On CW-complexes over groups with periodic cohomology

Abstract

If G has 4-periodic cohomology, then D2 complexes over G are determined up to polarised homotopy by their Euler characteristic if and only if G has at most two one-dimensional quaternionic representations. We use this to solve Wall's D2 problem for several infinite families of non-abelian groups and, in these cases, also show that any finite Poincaré 3-complex X with π 1 (X) = G admits a cell structure with a single 3-cell. The proof involves cancellation theorems for ZG modules where G has periodic cohomology.

Cancellation for $(G,n)$-complexes and the Swan finiteness obstruction

“…However, a solution in the case where G is finite abelian, which includes non-cancellation examples, follows from work of Browning [3], Dyer-Sieradski [13] and Metzler [26], and further examples of non-cancellation have appeared elsewhere in the literature [12,22]. These examples are of special interest due to their applications to smooth 4-manifolds [2,18,21], Wall's D2 problem [20,31] and combinatorial group theory [27].…”

“…This work can be viewed as an attempt to properly amalgamate the techniques and results obtained by Swan in [41] with the wider literature on applications of the Swan finiteness obstruction [20,28,31]. As such, we will rely heavily on calculations done in [41], though we will give alternate proofs where possible.…”

“…Hence rP pG,nq s has cancellation, by Lemma 2.4.Lemma 14.3]. We will need the following lemma, the proof of which is contained in the proof of[31, Theorem 6.9].…”

“…23,, the conditions of[31, Theorem 4.1] are met and so rpI, r ´1q b P s has cancellation. By Proposition 2.2, this implies that rP s has cancellation.Type II.…”

“…If G satisfies the Eichler condition, then ZG has projective cancellation, i.e. rP s has cancellation for all projective ZG modules P .If σ k pGq " 0, then ZG is a representative of the finiteness obstruction and we know already from[31, Theorem 6.3] that rZGs has cancellation if and only if m H pGq ď 2. It therefore suffices to restrict our attention to those groups with σ k pGq ‰ 0 and m H pGq ‰ 0 and so, by Theorem 1.11, it remains to prove Theorem 2.1 in the case where G has type II with m H pGq " 1, 2 or type IV and m H pGq " 2.In order to deal with these cases, we will appeal to the following which is a consequence of[31, Theorem 4.1].…”