The novikov conjecture and low-dimensional topology

Abstract

It is well known that many interesting manifolds can be obtained by cutting some standard manifold along a separating codimension-one submanifold and glueing the pieces together by a homeomorphism (or diffeomorphism) homotopic to the identity. (These manifolds will be simple homotopy equivalent.) For instance, in dimension at least five, all smooth homotopy spheres can be obtained from the standard sphere in this fashion. In [We 1] we studied the question of identifying those homotopy equivalences that can be so produced as well as related problems. In high dimensions, e.g. dimension at least five, a complete solution is often possible. Through dimension three, the result is trivial, no new homotopy equivalences are obtainable since for surfaces homeomorphisms homotopic to the identity are in fact isotopic to the identity. Thus, a gap is left in our knowledge in dimension four. It is known that for a large class of three-manifolds, homotopy implies isotopy for homeomorphisms, and no example is known of this failing for any threemanifold. This suggests that the situation for four-manifolds should be no different from that for three manifolds and should therefore be very different from the higher dimensional theory. Most of this paper is devoted to studying homotopy equivalences h : S 1 x L1 S~ where L1 and L2 are classical lens spaces. In w we review enough of [We 1] to get the flavor of the high dimensional theory and see why it would be anomalous for h not to be obtainable by cutting and pasting. (In fact, h x Is, : T 2 x Lx---~ T2xL2 can be obtained in such a manner.) In w we show by low dimensional techniques that if the codimension-one (three-) manifold cut along lies in the Poincar6 category then h cannot result. In w we remove this restriction by an algebraic technique that also shows that many other homotopy equivalences are not cut-pastable. It is in this algebra (w that the Novikov conjecture enters as an ingredient in calculating the image of the L-theory of three-manifold groups in the L-theory of a certain class of groups. This paper is an extension of part of the author's thesis. It is a pleasure to