Necessary and sufficient conditions for CW complexes to be homotopy equivalent to total spaces of S^{2k-1}-fibrations over S^{2k}, with order of attaching maps determined and homotopy types classified by stable classes for k not equal to 2 or 4.
Homotopy classification of $S^{2k-1}$-bundles over $S^{2k}$
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abstract
In this paper, we classify the homotopy types of the total spaces of $S^{2k-1}$-bundles (or fibrations) over $S^{2k}$ for $2\leq k\leq 6$. One of the two key new ingredients in the argument is the new necessary and sufficient conditions for a CW complex to be homotopy equivalent to the total space of a sphere bundle (fibration); the other is a formula relating the attaching map of the top cell of the total space and the characteristic map of a sphere bundle for $k=2,4$. When $k=4$, the classification results provide a negative answer to the conjecture in [6].
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Complexes equivalent to $S^{2k-1}$-fibrations over $S^{2k}$
Necessary and sufficient conditions for CW complexes to be homotopy equivalent to total spaces of S^{2k-1}-fibrations over S^{2k}, with order of attaching maps determined and homotopy types classified by stable classes for k not equal to 2 or 4.