Is paralellizability a topological invariant (Invariant under homemorphism)

general-topology manifolds vector-bundles topological-k-theory

This MO post is a motivation to ask:

Is paralellizability a topological invariant (Invariant under homeomorphism)?


Solution 1:

No. You can cook up counterexamples in high dimensions via surgery theory. The relevant paper here is Milnor, "Microbundles and differentiable structures", available here. The theorem you want is Corollary 6.4; as a corollary of this, Milnor gives an example of a (noncompact) smooth parallelizable manifold that carries a different, non-parallelizable differentiable structure.

You should be able to find closed examples using this corollary, but I don't know any off the top of my head.

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