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We construct families of non-trivial universal rigid secondary classes for foliations, and then discuss their application to prove that foliations are not homotopic. An observation of Lawson about the non-triviality of the normal Pontrjagin classes of foliations is extended, and then used to construct new families of examples of foliations with non-trivial rigid secondary classes. Examples are given of (abstractly constructed) foliations of compact manifolds with homotopic tangent bundles, but which are not homotopic as foliations.
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Steven Hurder (2024) studied this question.
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