Unoriented geometric functors

Abstract

Farrell and Hsiang [Farrell F. T. and Hsiang W. C.: Rational L-groups of Bieberbach groups. Comment. Math. Helv. 52 (1977), 89–109. MR 0448372 (56 #6679), p. 102] noticed that [Taylor Laurence R.: Surgery groups and inner automorphisms. Algebraic K-theory, III: Hermitian K-theory and geometric applications (Proc. Conf. Battelle Memorial Inst., Seattle, Wash., 1972). Springer, Berlin 1973, pp. 471–477. Lecture Notes in Math. 343. MR 0405460 (53 #9253)] implies that the geometric surgery groups defined in [Wall C. T. C.: Surgery on compact manifolds. Academic Press, London 1970. London Mathematical Society Monographs, No. 1. MR 0431216 (55 #4217), 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 non-orientable 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 [Taylor Laurence R.: Surgery groups and inner automorphisms. Algebraic K-theory, III: Hermitian K-theory and geometric applications (Proc. Conf. Battelle Memorial Inst., Seattle, Wash., 1972). Springer, Berlin 1973, pp. 471–477. Lecture Notes in Math. 343. MR 0405460 (53 #9253)].

2000 Mathematics Subject Classification: 55N25, 55N20.