Milnor showed that if the Euler class of an $S^3$ bundle over $S^4$ is $\pm 1$, then the total space is a homotopy sphere. How many $S^3$ bundles over $S^4$ do we have with the total space is homotopic (i.e. homeomorphic) to $S^7$, if known? (or any related reference)
total spaces of S3 bundles over S4 which are homotopic to S7
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general-topology
algebraic-topology
differential-topology
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0we know $\pi_3(SO(4))$= Z + Z so there are infinitely many different such bundles, but are all the total spaces of these bundles homeomorphic to S7? i.e. how may I compute the homology groups? – 2012-06-23