Novikov-Betti numbers and the fundamental group

Novikov homology, jump loci and Massey products

Let X be a finite CW-complex, denote its fundamental group by G. Let R be an n-dimensional complex repesentation of G. Any element A of the first cohomology group of X with complex coefficients gives rise to the exponential deformation of the representation R, which can be considered as a curve in the space of representations. We show that the cohomology of X with local coefficients corresponding to the generic point of this curve is computable from a spectral sequence starting from the cohomology of X with R-twisted coefficients. We compute the differentials of the spectral sequence in terms …

The co-rank of the fundamental group: the direct product, the first Betti number, and the topology of foliations

We study $b'_1(M)$, the co-rank of the fundamental group of a smooth closed connected manifold $M$. We calculate this value for the direct product of manifolds. We characterize the set of all possible combinations of $b'_1(M)$ and the first Betti number $b_1(M)$ by explicitly constructing manifolds with any possible combination of $b'_1(M)$ and $b_1(M)$ in any given dimension. Finally, we apply our results to the topology of Morse form foliations. In particular, we construct a manifold $M$ and a Morse form $ω$ on it for any possible combination of $b'_1(M)$, $b_1(M)$, $m(ω)$, and $c(ω)$, where…

Approximating Novikov-Shubin numbers of virtually cyclic coverings

We assign real numbers to finite sheeted coverings of compact CW complexes designed as finite counterparts to the Novikov-Shubin numbers. We prove an approximation theorem in the case of virtually cyclic fundamental groups employing methods from Diophantine approximation.

A geometric proof of Lück's vanishing theorem for the first $L^2$-Betti number of the total space of a fibration

A significant theorem of Lück says that the first $L^2$-Betti number of the total space of a fibration vanishes under some conditions on the fundamental groups. The proof is based on constructions on chain complexes. In the present paper, we translate the proof into the world of CW-complexes to make it more accessible.

The fundamental group and Betti numbers of toric origami manifolds

Toric origami manifolds are characterized by origami templates, which are combinatorial models built by gluing polytopes together along facets. In this paper, we examine the topology of orientable toric oigami manifolds with coorientable folding hypersurface. We determine the fundamental group. In our previous paper [HP], we studied the ordinary and equivariant cohomology rings of simply connected toric origami manifolds. We conclude this paper by computing some Betti numbers in the non-simply connected case.

Arnold conjecture over integers

For any closed symplectic manifold, we show that the number of 1-periodic orbits of a nondegenerate Hamiltonian thereon is bounded from below by a version of total Betti number over Z of the ambient space taking account of the total Betti number over Q and torsions of all characteristic. The proof is based on constructing a Hamiltonian Floer theory over the Novikov ring with integer coefficients, which generalizes our earlier work for constructing integer-valued Gromov-Witten type invariants. In the course of the construction, we build a Hamiltonian Floer flow category with compatible smooth g…

A spectral sequence to compute L2-Betti numbers of groups and groupoids

We construct a spectral sequence for L2-type cohomology groups of discrete measured groupoids. Based on the spectral sequence, we prove the Hopf-Singer conjecture for aspherical manifolds with poly-surface fundamental groups. More generally, we obtain a permanence result for the Hopf-Singer conjecture under taking fiber bundles whose base space is an aspherical manifold with poly-surface fundamental group. As further sample applications of the spectral sequence, we obtain new vanishing theorems and explicit computations of L2-Betti numbers of groups and manifolds and obstructions to the existe…

Large Random Simplicial Complexes, II; the fundamental groups

In our recent work we described conditions under which a multi-parameter random simplicial complex is connected and simply connected. We showed that the Betti numbers of multi-parameter random simplicial complexes in one specific dimension dominate significantly the Betti numbers in all other dimensions. In this paper we focus mainly on the properties of fundamental groups of multi-parameter random simplicial complexes, which can be viewed as a new class of random groups. We describe thresholds for nontrivially and hyperbolicity (in the sense of Gromov) for these groups. Besides, we find domai…

SU-bordism: structure results and geometric representatives

In the first part of this survey we give a modernised exposition of the structure of the special unitary bordism ring, by combining the classical geometric methods of Conner-Floyd, Wall and Stong with the Adams-Novikov spectral sequence and formal group law techniques that emerged after the fundamental 1967 work of Novikov. In the second part we use toric topology to describe geometric representatives in SU-bordism classes, including toric, quasitoric and Calabi-Yau manifolds.

曲率、直径与有界Betti数

Curvature, diameter and bounded Betti numbers

本文研究有界Betti数，利用Kapovitch与Wilking证明的广义Margulis引理，为闭黎曼流形有界Betti数的有界性定理提供现代支撑，并探讨相关拓扑意义。

First Betti numbers of orbits of Morse functions on surfaces

In this article we study algebraic properties of the specific class of groups $\mathcal{G}$ generated by direct products and wreath products. Such class of groups appears in calculation of fundamental groups of orbits of Morse functions on compact manifolds. We prove that for any group $G\in\mathcal{G}$ the ranks of the center $Z(G)$ and the quotient by commutator subgroup $G/[G,G]$ coincide. Moreover, this rank is a first Betti number of the orbit of Morse function.

Multiscale 2-Mapper——由第一 Betti 数引导的探索性数据分析

Multiscale 2-Mapper -- Exploratory Data Analysis Guided by the First Betti Number

本文针对 Mapper 参数选择难题构建 2-Mapper，提出由持续 Betti-1 信息引导的参数选择方法，并通过 Multiscale Mapper 分析覆盖影响，在含 Klein 瓶的三维形状数据上完成验证。

On $p$-adic limits of topological invariants

The purpose of this article is to define and study new invariants of topological spaces: the $p$-adic Betti numbers and the $p$-adic torsion. These invariants take values in the $p$-adic numbers and are constructed from a virtual pro-$p$ completion of the fundamental group. The key result of the article is an approximation theorem which shows that the $p$-adic invariants are limits of their classical analogues. This is reminiscent of Lück's approximation theorem for $L^2$-Betti numbers. After an investigation of basic properties and examples we discuss the $p$-adic analog of the Atiyah conject…

一些不纤维化于圆周的闭流形

Some closed manifolds that do not fibre over the circle

构造了在任意域上L^2-Betti数均为零但不虚纤维化于圆周的闭流形，基本群类最大，且维数可低至六。

The rational homology of real toric manifolds

This is an extended abstract for a talk given at the mini-workshop "Cohomology rings and fundamental groups of hyperplane arrangements, wonderful compactifications, and real toric varieties", held in Oberwolfach, September 30-October 6, 2012. We describe a formula (obtained in joint work with Alvise Trevisan) for computing the Betti numbers of real toric manifolds, based on a study of cohomology with coefficients in rank one local systems for various polyhedral products. Applying this formula, we recover Henderson's computation of the Betti numbers of real Hessenberg varieties. We conclude wit…

Macroscopic dimension and duality groups

We show that for a rationally inessential orientable closed $n$-manifold $M$ whose fundamental group $π$ is a duality group the macroscopic dimension of its universal cover is strictly less than $n$:$$ \dim_{MC}\Wi M<n.$$ As a corollary we obtain the following 0.1 Theorem. The inequality $ \dim_{MC}\Wi M<n$ holds for the universal cover of a closed spin $n$-manifold $M$ with a positive scalar curvature metric if the fundamental group $π_1(M)$ is a virtual duality group virtually satisfying the Analytic Novikov Conjecture.

基本群与具有有界关系长度的群表示

Fundamental groups and group presentations with bounded relator lengths

本文研究具有平凡第一贝蒂数的紧致测地空间在有限群等距作用下的直径比上界，并针对有限阿贝尔群的凯莱图，利用二维 CW 复形 $P_k(Γ(G,S))$ 给出拉普拉斯第一非平凡特征值、直径、切格常数和卡日丹常数的显式界。

Coarse fundamental groups and box spaces

We use a coarse version of the fundamental group first introduced by Barcelo, Kramer, Laubenbacher and Weaver to show that box spaces of finitely presented groups detect the normal subgroups used to construct the box space, up to isomorphism. As a consequence we have that two finitely presented groups admit coarsely equivalent box spaces if and only if they are commensurable via normal subgroups. We also provide an example of two filtrations $(N_i)$ and $(M_i)$ of a free group $F$ such that $M_i>N_i$ for all $i$ with $[M_i:N_i]$ uniformly bounded, but with $\Box_{(N_i)}F$ not coarsely equivale…

关于秩不超过七的光滑拟射影簇的 torsion-free nilpotent 基本群

On the torsion-free nilpotent fundamental groups of smooth quasi-projective varieties of rank up to seven

研究了光滑拟射影簇中 torsion-free nilpotent 基本群的结构，证明了其在特定秩条件下与某些 nilpotent Lie 群的同构关系，并回答了关于 nilpotent (quasi-)Kähler 群的问题。