In this chapter, we introduce Fox ideals Im(G0) for each m ≥ 0 by means of finitely generated free partial resolutions of the group G, and describe a practical method for computing Im(G) using representations of G. Fox ideals provide powerful tests for determining the rank of G (for m = 1), the deficiency of G (m = 2) and the homological dimension of G (m ≥ 2). These are derived in §§1–2, and some (partly new) applications are given.

N-torsion groups Nm(G), introduced in §3, are the direct analogues of the Whitehead group Wh(G), with ℤG replaced by ℤG/Im(G). In general, however, Nm(G) turns out to be much richer than Wh(G), and a practical evaluation method is described (using again representations of G), which yields non-trivial values in a large variety of cases. For m = 1, they distinguish Nielsen equivalence classes of generating systems of G, and hence isotopy or homeomorphy classes of Heegaard splitting of 3-manifolds (see §4). For m = 2 (or even m ≥ 2), N-torsion provides a crucial tool for the distinction of (simple)-homotopy classes of m-dimensional cell complexes (see §5). Here Fox ideals and N-torsion values are direct and natural generalizations of the bias modulus and the bias invariant respectively, as introduced in Chapter III (see 5.4 below).

Fox ideals and N-torsion and their preliminary versions have been discovered in different contexts and with varying degree of generality by several authors; see, for example, [Dy85], [Ho-AnLaMe91], [Ho-An88], [LuMo91] or [Me90]. Our treatment here is closest to [Lu9l2], where the independence from particular resolutions of G was achieved through matrix representations, and Im(G) and Nm(G) have been established as group theoretic invariants.