Vyacheslav S Krushkal, Frank Quinn
We present a new, more elementary proof of the Freedman-Teichner result that the geometric classification techniques (surgery, s-cobordism, and pseudoisotopy) hold for topological 4-manifolds with groups of subexponential growth. In an appendix Freedman and Teichner give a correction to their original proof, and reformulate the growth estimates in terms of coarse geometry.
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We address a long-standing and long-investigated problem in combinatorial topology, and break the exponential barrier for triangulations of real projective space, constructing a trianglation of $\mathbb{RP}^n$ of size $e^{(\frac{1}{2}+o(1))\sqrt{n}{\log n}}$.
We prove several cases of Zimmer's conjecture for actions of higher-rank cocompact lattices on low dimensional manifolds. For example, if $Γ$ is a cocompact lattice in $\mathrm{Sl}(n, \mathbb R)$, $M$ is a compact manifold, and $ω$ a volume form on $M$ we show that any homomorphism $ρ\colon Γ\rightarrow \mathrm{Diff}(M)$ has finite image if the dimension of $M$ is less than $n-1$ and that any homomorphism $ρ\colon Γ\rightarrow \mathrm{Diff}(M,ω)$ has finite image if the dimension of $M$ is less than $n$. The key step in the proof is to show any such action has uniform subexponential growth of …
4-manifold topology and collision rank of groups
本文引入碰撞秩概念，证明有限碰撞秩群在4-流形拓扑中是好的，并构造了碰撞秩为二的有限生成群，其例子来自Sturmian系统拓扑全群的换位子群，包含连续统多个两两不同构的无限简单可遗群。
We construct examples of nonresolvable generalized $n$-manifolds, $n\geq 6$, with arbitrary resolution obstruction, homotopy equivalent to any simply connected, closed $n$-manifold. We further investigate the structure of generalized manifolds and present a program for understanding their topology.
We prove that the deformation space $AH(M)$ of marked hyperbolic 3-manifolds homotopy equivalent to a fixed compact 3-manifold $M$ with incompressible boundary is locally connected at quasiconformally rigid points.
We survey work on the topology of the space AH(M) of all (marked) hyperbolic 3-manifolds homotopy equivalent to a fixed compact 3-manifold M with boundary. The interior of AH(M) is quite well-understood, but the topology of the entire space can be quite complicated. However, the topology is well-behaved at many points in the boundary of AH(M).
Let $M$ be a closed, oriented and smooth manifold of dimension $d$. Let $ŁM$ be the space of smooth loops in $M$. Chas and Sullivan introduced loop product, a product of degree $-d$ on the homology of $LM$. In this paper we show how for three manifolds the ``nontriviality'' of the loop product relates to the ``hyperbolicity'' of the underlying manifold. This is an application of the existing powerful tool and results in three dimensional topology such as the prime decomposition, torus decomposition, Seifert theorem, torus theorem.
The compact-open topology on the diffeomorphism or homeomorphism group of a smooth manifold without boundary is minimal in almost all dimensions
该研究证明维数非3的连通光滑流形微分同胚群上紧开拓扑是极小的，进而推得维数非3、4的流形同胚群上该拓扑极小，闭流形等满足自动连续性的情形下它还是唯一可分Hausdorff群拓扑。
Topological 4-dimensional surgery is conjectured to fail, in general, for free fundamental groups. M. Freedman and P. Teichner have shown that surgery problems with an arbitrary fundamental group have a solution, provided they satisfy a certain condition on Dwyer's filtration on second homology. We give a new geometric proof of this result, and analyze its relation to the canonical surgery problems.
This survey/expository article covers a variety of topics related to the "topology at infinity" of noncompact manifolds and complexes. In manifold topology and geometric group theory, the most important noncompact spaces are often contractible, so distinguishing one from another requires techniques beyond the standard tools of algebraic topology. One approach uses end invariants, such as the number of ends or the fundamental group at infinity; another approach seeks nice compactifications, then analyzes the boundaries. A thread connecting the two is shape theory. In these notes we provide a ca…
This article is the author's contribution to the volume "Problems on mapping class groups and related topics" which will be published in December 2006, with Benson Farb as Editor. Various individuals were invited by Farb to submit open problems which seemed interesting to them about surface mapping class groups. We singled out a particular recent development, Hempel's concept of the `Heegaard distance' of an element in the mapping class group, and used it to describe a number of open questions which relate to the correspondence between Heegaard surfaces in 3-manifolds and the mapping class gro…
Counterexamples in 4-manifold topology
本文以反例梳理4维流形上多种等价关系的蕴含与区别，证明如$h$-配边蕴含稳定同胚，并用反例说明其他蕴含失效，辅以流程图和表格总结。
Let M be a 4-manifold with residually finite fundamental group G having b_1(G) > 0. Assume that M carries a symplectic structure with trivial canonical class K = 0 in H^2(M). Using a theorem of Bauer and Li, together with some classical results in 4-manifold topology, we show that for a large class of groups $ is determined up to homotopy and, in favorable circumstances, up to homeomorphism by its fundamental group. This is analogous to what was proven by Morgan-Szabo in the case of b_1 = 0 and provides further evidence to the conjectural classification of symplectic 4-manifolds with K = 0$. A…
A property of $γ$ curves derived from topology of relative trisected 4-manifolds
本文研究了相对三重剖分的 4 维流形中 $γ$ 曲线的性质，证明了任意两个 $(g; 0, 0, k)$-三重剖分的闭流形可通过 Torelli 群元素沿 $γ$ 曲线切割重粘合关联，并推广至相对情形及与软木塞扭转的联系。
Let C be some class of objects equipped with a set of simplifying moves. When we apply these to a given object M in C as long as possible, we get a root of M. Our main result is that under certain conditions the root of any object exists and is unique. We apply this result to different situations and get several new results and new proofs of known results. Among them there are a new proof of the Kneser-Milnor prime decomposition theorem for 3-manifolds and different versions of this theorem for cobordisms, knotted graphs, and orbifolds.
Let M be a compact 3-manifold whose interior admits a complete hyperbolic structure. We let Lambda(M) be the supremum of the bottom eigenvalue of the Laplacian of N, where N varies over all hyperbolic 3-manifolds homeomorphic to the interior of M. Similarly, we let D(M) be the infimum of the Hausdorff dimensions of limit sets of Kleinian groups whose quotients are homeomorphic to the interior of M. We observe that Lambda(M)=D(M)(2-D(M)) if M is not handlebody or a thickened torus. We characterize exactly when Lambda(M)=1 and D(M)=1 in terms of the characteristic submanifold of the incompressib…
Let $P$ be a simple polytope of dimension $n$ with $m$ facets and $P_{v}$ be a polytope obtained from $P$ by cutting off one vertex $v$. Let $Z=Z(P)$ and $Z_{v}=Z(P_{v})$ be the corresponding moment-angle manifolds. In \cite{[GL]} S.Gitler and S.López conjectured that: $Z_{v}$ is diffeomorphic to $\partial[(Z-int(D^{n+m}))\times D^{2}]\sharp \mathop{\sharp} \limits_{j=1}^{m-n} \binom{m-n}{j} (S^{j+2}\times S^{m+n-j-1})$, and they have proved the conjecture in the case $m<3n$. In this paper we prove the conjecture in general case.
A group-theoretic framework for low-dimensional topology
本文提出一种群论框架，统一并推广了低维拓扑中三维和四维流形、结与结表面的对应关系。
Topology of boundary special generic maps into Euclidean spaces
研究边界特殊泛映射对源流形全局结构的限制，并应用于非奇异扩展问题，获得新的非奇异扩展结果。
We investigate relation between Dehn fillings and commensurability of hyperbolic 3-manifolds. The set consisting of the commensurability classes of hyperbolic 3-manifolds admits the quotient topology induced by the geometric topology. We show that this quotient space satisfies some separation axioms. Roughly speaking, this means that commensurablity classes are sparsely distributed in the space consisting of the hyperbolic 3-manifolds.
On Hasse norm principle for 3-manifolds in arithmetic topology
本研究借助算术拓扑中纽结与素数、3-流形与数环的类比，推导得出3-流形有限循环覆盖下的Hasse范数原理拓扑对应形式，拓展了该原理的适用范畴。
This article analyzes the interplay between symplectic geometry in dimension four and the invariants for smooth four-manifolds constructed using holomorphic triangles introduced in math.SG/0110169. Specifically, we establish a non-vanishing result for the invariants of symplectic four-manifolds, which leads to new proofs of the indecomposability theorem for symplectic four-manifolds and the symplectic Thom conjecture. As a new application, we generalize the indecomposability theorem to splittings of four-manifolds along a certain class of three-manifolds obtained by plumbings of spheres. This …
We study geometry, topology and deformation spaces of noncompact complex hyperbolic manifolds (geometrically finite, with variable negative curvature), whose properties make them surprisingly different from real hyperbolic manifolds with constant negative curvature. This study uses an interaction between Kähler geometry of the complex hyperbolic space and the contact structure at its infinity (the one-point compactification of the Heisenberg group), in particular an established structural theorem for discrete group actions on nilpotent Lie groups.
In the third part of this series of papers, we establish several topological results that will become important for studying the long-time behavior of Ricci flows with surgery. In the first part of this paper we recall some elementary observations in the topology of 3-manifolds. The main part is devoted to the construction of certain simplicial complexes in a given 3-manifold that exhibit useful intersection properties with embedded, incompressible solid tori. This paper is purely topological in nature and Ricci flows will not be used.
This is an expository note intended to illustrate current research in topological study of partially hyperbolic diffeomorphisms in dimension 3 with a beautiful result due to Margulis and Plante-Thurston on topological obstructions for a manifold to admit an Anosov flow.
We prove that the deformation space AH(M) of marked hyperbolic 3-manifolds homotopy equivalent to a fixed compact 3-manifold M with incompressible boundary is locally connected at minimally parabolic points. Moreover, spaces of Kleinian surface groups are locally connected at quasiconformally rigid points. Similar results are obtained for deformation spaces of acylindrical 3-manifolds and Bers slices.
On the topology of manifolds with positive intermediate curvature
本文研究正$m$-中间曲率与流形万有覆盖拓扑的关系，证明了低维及六维多数情形的猜想，并推出闭非球面6维流形不存在正$4$-中间曲率度量。
This article presents a new and more elementary proof of the main Seiberg-Witten-based obstruction to the existence of Einstein metrics on smooth compact 4-manifolds. It also introduces a new smooth manifold invariant which conveniently encapsulates those aspects of Seiberg-Witten theory most relevant to the study of Riemannian variational problems on 4-manifolds.
A smooth five-dimensional s-cobordism becomes a smooth product if stabilized by a finite number n of $S^2xS^2x[0,1]$'s. We show that for amenable fundamental groups, the minimal n is subextensive in covers, i.e., n(cover)/index(cover) has limit 0. We focus on the notion of sweepout width, which is a bridge between 4-dimensional topology and coarse geometry.
We prove that closed manifolds admitting a generic metric whose sectional curvature is locally quasi-constant are graphs of space forms. In the more general setting of QC spaces where sets of isotropic points are arbitrary, under suitable positivity assumption and for torsion-free fundamental groups they are still diffeomorphic to connected sums of spherical space forms and spherical bundles over the circle.
This is a survey of topological properties of open, complete nonpositively curved manifolds which may have infinite volume. Topics include topology of ends, restrictions on the fundamental group, as well as a review of known examples.
We construct a functor from the smooth 4-dimensional manifolds to the hyper-algebraic number fields, i.e. fields with non-commutative multiplication. It is proved that that the simply connected 4-manifolds correspond to the abelian extensions. We recover the Rokhlin and Donaldson's Theorems from the Galois theory of the non-commutative fields.
We show that there exist smooth, simply connected, four-dimensional spin manifolds which do not admit Einstein metrics, but nonetheless satisfy the strict Hitchin-Thorpe inequality. Our construction makes use of the Bauer/Furuta cohomotopy refinement of the Seiberg-Witten invariant, in conjunction with curvature estimates previously proved by the second author. These methods also easily allow one to construct examples of topological 4-manifolds which admit an Einstein metric for one smooth structure, but which have infinitely many other smooth structures for which no Einstein metric can exist.
We show a rigidity theorem for the Seiberg-Witten invariants mod 2 for families of spin 4-manifolds. A mechanism of this rigidity theorem also gives a family version of 10/8-type inequality. As an application, we prove the existence of non-smoothable topological families of 4-manifolds whose fiber, base space, and total space are smoothable as manifolds. These non-smoothable topological families provide new examples of 4-manifolds $M$ for which the inclusion maps $\operatorname{Diff}(M) \hookrightarrow \operatorname{Homeo}(M)$ are not weak homotopy equivalences. We shall also give a new series…
On genus theory for 3-manifolds in arithmetic topology
本文通过算术拓扑类比，证明Hilbert Satz 90的拓扑版本，并利用Hasse范数原理，给出积分同调3球面上有限阿贝尔分歧覆盖的Iyanaga--Tamagawa型亏格公式，与数论情形平行。
This paper introduces two-dimensional diagrams that are slight generalizations of moment map images for toric four-manifolds and catalogs techniques for reading topological and symplectic properties of a symplectic four-manifold from these diagrams. The paper offers a purely topological approach to toric manifolds as well as methods for visualizing other manifolds, including the K3 surface, and for visualizing certain surgeries. Most of the techniques extend to higher dimensions.
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.
We carry out the harmonic analysis on four Platonic spherical three-manifolds with different topologies. Starting out from the homotopies (Everitt 2004), we convert them into deck operations, acting on the simply connected three-sphere as the cover, and obtain the corresponding variety of deck groups. For each topology, the three-sphere is tiled into copies of a fundamental domain under the corresponding deck group. We employ the point symmetry of each Platonic manifold to construct its fundamental domain as a spherical orbifold. While the three-sphere supports an~orthonormal complete basis fo…
For a topological space $X$ we study continuous maps $f : X\to \mathbb R^m$ such that images of every pairwise distinct $k$ points are affinely (linearly) independent. Such maps are called affinely (linearly) $k$-regular embeddings. We investigate the cohomology obstructions to existence of regular embeddings and give some new lower bounds on the dimension $m$ as function of $X$ and $k$, for the cases $X$ is $\mathbb R^n$ or $X$ is an $n$-dimensional manifold. In the latter case, some nonzero Stiefel--Whitney classes of $X$ help to improve the bound.
Hardness of computation of quantum invariants on 3-manifolds with restricted topology
本文证明了在Reshetikhin-Turaev模型中，存在确定性多项式时间算法可将任意闭3-流形转换为具有相同量子不变量但拓扑受限的流形，表明计算量子不变量的难度在严格限制输入拓扑与组合结构时依然存在。
In this survey, we give an overview of Brieskorn manifolds and varieties, and their role in contact topology. We discuss open books, fillings and invariants such as contact and symplectic homology. We also present some new results involving exotic contact structures, invariants and orderability. The main tool for the required computations is a version of the Morse-Bott spectral sequence. We provide a proof for the particular version that is useful for us.
We show that any homomorphism from the homeomorphism group of a compact 2-manifold, with the compact-open topology, or equivalently, with the topology of uniform convergence, into a separable topological group is automatically continuous.
The topology of broken Lefschetz fibrations is studied by means of handle decompositions. We consider a slight generalization of round handles, and describe the handle diagrams for all that appear in dimension four. We establish simplified handlebody and monodromy representations for a certain subclass of broken Lefschetz fibrations/pencils, while showing that all near-symplectic closed 4-manifolds can be supported by these a la Auroux, Donaldson, Katzarkov. Various constructions of broken Lefschetz fibrations and a generalization of the symplectic fiber sum operation to the near-symplectic se…
Let $M$ be a simply-connected closed manifold of dimension $\geq 5$ which does not admit a metric with positive scalar curvature. We give necessary conditions for $M$ to admit a scalar-flat metric. These conditions involve the first Pontrjagin class and the cohomology ring of $M$. As a consequence any simply-connected scalar-flat manifold of dimension $\geq 5$ with vanishing first Pontrjagin class admits a metric with positive scalar curvature. We also describe some relations between scalar-flat metrics, almost complex structures and the free loop space.
On the topology of real Lagrangians in toric symplectic manifolds
本文研究环面辛流形中由矩多面体对合对称性产生的实拉格朗日子流形，建立Delzant构造的实类比，证明其微分同胚类型由组合数据决定，并实现环面辛del Pezzo曲面上所有可能的连通实拉格朗日微分同胚类型。
We give a purely topological definition of the perturbative quantum invariants of links and 3-manifolds associated with Chern-Simons field theory. Our definition is as close as possible to one given by Kontsevich. We will also establish some basic properties of these invariants, in particular that they are universally finite type with respect to algebraically split surgery and with respect to Torelli surgery. Torelli surgery is a mutual generalization of blink surgery of Garoufalidis and Levine and clasper surgery of Habiro.
This paper grew out of an attempt to find a suitable finite sheeted covering of an aspherical 3-manifold so that the cover either has infinite or trivial first homology group. With this motivation we define a new class of groups. These groups are in some sense eventually perfect. We prove results giving several classes of examples of groups which do (not) belong to this class. Also we prove some elementary results on these groups and state two conjectures. A direct application of one of the conjectures to the virtual Betti number conjecture of Thurston is mentioned.
In this paper we show that a simply connected 8-dimensional manifold M of positive sectional curvature and symmetry rank $\geq 2$ resembles a rank one symmetric space in several ways. For example, the Euler characteristic of M is equal to the Euler characteristic of S^8, H P^2 or C P^4. And if M is rationally elliptic then M is rationally isomorphic to a rank one symmetric space. For torsion-free manifolds we derive a much stronger classification. We also study the bordism type of 8-dimensional manifolds of positive sectional curvature and symmetry rank $\geq 2$. As an illustration we apply ou…
This book provides a self-contained introduction to the topology and geometry of surfaces and three-manifolds. The main goal is to describe Thurston's geometrisation of three-manifolds, proved by Perelman in 2002. The book is divided into three parts: the first is devoted to hyperbolic geometry, the second to surfaces, and the third to three-manifolds. It contains complete proofs of Mostow's rigidity, the thick-thin decomposition, Thurston's classification of the diffeomorphisms of surfaces (via Bonahon's geodesic currents), the prime and JSJ decomposition, the topological and geometric classi…
This paper investigates the question of which smooth compact 4-manifolds admit Riemannian metrics that minimize the L2-norm of the curvature tensor. Metrics with this property are called OPTIMAL; Einstein metrics and scalar-flat anti-self-dual metrics provide us with two interesting classes of examples. Using twistor methods, optimal metrics of the second type are constructed on the connected sums kCP_2 for k > 5. However, related constructions also show that large classes of simply connected 4-manifolds do not admit any optimal metrics at all. Interestingly, the difference between existence a…
We initiate a study of positive multisections of Lefschetz fibrations via positive factorizations in framed mapping class groups of surfaces. Using our methods, one can effectively capture various interesting symplectic surfaces in symplectic 4-manifolds as multisections, such as Seiberg-Witten basic classes and exceptional classes, or branched loci of compact Stein surfaces as branched coverings of the 4-ball. Various problems regarding the topology of symplectic 4-manifolds, such as the smooth classification of symplectic Calabi-Yau 4-manifolds, can be translated to combinatorial problems in…
On the topology and geometry of certain $13$-manifolds
本文分类上同调环同构于 $\mathrm{CP}^3\times S^7$ 的 $13$ 维流形，并证明其本身或与某个怪球连通和后具有非负截面曲率度量。
We give a bordism-theoretic characterisation of those closed almost contact (2q+1)-manifolds (with q > 2) which admit a Stein fillable contact structure. Our method is to apply Eliashberg's h-principle for Stein manifolds in the setting of Kreck's modified surgery. As an application, we show that any simply connected almost contact 7-manifold with torsion free second homotopy group is Stein fillable. We also discuss the Stein fillability of exotic spheres and examine subcritical Stein fillability.
Topology of $3$-manifolds with uniformly positive scalar curvature
本文分类了具有一致正数量曲率的3-流形，证明其等价于球面3-流形与S^1×S^2的无限连通和，并推广到带平均凸边界的情形。
Our main result gives an adjunction inequality for embedded surfaces in certain $4$-manifolds with contact boundary under a non-vanishing assumption on the Bauer--Furuta type invariants. Using this, we give infinitely many knots in $S^3$ that are not smoothly H-slice (that is, bounding a null-homologous disk) in many $4$-manifolds but they are topologically H-slice. In particular, we give such knots in the boundaries of the punctured elliptic surfaces $E(2n)$. In addition, we give obstructions to codimension-0 orientation-reversing embedding of weak symplectic fillings with $b_3=0$ into closed…
A conjecture of Kotschick predicts that a compact Kähler manifold $X$ fibres smoothly over the circle if and only if it admits a holomorphic one-form without zeros. In this paper we develop an approach to this conjecture and verify it in dimension two. In a joint paper with Hao, we use our approach to prove Kotschick's conjecture for smooth projective threefolds.
We give two applications of the 2-Engel relation, classically studied in finite and Lie groups, to the 4-dimensional topological surgery conjecture. The A-B slice problem, a reformulation of the surgery conjecture for free groups, is shown to admit a homotopy solution. We also exhibit a new collection of universal surgery problems, defined using ramifications of homotopically trivial links. More generally we show how n-Engel relations arise from higher order double points of surfaces in 4-space.
We provide obstructions to the existence of conformally Anosov Reeb flows on a 3-manifold that partially generalize similar obstructions to Anosov Reeb flows. In particular, we show $\mathbb{S}^3$ does not admit conformally Anosov Reeb flows. We also give a Riemannian geometric condition on a metric compatible with a contact structure implying that a Reeb field is Anosov. From this we can give curvature conditions on a metric compatible with a contact structure that implies universal tightness of the contact structure among other things.
Gromov's universal filling inequalities relate the filling radius and the filling volume of a Riemannian manifold to its volume. The main result of the present article is that in dimensions at least three the optimal constants in the filling inequalities depend only on dimension and orientability, not on the manifold itself. This contrasts with the analogous situation for the optimal systolic inequality, which does depend on the manifold.
For a non-compact n-manifold M let H(M) denote the group of homeomorphisms of M endowed with the Whitney topology and H_c(M) the subgroup of H(M) consisting of homeomorphisms with compact support. It is shown that the group H_c(M) is locally contractible and the identity component H_0(M) of H(M) is an open normal subgroup in H_c(M). This induces the topological factorization H_c(M) \approx H_0(M) \times \M_c(M) for the mapping class group \M_c(M) = H_c(M)/H_0(M) with the discrete topology. Furthermore, for any non-compact surface M, the pair (H(M), H_c(M)) is locally homeomorphic to (\square^w…
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