Manifolds with Few Cells and the Stable Homotopy of Spheres

Abstract

Let f-.S"^-1 ->■ S" and form the complex V(f) = 5" v Sk U/+[in,ij.]en+k where it£TTt(S{) is the canonical generator and [ , ] denotes Whitehead product.The complex V(f) is a Poincare duality complex.Under the assumption that / is in the stable range we show that V(f) has the homotopy type of a smooth, combinatorial or topological manifold iff the map/lies in the image of the O, PL or Top /-homomorphism respectively.Let 7rf denote the stable homotopy groups of spheres and suppose that <p e 7Tk_y.We may represent cp by a map f:Sn+lc~x -*■ Sn, and form the complex X(f) = S" Ur en+k.By studying the complex X(f) we may often detect that cp ^ 0. For example in this way one may study the Hopf invariant, ec-invariant, etc. [1], [2], [9].We may also form the complexwhere it e -n-fiS*) 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 cp and vice versa.The present note is devoted to a small stab in this direction.Our main result is the following:Theorem.Suppose that cp e 7rf_x.Represent cpbya map f: S^-1 -> Sn' n » k, and form the complexThen V(f) has the homotopy type of a closed topological, combinatorial or smooth manifold, iff cp lies in the image of the homomorphism JTop, /pl or J0 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

Locations

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