A Splitting Theorem for Manifolds and Surgery Groups

I. In this paper we state a splitting theorem for manifolds and describe some applications and consequences. We announced a weaker form of this result in August 1969 at the Georgia Topology Conference. Besides its immediate applications, especially in the classification of homotopy equivalent manifolds, it implies the existence of a Mayer-Vietoris sequence for surgery groups. Using this, many surgery groups can be systematically computed and more results on the classification of manifolds can be obtained. Details of the proofs and applications will be published elsewhere. This research was begun while I was working on my dissertation under the supervision of William Browder. His counsel and encouragement were invaluable and are deeply appreciated. Conversations with Friedhelm Waldhausen were also very useful. By manifold we will mean either a differentiable, piecewise linear or topological manifold. Where no restrictions are stated, all the propositions and terminology of this paper can be interpreted in any of these three contexts. Let X be a closed codimension one submanifold of F + 1 , a closed manifold, a n d / : W-+Y a homotopy equivalence of closed manifolds. We call ƒ "splittable" if it is homo topic to a map, which we will continue to cal l / , such that ƒ is transverse regular to X and f\f-X-*X a n d / | (W—f-X)—*Y—X are homotopy equivalences. If Y—Xhas two connected components, we will call them Fi and F2 and if it has one component we will let F i = Y—X and F2 = 0 . We call an inclusion of a group J? in a group G two-sided if the only double coset HzH, z&H, equal to its inverse, Hz~H, is the trivial double coset. For H normal in G this means G/H has no 2-torsion. If G is finite, H is two-sided in G if, and only if, H contains all the 2primary elements of G, In particular, any subgroup of a finite group of odd order is two-sided. If it\{X) —*TTI(Y) is an inclusion, it is two-