Manifolds with Few Cells and the Stable Homotopy of Spheres

Abstract

Let $f:{S^{n + k - 1}} \to {S^n}$ and form the complex $V(f) = {S^n} \vee {S^k}{ \cup _{f + [{i_n},{i_k}]}}{e^{n + k}}$ where ${i_t} \in {\pi _t}({S^t})$ is the canonical generator and [ , ] denotes Whitehead product. The complex $V(f)$ is a Poincaré 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 map $f$ lies in the image of the $O$, PL or Top $J$-homomorphism respectively.