Stable Modules and the D(2)-Problem

This 2003 book deals with two fundamental problems in low-dimensional topology with an eye on wider context.

F. E. A. Johnson (Author)

9780521537490, Cambridge University Press

Paperback, published 11 September 2003

280 pages, 20 b/w illus.
22.9 x 15.2 x 1.6 cm, 0.42 kg

"This book is well-written, nicely organized, and is a pleasure to read." Mathematical Reviews, American Mathematical Society

This 2003 book is concerned with two fundamental problems in low-dimensional topology. Firstly, the D(2)-problem, which asks whether cohomology detects dimension, and secondly the realization problem, which asks whether every algebraic 2-complex is geometrically realizable. The author shows that for a large class of fundamental groups these problems are equivalent. Moreover, in the case of finite groups, Professor Johnson develops general methods and gives complete solutions in a number of cases. In particular, he presents a complete treatment of Yoneda extension theory from the viewpoint of derived objects and proves that for groups of period four, two-dimensional homotopy types are parametrized by isomorphism classes of projective modules. This book is carefully written with an eye on the wider context and as such is suitable for graduate students wanting to learn low-dimensional homotopy theory as well as established researchers in the field.

1. Orders in semisimple algebras
2. Representation of finite groups
3. Stable modules and cancellation theorems
4. Relative homological algebra
5. The derived category of a finite group
6. k-invariants
7. Groups of periodic cohomology
8. Algebraic homotopy theory
9. Stability theorems
10. The D(2)-problem
11. Poincaré 3-complexes.

Subject Areas: Topology [PBP]