Homeomorphism Type Research Articles

Compactifications of moduli spaces and cellular decompositions

This paper studies compactifications of moduli spaces involving closed Riemann surfaces. The first main result identifies the homeomorphism types of these compactifications. The second main result introduces orbicell decompositions on these spaces using semistable ribbon graphs extending the earlier work of Looijenga.

Infinitely-ended hyperbolic groups with homeomorphic Gromov boundaries

Abstract We show that the Gromov boundary of the free product of two infinite hyperbolic groups is uniquely determined up to homeomorphism by the homeomorphism types of the boundaries of its factors. We generalize this result to graphs of hyperbolic groups over finite subgroups. Finally, we give a necessary and sufficient condition for the Gromov boundaries of any two hyperbolic groups to be homeomorphic (in terms of the topology of the boundaries of factors in terminal splittings over finite subgroups).

Regular circle actions on 2-connected 7-manifolds

A circle action S 1 × M → M on a manifold M is called regular if this action is free and the orbit space is a manifold. In this paper, we determine the homeomorphism (respectively, diffeomorphism) types of those 2-connected 7-manifolds (respectively, smooth 2-connected 7-manifolds) that admit regular circle actions (respectively, smooth regular circle actions).

Orbifold points on Teichmüller curves and Jacobians with complex multiplication

For each integer D≥5 with D≡0 or 1 mod4, the Weierstrass curve WD is an algebraic curve and a finite-volume hyperbolic orbifold which admits an algebraic and isometric immersion into the moduli space of genus two Riemann surfaces. The Weierstrass curves are the main examples of Teichmuller curves in genus two. The primary goal of this paper is to determine the number and type of orbifold points on each component of WD. Our enumeration of the orbifold points, together with Bainbridge [Geom. Topol. 11 (2007) 1887–2073] and McMullen [Math. Ann. 333 (2005) 87–130], completes the determination of the homeomorphism type of WD and gives a formula for the genus of its components. We use our formula to give bounds on the genus of WD and determine the Weierstrass curves of genus zero. We will also give several explicit descriptions of each surface labeled by an orbifold point on WD.

Topology of iterated $S^{1}$-bundles

In this paper we investigate what kind of manifolds arise as the total spaces of iterated $S^1$-bundles. A real Bott tower studied in \cite{CMO}, \cite{KM} and \cite{KN} is an example of an iterated $S^1$-bundle. We show that the total space of an iterated $S^1$-bundle is homeomorphic to an infra-nilmanifold. A real Bott manifold, which is the total space of a real Bott tower, provides an example of a closed flat Riemannian manifold. We also show that real Bott manifolds are the only closed flat Riemannian manifolds obtained from iterated $\bbr{P}^1$-bundles. Finally we classify the homeomorphism types of the total spaces of iterated $S^1$-bundles in dimension 3.

Wild singularities of flat surfaces

We consider flat surfaces and the points of their metric completions, particularly the singularities to which the flat structure of the surface does not extend. The local behavior near a singular point x can be partially described by a topological space L(x) which captures all the ways that x can be “approached linearly”. The homeomorphism type of L(x) is an affine invariant. When x is not a cone point or an infinite-angle singularity, we say it is wild; in this case it is necessary to add further metric data to L(x) to get a quantitative description of the surface near x.

Moduli of triangles in the Heisenberg group

We study triangles in the three-dimensional Heisenberg group, \({\mathbb{H}}\) , equipped with the Carnot-Caratheodory geometry. The set of ordered triangles in \({\mathbb{H}}\) (excluding certain degenerate triangles), up to congruence is naturally identified via a parametrization map to a fine moduli space of parameters. We determine the homeomorphism type of this moduli space and also that of the coarse moduli space of unordered triangles. We describe a boundary for the fine moduli space and construct a compactification for it, up to similarity under the non-isotropic dilation of \({\mathbb{H}}\) . Additionally, some trigonometric results for the Carnot-Caratheodory geometry of \({\mathbb{H}}\) are given: an angle deficit formula and an analog of the Law of Sines in Euclidean geometry.

Remarks on the classification of quasitoric manifolds up to equivariant homeomorphism

We give three sufficient criteria for two quasitoric manifolds M, M′ to be (weakly) equivariantly homeomorphic. We apply these criteria to count the weakly equivariant homeomorphism types of quasitoric manifolds with a given cohomology ring.

Erratic behavior of CAT(0) geodesics under G-equivariant quasi-isometries

We show that, given any connected, compact space \({Z \subset \mathbb{R}^n}\), there exists a group G acting geometrically on two CAT(0) spaces X and Y, a G-equivariant quasi-isometry \({f\colon X\rightarrow Y}\), and a geodesic ray c in X such that the closure of f (c), intersected with \({\partial Y}\), is homeomorphic to Z. This characterizes all homeomorphism types of “geodesic boundary images” that arise in this manner.

Realising end invariants by limits of minimally parabolic, geometrically finite groups

We shall show that for a given homeomorphism type and a set of end invariants (including the parabolic locus) with necessary topological conditions which a topologically tame Kleinian group with that homeomorphism type must satisfy, there is an algebraic limit of minimally parabolic, geometrically finite Kleinian groups which has exactly that homeomorphism type and end invariants. This shows that the Bers-Sullivan-Thurston density conjecture follows from Marden's conjecture proved by Agol, Calegari-Gabai combined with Thurston's uniformisation theorem and the ending lamination conjecture proved by Minsky, partially collaborating with Masur, Brock and Canary.

Curve complexes are rigid

Any quasi-isometry of the complex of curves is bounded distance from a simplicial automorphism. As a consequence, the quasi-isometry type of the curve complex determines the homeomorphism type of the surface.

Orbifold homeomorphism finiteness based on geometric constraints

We show that any collection of n-dimensional orbifolds with sectional curvature and volume uniformly bounded below, diameter bounded above, and with only isolated singular points contains orbifolds of only finitely many orbifold homeomorphism types. This is a generalization to the orbifold category of a similar result for manifolds proven by Grove, Petersen, and Wu. It follows that any Laplace isospectral collection of orbifolds with sectional curvature uniformly bounded below and having only isolated singular points also contains only finitely many orbifold homeomorphism types. The main steps of the argument are to show that any sequence from the collection has subsequence that converges to an orbifold, and then to show that the homeomorphism between the underlying spaces of the limit orbifold and an orbifold from the subsequence that is guaranteed by Perelman's stability theorem must preserve orbifold structure.

Cohomological rigidity and the number of homeomorphism types for small covers over prisms

In this paper, based upon the basic theory for glued manifolds in M.W. Hirsch (1976) [8, Chapter 8, §2 Gluing Manifolds Together], we give a method of constructing homeomorphisms between two small covers over simple convex polytopes. As a result we classify, up to homeomorphism, all small covers over a 3-dimensional prism P 3 ( m ) with m ⩾ 3 . We introduce two invariants from colored prisms and other two invariants from ordinary cohomology rings with Z 2 -coefficients of small covers. These invariants can form a complete invariant system of homeomorphism types of all small covers over a prism in most cases. Then we show that the cohomological rigidity holds for all small covers over a prism P 3 ( m ) (i.e., cohomology rings with Z 2 -coefficients of all small covers over a P 3 ( m ) determine their homeomorphism types). In addition, we also calculate the number of homeomorphism types of all small covers over P 3 ( m ) .

Weighted Sobolev spaces and embedding theorems

In the present paper we study embedding operators for weighted Sobolev spaces whose weights satisfy the well-known Muckenhoupt A p -condition. Sufficient conditions for boundedness and compactness of the embedding operators are obtained for smooth domains and domains with boundary singularities. The proposed method is based on the concept of 'generalized' quasiconformal homeomorphisms (homeomorphisms with bounded mean distortion). The choice of the homeomorphism type depends on the choice of the corresponding weighted Sobolev space. Such classes of homeomorphisms induce bounded composition operators for weighted Sobolev spaces. With the help of these homeomorphism classes the embedding problem for non-smooth domains is reduced to the corresponding classical embedding problem for smooth domains. Examples of domains with anisotropic Holder singularities demonstrate the sharpness of our machinery comparatively with known results.

Infinitely generated Lawson homology groups on some rational projective varieties

We construct rational projective 4-dimensional varieties with the property that certain Lawson homology groups tensored with Q are infinite dimensional Q-vector spaces. More generally, for each pair of integers p and k, with k > 0, p > 0, we find a projective variety Y such that LpH 2p+k (Y) is infinitely generated. We also construct two singular rational projective 3-dimensional varieties Y and Y' with the same homeomorphism type but different Lawson homology groups; specifically, L 1 H 3 (Y) is not isomorphic to L 1 H 3 (Y') even up to torsion.

Constructing infinitely many smooth structures on small 4-manifolds

The purpose of this article is two-fold. First we outline a general construction scheme for producing simply connected minimal symplectic -manifolds with small Euler characteristics. Using this scheme, we illustrate how to obtain irreducible symplectic -manifolds homeomorphic but not diffeomorphic to ℂ ℙ 2 # ( 2 k + 1 ) ℂ ℙ ¯ 2 for k = 1, …, 4, or to 3 ℂ ℙ 2 # ( 2 l + 3 ) ℂ ℙ ¯ 2 for l = 1, …, 6. Secondly, for each of these homeomorphism types, we show how to produce an infinite family of pairwise nondiffeomorphic nonsymplectic 4-manifolds belonging to it. In particular, we prove that there are infinitely many exotic irreducible nonsymplectic smooth structures on, ℂ ℙ 2 # 3 ℂ ℙ ¯ 2 , 3 ℂ ℙ 2 # 5 ℂ ℙ ¯ 2 and 3 ℂ ℙ 2 # 7 ℂ ℙ ¯ 2 .

Slow quasiregular mappings and universal coverings

We define slow quasiregular mappings and study cohomology and universal coverings of closed manifolds receiving slow quasiregular mappings. We show that closed manifolds receiving a slow quasiregular mapping from a punctured ball have the de Rham cohomology type of either Sn or Sn-1×S1. We also show that in the case of manifolds of the cohomology type of Sn-1×S1, the universal covering of the manifold has exactly two ends, and the lift of the slow mapping into the universal covering has a removable singularity at the point of punctuation. We also obtain exact growth bounds and a global homeomorphism–type theorem for slow quasiregular mappings into the manifolds of the cohomology type Sn-1×S1

Reverse engineering small 4–manifolds

We introduce a general procedure called `reverse engineering' that can be used to construct infinite families of smooth 4-manifolds in a given homeomorphism type. As one of the applications of this technique, we produce an infinite family of pairwise nondiffeomorphic 4-manifolds homeomorphic to CP^2#3(-CP^2).

Level Structures on the Weierstrass Family of Cubics

Let W → 𝔸 2 be the universal Weierstrass family of cubic curves over ℂ. For each N ≥ 2, we construct surfaces parameterizing the three standard kinds of level N structures on the smooth fibers of W. We then complete these surfaces to finite covers of 𝔸 2. Since W → 𝔸 2 is the versal deformation space of a cusp singularity, these surfaces convey information about the level structure on any family of curves of genus g degenerating to a cuspidal curve. Our goal in this note is to determine for which values of N these surfaces are smooth over (0, 0). From a topological perspective, the results determine the homeomorphism type of certain branched covers of S 3 with monodromy in SL2 (ℤ/N).

Reduced Delzant spaces and a convexity theorem

The convexity theorem of Atiyah and Guillemin–Sternberg says that any connected compact manifold with Hamiltonian torus action has a moment map whose image is the convex hull of the image of the fixed point set. Sjamaar–Lerman proved that the Marsden–Weinstein reduction of a connected Hamitonian G -manifold is a stratified symplectic space. Suppose 1 → A → G → T → 1 is an exact sequence of compact Lie groups and T is a torus. Then the reduction of a Hamiltonian G -manifold with respect to A yields a Hamiltonian T -space. We show that if the A -moment map is proper, then the convexity theorem holds for such a Hamiltonian T -space, even when it is singular. We also prove that if, furthermore, the T -space has dimension 2 dim T and T acts effectively, then the moment polytope is sufficient to essentially distinguish their homeomorphism type, though not their diffeomorphism types. This generalizes a theorem of Delzant in the smooth case. This paper is a concise version of a companion paper [B. Lian. B. Song, A convexity theorem and reduced Delzant spaces, math.DG/0509429].