Manifolds with few cells and the stable homotopy of spheres

Smith, Larry

doi:10.1090/S0002-9939-1972-0296957-5

Manifolds with few cells and the stable homotopy of spheres
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Proc. Amer. Math. Soc. 31 (1972), 279-284 Request permission

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.

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