THE REDUCIBILITY OF THOM COMPLEXES AND SURGERY ON MAPS OF DEGREE d

1. Statement of results. In this paper we announce a substantial extension of the obstruction theory developed in [ l ] . Detailed proofs and further properties will appear elsewhere. Basically what we have accomplished is to put the theory in [l ] in a more natural setting and extend it to the metastable range. We also apply it to the question of when the Thorn complex JT(£) of a vector bundle £ over a manifold is reducible and hence get an obstruction theory for imbedding 1-connected manifolds in the metastable range. All manifolds will be oriented, compact, and differentiable. Bundles are also assumed to be oriented. Let 6 denote the trivial g-plane bundle over a space and set wr,t =TrS ~. THEOREM 1 ( T H E REDUCIBILITY THEOREM). Let % be a k-plane bundle over a closed connected manifold M and assume 2k^n+3. Then (a) The Thorn complex T(£@0) is reducible whenever a sequence of obstructions öt(£) £i?*(ikf ; 7rw+&,t_i), Ki^n, vanishes. In the stable range, k^n-^-3, the converse is also true, namely, the reducibility of T(%®6) implies the vanishing of the obstructions 0»(£). (b) Let s^.H^M; Tn+kj-^-^H^M; Tn+k+i,i-i) be the homomorphism induced by the suspension homomorphism 7rw+fc,»-i—>7rn+k+i,i-iThen *>.•«)) =o&®ffi).