Thom Complexes

THE spaces which form the title of this paper were introduced by Thorn in (16) as a tool in his study of differentiable manifolds. In addition certain special Thorn complexes have been studied by James (10) in connexion with Stiefel manifolds (cf. (8), (9)). The purpose of this paper is to prove a number of general results on Thorn complexes, and to deduce the main theorems of James (9), (10) as immediate consequences. Our main result (3.3) is a duality theorem (in the Whitehead-Spanier $-theory) for Thorn complexes over differentiable manifolds.f Besides its application this is a result of some independent interest, since it provides a satisfactory place for manifolds in $-theory. In § 1 we introduce, for a finite CW-complex X, a finite group J(X): the group of orthogonal sphere bundles over X under 'stable fibre homotopy equivalence'. If X is a sphere then J(X) is just the image of the stable 'J-homomorphism' in the appropriate dimension. A general study of J(X) by the methods of (2) will be given in a future publication. In § 2 we consider the $-type of Thorn complexes over X and examine the relation with the group J{X) of § 1. The main duality theorem is established in § 3. As an application we prove a result which was conjectured in (11): the stable fibre homotopy type of the tangent sphere bundle of a differentiable manifold X depends only on the homotopy type of X. The stunted projective spaces introduced by James in (10) are studied in § 4, and their identification as Thorn complexes is established. In § 5 corresponding results are proved for the quasi-projective spaces of (10). Applying the general results of §§ 1-3 to the spaces of §§ 4, 5 we deduce in § 6 the main theorems of James J (9), (10).