11 - Poincaré 3-complexes

Summary

A finite Poincaré complex M of dimension n is said to be of standard form when it can be described thus

where K is a finite complex of dimension ≤ n -1 and α : S^n-l → K is a continuous map. As we described in the Introduction, Wall showed in [72] that, when n ≤ 4, every finite Poincaré n-complex is homotopy equivalent to one in standard form. Since the question in dimension 3 was the genesis of the D(2)-problem, it is appropriate to conclude our investigation of the D(2)-problem for finite fundamental groups by asking how far it takes us in the direction of obtaining standard forms in dimension 3. We prove:

Theorem V: Let G be a finite group; then G has a standard Poincare 3-form if and only if there is a finite presentation G of G with. Moreover, G then necessarily has free period 4, and the presentation G is automatically balanced.

If G is a finite group of free period 4, which also has the D(2)-property, it is straightforward to see that there exists a finite Poincare 3-complex M of standard form such that π₁(M) ≌ G. However, the intransigence of the D(2)-problem suggests the possibility that Poincaré 3-complexes might need to have more than one 3-cell. To an extent, this is supported by a consideration of classical examples. On general grounds, it can be shown that smooth 3-manifolds admit cellular representations with just one top-dimensional cell. Nevertheless, their natural representations are often more complicated.