scholarly journals On the Berstein–Svarc theorem in dimension 2

2009 ◽  
Vol 146 (2) ◽  
pp. 407-413 ◽  
Author(s):  
ALEXANDER N. DRANISHNIKOV ◽  
YULI B. RUDYAK
Keyword(s):  
Fundamental Group ◽  
Cohomological Dimension ◽  
Dimension 2 ◽  
Orientable Manifolds

AbstractWe prove that for any group π with cohomological dimension at least n the nth power of the Berstein class of π is nontrivial. This allows us to prove the following Berstein–Svarc theorem for all n:Theorem. For a connected complex X with dim X = cat X = n, we have$\ber_X^n$ ≠ 0 where$\ber_X$is the Berstein class of X.Previously it was known for n ≥ 3.We also prove that, for every map f: M → N of degree ±1 of closed orientable manifolds, the fundamental group of N is free provided that the fundamental group of M is.

A group with zero homology

1968 ◽  
Vol 64 (3) ◽  
pp. 599-602 ◽  
Author(s):  
D. B. A Epstein
Keyword(s):  
Fundamental Group ◽  
Cohomological Dimension ◽  
Dimensional Manifold ◽  
Finitely Generated ◽  
3 Dimensional ◽  
Identity Group ◽  
Dimension 2

In this paper we describe a group G such that for any simple coefficients A and for any i > 0, Hi(G; A) and Hi(G; A) are zero. Other groups with this property have been found by Baumslag and Gruenberg (1). The group G in this paper has cohomological dimension 2 (that is Hi(G; A) = 0 for any i > 2 and any G-module A). G is the fundamental group of an open aspherical 3-dimensional manifold L, and is not finitely generated. The only non-trivial part of this paper is to prove that the fundamental group of the 3-manifold L, which we shall construct, is not the identity group.

scholarly journals Minimal 4-manifolds for groups of cohomological dimension 2

1994 ◽  
Vol 37 (3) ◽  
pp. 455-461
Author(s):  
Jonathan A. Hillman
Keyword(s):  
Fundamental Group ◽  
Cohomological Dimension ◽  
Augmentation Ideal ◽  
Euler Characteristics ◽  
Canonical Orientation ◽  
Dimension 2 ◽  
Mac Lane

We show that if π is a group with a finite 2-dimensional Eilenberg-Mac Lane complex then the minimum of the Euler characteristics of closed 4-manifolds with fundamental group π is 2χ(K(π, 1)). If moreover M is such a manifold realizing this minimum then π2(M) ≅ Similarly, if π is a PD3-group and w1(M) is the canonical orientation character of π then χ(M)≧l and π2(M) is stably isomorphic to the augmentation ideal of Z[π].

BOUNDARY OF CAT(-1) COXETER GROUPS

1999 ◽  
Vol 09 (02) ◽  
pp. 169-178 ◽  
Author(s):  
N. BENAKLI
Keyword(s):  
Coxeter Groups ◽  
Cohomological Dimension ◽  
Existence Problem ◽  
Joint Work ◽  
Coxeter Graph ◽  
Combinatorial Properties ◽  
Jsj Decomposition ◽  
Complete Bipartite Graphs ◽  
Dimension 2 ◽  
Virtual Cohomological Dimension

In this paper, we study the topological properties of the hyperbolic boundaries of CAT(-1) Coxeter groups of virtual cohomological dimension 2. We will show how these properties are related to combinatorial properties of the associated Coxeter graph. More precisely, we investigate the connectedness, the local connectedness and the existence problem of local cut points. In the appendix, in a joint work with Z. Sela, we will construct the JSJ decomposition of the Coxeter groups for which the corresponding Coxeter graphs are complete bipartite graphs.

Zariski Hyperplane Section Theorem for Grassmannian Varieties

2003 ◽  
Vol 55 (1) ◽  
pp. 157-180 ◽  
Author(s):  
Ichiro Shimada
Keyword(s):  
Fundamental Group ◽  
Central Extension ◽  
Projective Variety ◽  
Hyperplane Section ◽  
General Linear ◽  
Linear Automorphism ◽  
Section Theorem ◽  
Dimension 2

AbstractLet ϕ: X → M be a morphism from a smooth irreducible complex quasi-projective variety X to a Grassmannian variety M such that the image is of dimension ≥ 2. Let D be a reduced hypersurface in M, and γ a general linear automorphism of M. We show that, under a certain differentialgeometric condition on ϕ(X) and D, the fundamental group π1((γ ○ ϕ)−1 (M \ D)) is isomorphic to a central extension of π1(M \ D) × π1(X) by the cokernel of π2(ϕ) : π2(X) → π2(M).

scholarly journals Topological 4-manifolds with right-angled Artin fundamental groups

2019 ◽  
Vol 11 (04) ◽  
pp. 777-821
Author(s):  
Ian Hambleton ◽  
Alyson Hildum
Keyword(s):  
Fundamental Group ◽  
Cohomological Dimension ◽  
Fundamental Groups ◽  
Connected Sum ◽  
Dimension Formula ◽  
Assembly Map ◽  
Torsion Free

We classify closed, spin[Formula: see text], topological [Formula: see text]-manifolds with fundamental group [Formula: see text] of cohomological dimension [Formula: see text] (up to [Formula: see text]-cobordism), after stabilization by connected sum with at most [Formula: see text] copies of [Formula: see text]. In general, we must also assume that [Formula: see text] satisfies certain [Formula: see text]-theory and assembly map conditions. Examples for which these conditions hold include the torsion-free fundamental groups of [Formula: see text]-manifolds and all right-angled Artin groupswhose defining graphs have no [Formula: see text]-cliques.

Sign in / Sign up

Export Citation Format

Share Document