TY - CHAP
T1 - A survey of recent progress on some problems in 2-dimensional topology
AU - Harlander, Jens
N1 - Publisher Copyright:
© Cambridge University Press 2018.
PY - 2017/1/1
Y1 - 2017/1/1
N2 - Introduction A primary theme in algebraic topology is the characterization of the properties of a space or group in homological terms. Specific examples are Wall's questions concerning finiteness properties of spaces, or the Eilenberg-Ganea problem that compares the cohomological dimension of a group with its geometric dimension. Relevant to both examples is the question: can a partial resolution of length n be geometrically realized? Any partial resolution of length n is chain homotopically equivalent to one of length n with geometric (n −1)-skeleton. For n ≥ 3, the (n −1)-skeleton of the universal covering of a classifying space is simply connected, and this fact can be used to obtain a geometric realizations of the original partial resolution. For n = 2, however, simple connectivity fails and the realization question translates into difficult questions concerning presentations of groups. These questions are the topic of this chapter. We will introduce the geometric realization problem in dimension 2, the relation lifting problem and the relation gap problem. The first in this list of problems is the most fundamental, while the second and third can be stated purely in terms of combinatorial group theory. The final section in this chapter is an appendix jointly written by F. Rudolf Beyl and the author. It gives a brief overview of the state of the geometric realization problem for finite groups. This first chapter also serves as an introduction to two later chapters: Chapter 6 is a detailed account of the relation gap problem, and Chapter 7 discusses the relation gap problem in the context of free products. Basic definitions Let G be a group and g be a generating set for G. Let x be a set of letters and be a bijection. This bijection extends to a group epimorphism from the free group F = F (x) onto G. The kernel N of is called the relation group associated with the generating set g of G.
AB - Introduction A primary theme in algebraic topology is the characterization of the properties of a space or group in homological terms. Specific examples are Wall's questions concerning finiteness properties of spaces, or the Eilenberg-Ganea problem that compares the cohomological dimension of a group with its geometric dimension. Relevant to both examples is the question: can a partial resolution of length n be geometrically realized? Any partial resolution of length n is chain homotopically equivalent to one of length n with geometric (n −1)-skeleton. For n ≥ 3, the (n −1)-skeleton of the universal covering of a classifying space is simply connected, and this fact can be used to obtain a geometric realizations of the original partial resolution. For n = 2, however, simple connectivity fails and the realization question translates into difficult questions concerning presentations of groups. These questions are the topic of this chapter. We will introduce the geometric realization problem in dimension 2, the relation lifting problem and the relation gap problem. The first in this list of problems is the most fundamental, while the second and third can be stated purely in terms of combinatorial group theory. The final section in this chapter is an appendix jointly written by F. Rudolf Beyl and the author. It gives a brief overview of the state of the geometric realization problem for finite groups. This first chapter also serves as an introduction to two later chapters: Chapter 6 is a detailed account of the relation gap problem, and Chapter 7 discusses the relation gap problem in the context of free products. Basic definitions Let G be a group and g be a generating set for G. Let x be a set of letters and be a bijection. This bijection extends to a group epimorphism from the free group F = F (x) onto G. The kernel N of is called the relation group associated with the generating set g of G.
UR - http://www.scopus.com/inward/record.url?scp=85047513319&partnerID=8YFLogxK
U2 - 10.1017/9781316555798.002
DO - 10.1017/9781316555798.002
M3 - Chapter
AN - SCOPUS:85047513319
SN - 9781316600900
SP - 1
EP - 26
BT - Advances in Two-Dimensional Homotopy and Combinatorial Group Theory
ER -