Minimal algebraic complexes overD_4n

Abstract

We show that cancellation of free modules holds in the stable class 3 .Z/ over dihedral groups of order 4n. In light of a recent result on realizing k-invariants for these groups, this completes the proof that all dihedral groups satisfy the D(2) property.

Two remarks on Wall's D2 problem

“…This result is an application of [11, theorem 2•1]. The result was known for cyclic and dihedral groups (see [23,26,28]), but the argument given here is more uniform and the tetrahedral, octahedral and isosahedral groups do not seem to have been covered before. [33, theorem 4] shows that there exist finite D2-complexes X , with π 1 (X ) = G and χ(X ) = μ 2 (G) realizing this minimum value, for every finitely presented group G. Since μ 2 (G) 1 − def(G) by Swan [31, proposition 1], a necessary condition for any group G to have the D2-property is that μ 2 (G) = 1 − def(G).…”

“…This result is an application of [11, Theorem 2.1]. The result was known for cyclic and dihedral groups (see [23], [28], [26]), but the argument given here is more uniform and the tetrahedral, octahedral and isosahedral groups do not seem to have been covered before. Remark 1.2.…”

An exotic presentation of Q_28

“…To verify that every Y of minimal Euler characteristic is homotopy equivalent to E n,r for some r ∈ R n , it is sufficient to show that every minimal algebraic 2-complex over Z[Q 4n ] is homotopy equivalent to such a E n,r [13,15,19,20]. This task breaks down into two steps, paralleling the solution of the D(2)-problem for dihedral groups [12,15,18,27,23].…”

“…Note that without loss of generality such a 2-complex is (the Cayley complex of) a finite presentation of π 1 (Y ). It has been show that such a space Y cannot have certain fundamental groups such as cyclic groups, products of the form C ∞ × C n [7] or dihedral groups [12,15,18,27,23].…”

On CW-complexes over groups with periodic cohomology

“…m H (G) = 0. The work of W. J. Browning [3] can be applied in these cases and this has led to proofs of the D2 property for finite abelian groups [3], [5], [11], dihedral groups [25], the polyhedral groups T , O, I [15] (exhausting the finite subgroups of SO(3)) and various metacyclic groups [21], [33], [46]. It has also been shown for various infinite abelian groups [12], [13] and free groups [20].…”