Tahl Nowik
We study finite order invariants of null-homotopic immersions of a closed orientable surface into an aspherical orientable 3-manifold. We give the foundational constructions, and classify all order one invariants.
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We show that any open aspherical manifold of dimension n>3 is tangentially homotopy equivalent to an n-manifold whose universal cover is not homeomorphic to the Euclidean space.
Aspherical manifolds which do not have Bounded Index Property
本文证明某些非球面流形不具有有界指数性质，从而否定了Jiang提出的相关问题。
In this paper, Problem 4.17 on R. Kirby's problem list is solved by constructing infinitely many aspherical 4-manifolds that are homology 4-spheres
We provide upper bounds on the size of the homology of a closed aspherical Riemannian manifold that only depend on the systole and the volume of balls. Further, we show that linear growth of mod p Betti numbers or exponential growth of torsion homology imply that a closed aspherical manifold is "large".
Aspherical 4-manifolds with positive Euler characteristic and their geography
本文显式构造了任意正整数n下满足χ=σ=n的闭定向非球面光滑4维流形，证明了Edmonds猜想，否定了非球面4维流形上的实Bogomolov-Miyaoka-Yau不等式，并回答了相关公开问题。
We show that in all dimensions >7 there are closed aspherical manifolds whose fundamental groups have nontrivial center but do not possess any topological circle actions. This disproves a conjectured converse (proposed by Conner and Raymond) to a classical theorem of Borel.
We prove a lower bound on the number of maximally broken trajectories of the negative gradient flow of a Morse-Smale function on a closed aspherical manifold in terms of integral (torsion) homology.
The Borel Conjecture predicts that closed aspherical manifolds are topological rigid. We want to investigate when a non-aspherical oriented connected closed manifold M is topological rigid in the following sense. If f: N --> M is an orientation preserving homotopy equivalence with a closed oriented manifold as target, then there is an orientation preserving homeomorphism h: N --> M such that h and f induce up to conjugation the same maps on the fundamental groups. We call such manifolds Borel manifolds. We give partial answers to this questions for S^k x S^d, for sphere bundles over aspherical…
Exotic diffeomorphisms on closed aspherical $4$-manifolds
该研究通过Davis反射技巧构造闭非球面4-流形的微分同胚，得到拓扑同痕于恒等但光滑不同痕于恒等的首批此类流形例子，丰富了微分拓扑相关研究。
Let M be a closed, orientable, irreducible, non-simply connected 3-manifold. We prove that if M admits a sequence of Riemannian metrics whose sectional curvature is locally controlled and whose thick part becomes asymptotically hyperbolic and has a sufficiently small volume, then M is Seifert fibred or contains an incompressible torus. This result gives an alternative approach for the last step in Perelman's proof of the Geometrisation Conjecture for aspherical 3-manifolds.
We fix a null-homologous, homotopically essential knot $J$ in a 3-manifold with PTFA fundamental group and study concordance of knots that are homotopic to $J$. We construct an infinite family of knots that are characteristic to $J$, and then use $L^2$-methods to show that they are not concordant to $J$.
An explicit construction of closed, orientable, smooth, aspherical 4-manifolds with any odd Euler characteristic greater than 12 is presented. The manifolds constructed here are all Haken manifolds in the sense of B. Foozwell and H. Rubinstein and can be systematically reduced to balls by suitably cutting them open along essential codimension-one submanifolds. It is easy to construct examples with even Euler characteristic from products of surfaces. And Euler characteristics divisible by 3 are know to arise from complex algebraic geometry considerations. Examples with Euler characteristic 1, 5…
Actions of large finite groups on aspherical manifolds
本文研究有限群对闭连通非球面流形的作用，在基本群外自同构群的假设下证明同胚群为Jordan群、给出离散对称度的界等，并验证局部齐性空间满足相关假设。
For a Lie group $G=\R^{n}\ltimes_ϕ\R^{m}$ with the semi-simple action $ϕ:\R^{n}\to {\rm Aut}(\R^{m})$, we show that if $Γ$ is a finite extension of a lattice of $G$ then $K(Γ, 1)$ is formal. Moreover we show that a compact symplectic aspherical manifold with the fundamental group $Γ$ satisfies the hard Lefschetz property. By those results we give many examples of formal solvmanifolds satisfying the hard Lefschetz property but not admitting Kähler structures.
Although Kirby and Siebenmann showed that there are manifolds that do not admit PL structures, the possibility remained that all manifolds could be triangulated. In the late seventies Galewski and Stern and independently Matumoto showed that non-triangulable manifolds exist in all dimensions > 4 if and only if homology 3-spheres with certain properties do not exist. In 2013 Manolescu showed that, indeed, there were no such homology 3-spheres and hence, non-triangulable manifolds exist in each dimension >4. It follows from work of Freedman in 1982 that there are 4-manifolds that cannot be trian…
We establish isosystolic inequalities for a class of manifolds which includes the aspherical manifolds. In particular, we relate the systolic volume of aspherical manifolds first to their minimal entropy, then to the algebraic entropy of their fundamental groups.
On Lipschitz cohomology of aspherical manifolds
本文引入了带局部系数的群的 Lipschitz 上同调类概念，并将群 $Γ$ 的 Novikov 高阶符号猜想归结为判断 Berstein-Schwarz 类是否为 Lipschitz 的问题。
This paper contains examples of closed aspherical manifolds obtained as a by-product of recent work by the author [arXiv:math.GR/0509490] on the relative strict hyperbolization of polyhedra. The following is proved. (I) Any closed aspherical triangulated n-manifold M^n with hyperbolic fundamental group is a retract of a closed aspherical triangulated (n+1)-manifold N^(n+1) with hyperbolic fundamental group. (II) If B_1,...,B_m are closed aspherical triangulated n-manifolds, then there is a closed aspherical triangulated manifold N of dimension n+1 such that N has nonzero simplicial volume, N r…
We show that a generic Hamiltonian diffeomorphism on a closed symplectic manifold which is symplectically aspherical has at least the stable Morse number of fixed points - this is in line with a conjecture by Arnold.
We show that the aspherical manifolds produced via the relative strict hyperbolization of polyhedra enjoy many group-theoretic and topological properties of open finite volume negatively pinched manifolds, including relative hyperbolicity, nonvanishing of simplicial volume, co-Hopf property, finiteness of outer automorphism group, absence of splitting over elementary subgroups, and acylindricity. In fact, some of these properties hold for any compact aspherical manifold with incompressible aspherical boundary components, provided the fundamental group is hyperbolic relative to fundamental grou…
Some properties of non-orientable 3-manifolds are shown. The semi-group of cobordism of immersions of surfaces in such manifolds is computed and proven actually to be a group. Explicit invariants are provided.
We prove that every finitely generated group with recursive aspherical presentation embeds into a group with finite aspherical presentation. This and several known facts about groups and manifolds imply that there exists a 4-dimensional closed aspherical manifold $M$ such that the fundamental group $π_1(M)$ coarsely contains an expander, and so it has infinite asymptotic dimension, is not coarsely embeddable into a Hilbert space, and does not satisfy the Baum-Connes conjecture with coefficients. Closed aspherical manifolds with any of these properties were previously unknown.
This is a survey on known results and open problems about closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. Many examples come from certain kinds of non-positive curvature conditions. The property aspherical which is a purely homotopy theoretical condition implies many striking results about the geometry and analysis of the manifold or its universal covering, and the ring theoretic properties and the K- and L-theory of the group ring associated to its fundamental group. The Borel Conjecture predicts that closed aspherical manifolds are to…
Aspherical 4-manifolds with elementary amenable fundamental group
分类了带边界的紧致不可球化4维流形的初等可解基本群，证明了它们要么是多重循环群，要么是可解的Baumslag-Solitar群，并由此得到拓扑刚性及边界分类结果。
Tautological classes, or generalised Miller-Morita-Mumford classes, are basic characteristic classes of smooth fibre bundles, and have recently been used to describe the rational cohomology of classifying spaces of diffeomorphism groups for several types of manifolds. We show that rationally tautological classes depend only on the underlying topological block bundle, and use this to prove the vanishing of tautological classes for many bundles with fibre an aspherical manifold.
Symplectically aspherical Kähler manifolds, scalar curvature, and the fundamental group
研究拉回至万有覆盖上为恰当形式的凯勒形式的闭光滑流形，即辛非球面凯勒流形，探讨其拓扑几何性质，如大基本群、无正标量曲率凯勒度量，还研究相关猜想及复几何性质。
The paper was withdrawn due to a gap in the proof of Lemma 3.
In this note, we investigate the relation between double points and complex points of immersed surfaces in almost-complex 4-manifolds and show how estimates for the minimal genus of embedded surfaces lead to inequalities between the number of double points and the number of complex points of an immersion. We also provide a generalization of a classical genus estimate due to V.A. Rokhlin to the case of immersed surfaces.
Almost-concordance of knots in aspherical 3-manifolds
本文研究非球面3-流形中纽结的几乎协调问题，提出方法验证相关猜想在非平凡自由同伦类下成立，通过推广Milnor链环不变量区分几乎协调类，证明其在大量开放情形下成立。
Aspherical manifolds with nonvanishing tautological classes
本文构造了一类具有非平凡中心基本群的非球面流形，其无穷多个示性类非零且非幂零，并给出不紧致流形边界的例子，从而反驳了 Hebestreit--Land--Lück--Randal-Williams 猜想。
Aspherical 4-Manifolds, Complex Structures, and Einstein Metrics
本文证明正欧拉示性数的扩展图4-流形及闭实双曲4-流形均不支持复结构，并构造无穷多个支持近复结构的此类流形。
Exotic aspherical 4-manifolds
本研究构造了同胚但不微分同胚的闭非球面光滑4-流形，为四维Borel猜想的光滑类比提供反例，核心方法是将反射群技巧应用于带边怪异4-流形对。
On quasi-holomorphic homotopies of immersions of 3-manifolds into 5-manifolds
提出拟全伦同伦概念扩展正则同伦，允许通过孤立奇点（交叉帽），并描述其局部形式及与全纯映射芽的联系，主要结果给出浸入像补基本群在拟全伦同伦下的变化规律。
Sz\H ucs proved in 2000 that the $r$-tuple-point manifold of a generic immersion is cobordant to the $Σ^{1_{r-1}}$-point manifold of its generic projection. Here we slightly extend this by showing that the natural mappings of these manifolds are bordant to each other. The main novelty of our approach is that we construct the bordism explicitly.
The signature of geometrically decomposable aspherical 4-manifolds
本文构造了具有非零符号差的几何可分解非球面4流形，证明其满足Bogomolov--Miyaoka--Yau型不等式，并给出达到等号的非几何例子及更高图4流形中严格不等式的无穷多实例。
On totally c-symplectically aspherical manifolds
该研究在≥6的偶数维构造了具非平凡第二同伦群的全辛非球面凯勒流形例子，在四维构造了具非平凡第二同伦群的全c-辛非球面近辛流形例子，拓展了相关流形的存在性结果。
Realisation of cycles by aspherical manifolds
本文为 Steenrod 闭链实现问题提出纯组合方法，证明任意拓扑空间的 n 维同调类可由实对称三对角矩阵同谱流形的有限叶覆盖像实现，且弧连通空间的同调类可由无球面流形的连续像实现。
We show that closed aspherical manifolds supporting an affine structure, whose holonomy map is injective and contains a pure translation, must have vanishing simplicial volume. This provides some further evidence for the veracity of the Auslander Conjecture. Along the way, we provide a simple cohomological criterion for aspherical manifolds with normal amenable subgroups of $π_1$ to have vanishing simplicial volume. This answers a special case of a question due to Lück.
In this paper we show that for a large class C of 4-manifolds each member of C has only finitely many simple homotopy type upto s-cobordism. This result generalizes a similar result of Hillman for certain complex surfaces. We also present a correction in the proof of Hillman's result.
We exhibit geometric situations, where higher indices of the spinor Dirac operator on a spin manifold $N$ are obstructions to positive scalar curvature on an ambient manifold $M$ that contains $N$ as a submanifold. In the main result of this note, we show that the Rosenberg index of $N$ is an obstruction to positive scalar curvature on $M$ if $N \hookrightarrow M \twoheadrightarrow B$ is a fiber bundle of spin manifolds with $B$ aspherical and $π_1(B)$ of finite asymptotic dimension. The proof is based on a new variant of the multi-partitioned manifold index theorem which might be of independe…
We give geometric formulae which enable us to detect (completely in some cases) the regular homotopy class of an immersion with trivial normal bundle of a closed oriented 3-manifold into 5-space. These are analogues of the geometric formulae for the Smale invariants due to Ekholm and the second author. As a corollary, we show that two embeddings into 5-space of a closed oriented 3-manifold with no 2-torsion in the second cohomology are regularly homotopic if and only if they have Seifert surfaces with the same signature. We also show that there exist two embeddings $F_0$ and $F_8 : T^3 \hookri…
An aspherical 5-manifold with perfect fundamental group
本文构造了具有完美基群的非球面闭定向5维流形，完成了与D.H.Kochloukova和I.Lima合作研究PD_n群的pro-p完成为pro-p庞加莱对偶群的研究，探讨了是否存在维度下降1的例子。
We consider the relation between simplicial volume and two of its variants: the stable integral simplicial volume and the integral foliated simplicial volume. The definition of the latter depends on a choice of a measure preserving action of the fundamental group on a probability space. We show that integral foliated simplicial volume is monotone with respect to weak containment of measure preserving actions and yields upper bounds on (integral) homology growth. Using ergodic theory we prove that simplicial volume, integral foliated simplicial volume and stable integral simplicial volume coinc…
We study aspherical manifolds that do not support Anosov diffeomorphisms. Weakening conditions of Gogolev and Lafont, we show that the product of an infranilmanifold with finitely many aspherical manifolds whose fundamental groups have trivial center and finite outer automorphism group does not support Anosov diffeomorphisms. In the course of our study, we obtain a result of independent group theoretic and topological interest on the stability of the Hopf property, namely, that the product of finitely many Hopfian groups with trivial center is Hopfian.
On the automorphism group of parabolic structures and closed aspherical manifolds
本文研究闭非球面抛物 ${\sfG}$-流形的自同构群，证明某些抛物结构会强烈限制流形拓扑，并刻画了具有几乎可解基本群的标准 $CR$-流形与四元数接触流形的商结构。
First order invariants of generic immersions of manifolds of dimension nm-1 into manifolds of dimension n(m+1)-1, m,n>1 are constructed using the geometry of self-intersections. The range of one of these invariants is related to Bernoulli numbers. As by-products some geometrically defined invariants of regular homotopy are found.
A well-known question by Gromov asks whether the vanishing of the simplicial volume of oriented closed connected aspherical manifolds implies the vanishing of the Euler characteristic. We study various versions of Gromov's question and collect strategies towards affirmative answers and strategies towards negative answers to this problem. Moreover, we put Gromov's question into context with other open problems in low- and high-dimensional topology. A special emphasis is put on a comparative analysis of the additivity properties of the simplicial volume and the Euler characteristic for manifolds…
We define a new class of irreducible groups, called groups not infinite-index presentable by products or not IIPP. We prove that certain aspherical manifolds with fundamental groups not IIPP do not admit maps of non-zero degree from direct products. This extends previous results of Kotschick and Loeh, providing new classes of aspherical manifolds - beyond those non-positively curved ones which were predicted by Gromov - that do not admit maps of non-zero degree from direct products. A sample application is that an aspherical geometric 4-manifold admits a map of non-zero degree from a direct pr…
We use the reflection group trick to glue manifolds with corners that are Borel-Serre compactifications of locally symmetric spaces of noncompact type and obtain aspherical manifolds. We call these \emph{piecewise locally symmetric} manifolds. This class of spaces provide new examples of aspherical manifolds whose fundamental groups have the structure of a complex of groups. These manifolds typically do not admit a locally $\CAT(0)$ metric. We prove that any self homotopy equivalence of such manifolds is homotopic to a homeomorphism. We compute the group of self homotopy equivalences of such a…
Aspherical manifolds with boundary
本文系统研究带边界的紧致非球面流形，提出同调判据判定闭流形何时能作为非球面流形的边界，并通过两步法（构造庞加莱对和手术理论）实现，在交换基本群情形给出示例。
Positive scalar curvature metrics and aspherical summands
本文证明了在3、4、5维中，闭合非球面流形与非紧流形的连通和不接受具有非负标量曲率的完整度量，并验证了Gromov的非紧支配猜想。
Let $\mathrm{SL}_{n}(\mathbb{Z})$ $(n\geq 3)$ be the special linear group and $M^{r}$ be a closed aspherical manifold. It is proved that when $r<n,$ a group action of $\mathrm{SL}_{n}(\mathbb{Z})$ on $M^{r}$ by homeomorphisms is trivial if and only if the induced group homomorphism $\mathrm{SL}_{n}(% \mathbb{Z})\rightarrow \mathrm{Out}(π_{1}(M))$ is trivial. For (almost) flat manifolds, we prove a similar result in terms of holonomy groups. Especially, when $π_{1}(M)$ is nilpotent, the group $\mathrm{SL}_{n}(% \mathbb{Z})$ cannot act nontrivially on $M$ when $r<n.$ This confirms a conjecture r…
We show that the Thurston seminorms of all finite covers of an aspherical 3-manifold determine whether it is a graph manifold, a mixed 3-manifold or hyperbolic.
In this paper we define two regular homotopy invariants c and i for immersions of oriented 3-manifolds into R^5 in a geometric manner. The pair (c(f),i(f)) completely describes the regular homotopy class of the immersion f. The invariant i corresponds to the 3-dimensional obstruction that arises from Hirsch-Smale theory and extends the one defined in [10] for immersions with trivial normal bundle.
We obtain restrictions on the topology of a closed connected manifold B that bounds a (possibly noncompact) manifold whose interior V admits a complete Riemannian metric of nonpositive sectional curvature. If G denotes the fundamental group of B, then a sample result is that B must be aspherical and incompressible if one of the following is true: (1) V has finite volume and G is virtually nilpotent, (2) G is virtually nilpotent and has no proper torsion-free quotients, (3) G is isomorphic to a uniform, irreducible lattice of real rank > 1.
We give complete geometric invariants of cobordisms of fold maps with oriented singular set and cobordisms of even codimensional fold maps. These invariants are given in terms of cobordisms of stably framed manifolds and cobordisms of immersions with prescribed normal bundles defined by the author in his earlier works.
In this short note we prove the Borel conjecture for a family of aspherical manifolds that includes higher graph manifolds.
The cobordism group $N(M^n)$ of codimension-one immersions in the $n$-manifold $M^n$ has a natural filtration induced by any cellular decomposition. The problem addressed in this paper is the explicit computation of the graded group $gr^*N(M^n)$. We introduce some new invariants for immersions enlightening the Atiyah-Hirzebruch spectral sequence associated to $N(M)$, which are of combinatorial-geometric nature. Explicit computations are developed for $n\leq7$, and the group structure is also investigated for orientable 4-manifolds.
On a problem of Hopf for circle bundles over aspherical manifolds with hyperbolic fundamental groups
本文完全刻画了具有双曲基本群的闭可定向非球面流形上圆丛的绝对度大于一的自映射存在性，证明其等价于丛虚拟平凡，并由此验证了Hopf问题的强版本，同时构造了关于映射度单调的新数值不变量。
The Cannon Conjecture for a torsionfree hyperbolic group G with boundary homeomorphic to S^2 says that G is the fundamental group of an aspherical closed 3-manifold M. It is known that then M is a hyperbolic 3-manifold. We prove the stable version that for any closed manifold N of dimension greater or equal to 2 there exists a closed manifold M together with a simple homotopy equivalence from M to the cartesian product of N and BG. If N is aspherical and pi_1(N) satisfies the Farrell-Jones Conjecture, then M is unique up to homeomorphism.
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