Surgery up to homotopy equivalence for nonpositively curved manifolds

Abstract

Let M' be a smooth closed manifold which admits a metric of nonpositive curvature. We show, using a theorem of Farrell and Hsiang, that if n + k ? 6, then the surgery obstruction map [M X D , a; G/TOP] Lh+k(7TIM, w1(M)) is injective, where Lh are the obstruction groups for surgery up to homotopy equivalence. Farrell and Hsiang have shown that if Mn is a closed nonpositively curved Riemannian manifold and n + k > 6, then the surgery map [Mn X Dk , a; G/TOP, *] -*Ln+k(7T1M, wI(M)) for topological surgery in a split monomorphism [1]. This note considers the implications of their result of h-surgery, i.e. surgery up to homotopy equivalence rather than simple homotopy equivalence. LEMMA. Let QP be a compact manifold such that Os: [QP, a; G/TOP, *] Ls(7T,Q, wl(Q)) is monic, let Nn be a compact, connected submanifold of QP which has a normal microbundle vQ(N), and let f: UP -* QP be a simple homotopy equivalence which is a homeomorphism near the boundary. Then any normal map f Iv: V= f -(N) N induced by making f transverse to N is normally cobordant to a homeomorphism. PROOF. Recall that the long exact sequence of topological surgery sa(QP x Dka)i DQP X Dk a; G/TOP, *] -4LP+k(lQ, WI(Q)) 8a(QP X Dk ) -Q, _) [Q a; G/TOP, * ] -LP(7T1Q, WI(Q)) (where a = s or h denotes surgery up to simple homotopy equivalence or up to homotopy equivalence, respectively) is a long exact sequence of groups and group homomorphisms (see [4]; also see [2] for surgery in the topological category). Let x = [ f: U Q] E Ss(Q, a) and let i: N Q be the inclusion map. Then the element [f Iv: V N] E [N, a; G/TOP, * ] defined above is i*(,qs(x)), where i*: [Q, a; G/TOP, * ] -* [N, a; G/TOP, * ]. Since Os is a monomorphism, image('q5) = 0 in [Q, a; G/TOP, *], and since i* is a homomorphism of groups we see that Received by the editors December 15, 1982 and, in revised form, August 15, 1983. 1980 Mathematics Subject Classification. Primary 57R67, 57R65. C1984 American Mathematical Society 0002-9939/84 $1.00 + $.25 per page