Some Aspects of Efficiency

Two remarks on Wall's D2 problem

“…In this notation, d F (R) is the minimum number of normal generators for R, and d G (R/[R.R]) is the minimum number of ZG-module generators for the module R ab . The survey articles of Harlander [13,14] provide some key examples (such as those constructed by Bridson and Tweedale [3]), and a guide to the literature. A connection to the D2 problem is provided by the following result: Theorem 3.2 (Dyer [13, Theorem 3.5]).…”

Quillen’s plus construction and the D(2) problem

“…That is, given a finite cohomologically 2-dimensional 3-complex X , if we can show that every finite 2-complex Y with 1 .Y / D 1 .X / satisfies .X / < .Y / then clearly X cannot be homotopy equivalent to any such Y . It has been shown that certain constructions involving presentations of a group would allow one to construct such a space [6,Theorem 3.5]. A candidate for such a space is given in [2].…”

“…This allows us to generalize existing approaches to the D(2) problem such as Johnson [8, Theorem I] and Harlander [6,Theorem 3.5].…”

On the $K$-theory and homotopy theory of the Klein bottle group

“…More information on the history of the geometric realization problem and the D(2)-problem can be found in the introduction of Johnson's book [10]. See also [6].…”