Manifolds with few cells and the stable homotopy of spheres

Abstract

Let f: Sn+k1 -+ Sn and form the complex V(f) = Sn V Sk Uf+[ifn,ik] en+k where it E jrT(SI) is the canonical generator and [ , ] denotes Whitehead product. The complex V(f) is a Poincare duality complex. Under the assumption that f is in the stable range we show that V(f) has the homotopy type of a smooth, combinatorial or topological manifold iff the mapf lies in the image of the 0, PL or Top J-homomorphism respectively. Let VrS denote the stable homotopy groups of spheres and suppose that e E 77s We may represent p by a map f: Sn+k-1 Sn, and form the complex X(f) = Sn uf en+k. By studying the complex X(f) we may often detect that p $ 0. For example in this way one may study the Hopf invariant, ec-invariant, etc. [1], [2], [9]. We may also form the complex V(f) = Sn V Sk Uf+[i ,ik3 en+k where it E 77t(St) is the canonical generator. The complex V(f) is in fact a Poincare duality complex, and it is therefore to be hoped that by asking questions about smoothing V( f) we may discover information about the element p and vice versa. The present note is devoted to a small stab in this direction. Our main result is the following: THEOREM. Suppose that p E 77e 1. Represent p by a mapf: Sn+k-_ S n >> k, andform the complex V(f) = Sn V Sk Uf?[if,ik] en+k. Then V( f ) has the homotopy type of a closed topological, combinatorial or smooth manifold, if 9 lies in the image of the homomorphism JTOP, JPL or JO respectively. We make no claim for originality for the preceding theorem. The result should be well known on the basis of the results of [5], but it does Received by the editors November 6, 1970. AMS 1969 subject classifcations. Primary 5540, 5560.