Unoriented Geometric Functors

Farrell and Hsiang [2, p. 102] noticed that [5] implies that the geometric surgery groups defined in [6, Chapter 9] do not have the naturality Wall claims for them. Augmenting Wall’s definitions using spaces over RP∞ and line bundles they fixed the problem. The definition of geometric Wall groups involves homology with local coefficients and these also lack Wall’s claimed naturality. One would hope that a geometric bordism theory involving nonorientable manifolds would enjoy the same naturality as that enjoyed by homology with local Z coefficients. A setting for this naturality entirely in terms of local Z coefficients is presented in this paper. Applying this theory to the example of non-orientable Wall groups restores much of the elegance of Wall’s original approach. Furthermore, a geometric determination of the map induced by conjugation by a group element is given as well as a discussion of further cases beyond the reach of [5]. 1. A review of local coefficients A local coefficient system is a functor from the path groupoid, Π (X), of a space X to some category, [4, page 58]. Two coefficient systems are equivalent if there is a natural transformation between the two functors. In the case of interest here the category is the group Z and such a local coefficient system will be called a Z-system. Definition 1.1. For any space X, let Z (X) denote the category whose objects are Z-systems on X and whose morphisms are the natural transformations between them. The data for such a system on a space X can be packaged as a function Λ: Π (X) → {±1} which is a homomorphism of groupoids. One such system is the trivial Z-system which assigns +1 to every path. A natural transformation, or morphism, or just map, between Λ0 and Λ1 is a function ζ : X → {±1} such that for every path λ ∈ Π (X), Λ0(λ) · Λ1(λ) = ζ (