Qayum Khan
Let $G$ be a compact Lie group. (Compact) topological $G$-manifolds have the $G$-homotopy type of (finite-dimensional) countable $G$-CW complexes (2.5). This partly generalizes Elfving's theorem for locally linear $G$-manifolds [Elf96], wherein the Lie group $G$ is linear (such as compact).
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We prove that for every countable group G there exists a hyperbolic 3-manifold M such that the isometry group of M, the mapping class group of M, and the outer automorphism group of the fundamental group of M are isomorphic to G.
Let $G$ be a matrix group. Topological $G$-manifolds with Palais-proper action have the $G$-homotopy type of countable $G$-CW complexes (3.2). This generalizes E Elfving's dissertation theorem for locally linear $G$-manifolds (1996). Also we improve the Bredon--Floyd theorem from compact groups $G$ (1960).
The purpose of this paper is to present, for all $n\ge 3$, very simple examples of continuous maps $f:M^{n-1} \to M^{n}$ from closed $(n-1)$-manifolds $M^{n-1}$ into closed $n$-manifold $M^n$ such that even though the singular set $S(f)$ of $f$ is countable and dense, the map $f$ can nevertheless be approximated by an embedding, i.e. $f$ is a {\sl near-embedding}.
The Hilbert-Smith conjecture states, for any connected topological manifold $M$, any locally compact subgroup of $\mathrm{Homeo}(M)$ is a Lie group. We generalize basic results of Segal-Kosniowski-tomDieck (2.6), James-Segal (2.12), G Bredon (3.7), Jaworowski-Antonyan et al. (5.5), and E Elfving (7.3). The last is our main result: for any Lie group $G$, any Palais-proper topological $G$-manifold has the equivariant homotopy type of a countable proper $G$-CW complex. Along the way, we verify an $n$-classifying space for principal $G$-bundles (5.10).
Relationships between a chaotic behavior and closely related properties of topological transitivity, sensitivity to initial conditions, density of closed orbits of homeomorphism groups and their countable products are investigated. We construct numerous new examples of chaotic groups of homeomorphisms of countable products of various metrizable topological spaces, including infinite-dimensional topological manifolds, whose factors can be as noncompact surfaces, so triangulable closed manifolds of an arbitrary dimension.
Uniform first order interpretation of the second order theory of countable groups of homeomorphisms
该研究证明了流形同胚群的一阶理论可统一解释其二阶理论，并揭示了同胚群的可定义子集及相关计算理论结果。
Borel classification of simplicial complexes and non-compact $2$- and $3$-manifolds
该研究推广了斯通对偶，获得单纯复形与非紧2、3维流形的同胚/PL同胚完全不变量，证明相关同胚关系可通过可数结构分类且与可数图同构关系Borel双可归约。
There are many "minimax" complexity functions in mathematics: width of a tree or a link, Heegaard genus of a 3-manifold, the Cheeger constant of a Riemannian manifold. We define such a function w, "width", on countable (or finite) groups and show w(Z^k) = k-1.
This paper concerns the class of contractible open 3-manifolds which are ``locally finite strong end sums'' of eventually end-irreducible Whitehead manifolds. It is shown that whenever a 3-manifold in this class is a covering space of another 3-manifold the group of covering translations must be a free group. It follows that such a 3-manifold cannot cover a closed 3-manifold. For each countable free group a specific uncountable family of irreducible open 3-manifolds is constructed whose fundamental groups are isomorphic to the given group and whose universal covering spaces are in this class a…
Based in the isomorphism between Lie algebroid cohomology and piecewise smooth cohomology, it is proved that the Rham cohomology of a locally trivial Lie groupoid $G$ on a smooth manifold $M$ is isomorphic to the piecewise Rham cohomology of $G$, in which $G$ and $M$ are manifolds without boundary and $M$ is smoothly triangulated by a finite simplicial complex $K$ such that, for each simplex $Δ$ of $K$, the inverse images of $Δ$ by the source and target mappings of $G$ are transverses submanifolds in the ambient space $G$. As a consequence, it is shown that the piecewise de Rham cohomology of …
We study Lie foliations on compact manifolds, in case the Lie group is compact. Our main results improve Tischler classical result on the existence of fibration and, as an application, we study the case the manifold has an amenable fundamental group.
In this paper, we show that there is a close relationship between generalized subtangent manifolds and Lie groupoids. We obtain equivalent assertions among the integrability conditions of generalized almost subtangent manifolds, the condition of compatibility of source and target maps of symplectic groupoids with symplectic form and generalized subtangent maps.
We present a simple approach to questions of topological orbit equivalence for actions of countable groups on topological and smooth manifolds. For example, for any action of a countable group $Γ$ on a topological manifold where the fixed sets for any element are contained in codimension two submanifolds, every orbit equivalence is equivariant. Even in the presence of larger fixed sets, for actions preserving rigid geometric structures our results force sufficiently smooth orbit equivalences to be equivariant. For instance, if a countable group $Γ$ acts on ${\mathbb T}^n$ and the action is $C^…
This paper concerns the topology of isospectral real manifolds of certain Jacobi elements associated with real split semisimple Lie algebras. The manifolds are related to the compactified level sets of the generalized (nonperiodic) Toda lattice equations defined on the semisimple Lie algebras. We then give a cellular decomposition and the associated chain complex of the manifold by introducing colored Dynkin diagrams which parametrize the cells in the decomposition. We also discuss the Morse chain complex of the manifold.
Let $M$ be a manifold, $N$ a 1-dimensional manifold. Assuming $r \neq \dim(M)+1$, we show that any nontrivial homomorphism $ρ: \text{Diff}^r_c(M)\to \text{Homeo}(N)$ has a standard form: necessarily $M$ is $1$-dimensional, and there are countably many embeddings $ϕ_i: M\to N$ with disjoint images such that the action of $ρ$ is conjugate (via the product of the $ϕ_i$) to the diagonal action of $\text{Diff}^r_c(M)$ on $M \times M \times ...$ on $\bigcup_i ϕ_i(M)$, and trivial elsewhere. This solves a conjecture of Matsumoto. We also show that the groups $\text{Diff}^r_c(M)$ have no countable ind…
Manifold classification from the descriptive viewpoint
本文从描述集合论角度研究流形和李群离散子群的分类问题，揭示了流形同胚问题的Borel复杂性及双曲流形等距关系的分类特性。
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