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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Manifolds with few cells and the stable homotopy of spheres


Author: Larry Smith
Journal: Proc. Amer. Math. Soc. 31 (1972), 279-284
MSC: Primary 55E45; Secondary 57C10
DOI: https://doi.org/10.1090/S0002-9939-1972-0296957-5
MathSciNet review: 0296957
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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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DOI: https://doi.org/10.1090/S0002-9939-1972-0296957-5
Keywords: Smoothing Poincare complexes, the $ J$-homomorphism
Article copyright: © Copyright 1972 American Mathematical Society

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