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 Poincare's conjecture and the homeotopy group of a closed, orientable 2-manifold

1974 ◽  
Vol 17 (2) ◽  
pp. 214-221 ◽  
Author(s):  
Joan S. Birman
Keyword(s):  
Dimensional Manifold ◽  
3 Dimensional ◽  
Dimension 2 ◽  
Poincare Conjecture ◽  
Simply Connected

In 1904 Poincaré [11] conjectured that every compact, simply-connected closed 3-dimensional manifold is homeomorphic to a 3-sphere. The corresponding result for dimension 2 is classical; for dimension ≧ 5 it was proved by Smale [12] and Stallings [13], but for dimensions 3 and 4 the question remains open. It has been discovered in recent years that the 3-dimensional Poincaré conjecture could be reformulated in purely algebraic terms [6, 10, 14, 15] however the algebraic problems which are posed in the references cited above have not, to date, proved tractable.

Pro-p completions of groups of cohomological dimension 2

2016 ◽  
Vol 26 (03) ◽  
pp. 551-564
Author(s):  
Dessislava H. Kochloukova
Keyword(s):  
Free Group ◽  
Cohomological Dimension ◽  
Finitely Generated ◽  
Limit Group ◽  
Finitely Presented Group ◽  
Dimension Formula ◽  
Dimension 2 ◽  
Finitely Presented

We study when an abstract finitely presented group [Formula: see text] of cohomological dimension [Formula: see text] has pro-[Formula: see text] completion [Formula: see text] of cohomological dimension [Formula: see text]. Furthermore, we prove that for a tree hyperbolic limit group [Formula: see text] we have [Formula: see text] and show an example of a hyperbolic limit group [Formula: see text] that is not free and [Formula: see text] is free pro-[Formula: see text]. For a finitely generated residually free group [Formula: see text] that is not a limit group, we show that [Formula: see text] is not free pro-[Formula: see text].

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].

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.

scholarly journals Triangulations with few vertices of manifolds with non-free fundamental group

2019 ◽  
Vol 149 (6) ◽  
pp. 1453-1463
Author(s):  
Petar Pavešić
Keyword(s):  
Fundamental Group ◽  
Lower Bounds ◽  
Category Theory ◽  
Homology Sphere ◽  
Dimensional Manifold ◽  
Lusternik Schnirelmann ◽  
Few Vertices

AbstractWe study lower bounds for the number of vertices in a PL-triangulation of a given manifold M. While most of the previous estimates are based on the dimension and the connectivity of M, we show that further information can be extracted by studying the structure of the fundamental group of M and applying techniques from the Lusternik-Schnirelmann category theory. In particular, we prove that every PL-triangulation of a d-dimensional manifold (d ⩾ 3) whose fundamental group is not free has at least 3d + 1 vertices. As a corollary, every d-dimensional homology sphere that admits a combinatorial triangulation with less than 3d vertices is PL-homeomorphic to Sd. Another important consequence is that every triangulation with small links of M is combinatorial.

Correlation based swarm trackers for 3-dimensional manifold mesh formation

2009 ◽  
Author(s):  
Charles Casey ◽  
Laurence G. Hassebrook ◽  
Priyanka Chaudhary
Keyword(s):  
Dimensional Manifold ◽  
3 Dimensional
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