lens spaces up to orientation-preserving homeomorphisms. The chief ingredient in our proof is a study of the Alexander polynomial of ` symmetric' links in $S^3$.

Fixed-point index, the Incompatibility Theorem, and torus parametrization

The fixed-point index of a homeomorphism of Jordan curves measures the number of fixed-points, with multiplicity, of the extension of the homeomorphism to the full Jordan domains in question. The now-classical Circle Index Lemma says that the fixed-point index of a positive-orientation-preserving homeomorphism of round circles is always non-negative. We begin by proving a generalization of this lemma, to accommodate Jordan curves bounding domains which do not disconnect each other. We then apply this generalization to give a new proof of Schramm's Incompatibility Theorem, which was used by Sch…

A local to global argument on low dimensional manifolds

For an oriented manifold $M$ whose dimension is less than $4$, we use the contractibility of certain complexes associated to its submanifolds to cut $M$ into simpler pieces in order to do local to global arguments. In particular, in these dimensions, we give a different proof of a deep theorem of Thurston in foliation theory which says that the natural map between classifying spaces $\mathrm{B}\text{Homeo}^δ(M)\to \mathrm{B}\text{Homeo}(M)$ induces a homology isomorphism where $\text{Homeo}^δ(M)$ denotes the group of homeomorphisms of $M$ made discrete. Our proof shows that in low dimensions, …

A short proof of a theorem of Morton Brown on chains of cells

Suppose that a topological space $X$ is the union of an increasing sequence of open subsets each of which is homeomorphic to the Euclidean space $R^n$. Then $X$ itself is homeomorphic to $R^n$. This is an old theorem of Morton Brown. We observe that this theorem is an immediate consequence of other two theorems of Morton Brown concerning near homeomorphisms and cellular sets.

2-manifold recognition is in logspace

We prove that the homeomorphism problem for 2-manifolds can be decided in logspace. The proof relies on Reingold's logspace solution to the undirected $s,t$-connectivity problem in graphs.

Global fixed points for centralizers and Morita's Theorem

We prove a global fixed point theorem for the centralizer of a homeomorphism of the two dimensional disk $D$ that has attractor-repeller dynamics on the boundary with at least two attractors and two repellers. As one application, we show that there is a finite index subgroup of the centralizer of a pseudo-Anosov homeomorphism with infinitely many global fixed points. As another application we give an elementary proof of Morita's Theorem, that the mapping class group of a closed surface $S$ of genus $g$ does not lift to the group of diffeormorphisms of $S$ and we improve the lower bound for $g$…

Generating the Torelli group

We give a new proof of the theorem of Birman-Powell that the Torelli subgroup of the mapping class group of a closed orientable surface of genus at least 3 is generated by simple homeomorphisms known as bounding pair maps. The key ingredient is a proof that the subcomplex of the curve complex of the surface spanned by curves within a fixed homology class is connected.

Reconstructing maps out of groups

We show that, in many situations, a homeomorphism $f$ of a manifold $M$ may be recovered from the (marked) isomorphism class of a finitely generated group of homeomorphisms containing $f$. As an application, we relate the notions of {\em critical regularity} and of {\em differentiable rigidity}, give examples of groups of diffeomorphisms of 1-manifolds with strong differential rigidity, and in so doing give an independent, short proof of a recent result of Kim and Koberda that there exist finitely generated groups of $C^α$ diffeomorphisms of a 1-manifold $M$, not embeddable into $\mathrm{Diff}…

Global fixed points of mapping class group actions and a theorem of Markovic

We give a short and elementary proof of the non-realizability of the mapping class group via homeomorphisms. This was originally established by Markovic, resolving a conjecture of Thurston. With the tools established in this paper, we also obtain some rigidity results for actions of the mapping class group on Euclidean spaces.

Une nouvelle preuve du theoreme de point fixe de Handel

M Handel has proved in [Topology 38 (1999) 235--264] a fixed point theorem for an orientation preserving homeomorphism of the open unit disk, that may be extended to the closed disk and that satisfies a linking property of orbits. We give here a new proof of Handel's fixed point theorem, based on Brouwer theory and some plane topology arguments. We will slightly improve the theorem by proving the existence of a simple closed curve of index 1. This index result was known to be true under an additional hypothesis and has been used by different authors (J Franks [NYJM 2 (1996) 1--19, Trans.AMS 34…

瑟斯顿地震定理的反德西特证明

The Anti-de Sitter proof of Thurston's earthquake theorem

本文利用 $\mathrm{PSL}(2,\mathbb R)$ 的双不变几何（反德西特三维空间）给出瑟斯顿地震定理的一个初等证明，说明圆周保向同胚可扩张为双曲平面上的地震。