Homology, Homotopy and Applications, vol. 13(2), 2011, pp.63–72 ON THE K -THEORY AND HOMOTOPY THEORY OF THE KLEIN BOTTLE GROUP JENS HARLANDER and ANDREW MISSELDINE (communicated by Graham Ellis) Abstract We construct infinitely many chain homotopically distinct algebraic 2-complexes for the Klein bottle group and give vari- ous topological applications. We compare our examples to other examples in the literature and address the question of geometric realizability. 1. Introduction Let G be a group. A (G,n)-complex X is a finite n-dimensional CW-complex with π 1 (X)= G and π i (X) = 0 for 2  i<n. Its directed Euler characteristic is the alternating sum χ(X)= c n − c n−1 + ···± c 0 , where c i is the number of i-cells. An algebraic (G,n)-complex X is an exact sequence ZG m n → ZG m n−1 →···→ ZG m 0 → Z → 0. Thus, an algebraic (G,n)-complex is a partial free resolution of the trivial module Z of length n. Its directed Euler characteristic is defined as the alternating sum χ(X )= m n − m n−1 + ···± m 0 . Note that the augmented cellular chain complex C ∗ (  X) → Z → 0 of the universal covering  X of a (G,n)-complex X is an algebraic (G,n)- complex. The geometric realization problem poses the question: Given an algebraic (G, 2)-complex X , does there exist a (G, 2)-complex X so that X and C ∗ (  X) → Z → 0 are chain homotopy equivalent? The realization question came out of work of Wall [21]. Closely related to the geometric realization problem is Wall’s D(2) problem: Suppose that X is a finite 3-complex such that H 3 (  X, Z)= H 3 (X, B) = 0 for all local coefficient systems B on X. Is X homotopy equivalent to a finite 2-complex? In [10, appendix B], Johnson proved the following realization theorem: Let G be a finitely presented group of type FL(3); then the D(2)-property holds for G if and only if every algebraic (G, 2)-complex admits a geometric realization. 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]. The theorem below is the main contribution of this article. Received January 5, 2011, revised April 24, 2011; published on September 21, 2011. 2000 Mathematics Subject Classification: 57M20, 57M05, 20C07, 18G35. Key words and phrases: 2-complex, algebraic 2-complex, chain complex, stably free module, relation module, geometric realization. Article available at http://intlpress.com/HHA/v13/n2/a5 and doi:10.4310/HHA.2011.v13.n2.a5 Copyright c  2011, International Press. Permission to copy for private use granted.