Skip to main content
MathOverflow

Return to Answer

added 387 characters in body
Source Link
Igor Belegradek
Igor Belegradek
  • 29.1k
  • 2
  • 80
  • 176

It is proved in [Kaminker, J., Proc. Amer. Math. Soc. 41 (1973), 305–308] that the tangent sphere bundle of a closed smooth H-manifold is (unstably) fibre homotopy trivial. On other other hand, surgery theory allows to construct H-manifolds with non-trivial rational Pontrjagin classes, see e.g. [Victor Belfi, Pacific J. Math. Volumevol 36, Number 3 (1971), 615-621] but this is a standard surgery-theoretic argument.

Finally, in [Milnor-Spanier, Pacific J. Math. vol 10, Number 2 (1960), 585-590] it is shown that the tangent bundle to $S^n$ is fiber homotopy trivial if and only if $n=1,3,7$, which are exactly the parallelizable spheres (by Bott-Milnor).

Thus the above is a complete answer to your question. This phenomenon you ask for never occus for spheres but occurs for lots of H-manifolds.

It is proved in [Kaminker, J., Proc. Amer. Math. Soc. 41 (1973), 305–308] that the tangent sphere bundle of a closed smooth H-manifold is (unstably) fibre homotopy trivial. On other other hand, surgery theory allows to construct H-manifolds with non-trivial rational Pontrjagin classes, see e.g. [Victor Belfi, Pacific J. Math. Volume 36, Number 3 (1971), 615-621] but this is a standard surgery-theoretic argument.

It is proved in [Kaminker, J., Proc. Amer. Math. Soc. 41 (1973), 305–308] that the tangent sphere bundle of a closed smooth H-manifold is (unstably) fibre homotopy trivial. On other other hand, surgery theory allows to construct H-manifolds with non-trivial rational Pontrjagin classes, see e.g. [Victor Belfi, Pacific J. Math. vol 36, Number 3 (1971), 615-621] but this is a standard surgery-theoretic argument.

Finally, in [Milnor-Spanier, Pacific J. Math. vol 10, Number 2 (1960), 585-590] it is shown that the tangent bundle to $S^n$ is fiber homotopy trivial if and only if $n=1,3,7$, which are exactly the parallelizable spheres (by Bott-Milnor).

Thus the above is a complete answer to your question. This phenomenon you ask for never occus for spheres but occurs for lots of H-manifolds.

deleted 4 characters in body
Source Link
Igor Belegradek
Igor Belegradek
  • 29.1k
  • 2
  • 80
  • 176

It is proved in [Kaminker, J., Proc. Amer. Math. Soc. 41 (1973), 305–308] that the tangent sphere bundle of a connected, closed smooth H-manifold is (unstably) fibre homotopy trivial. On other other hand, surgery theory allows to construct H-manifolds with non-trivial rational Pontrjagin classes, see e.g. [Victor Belfi, Pacific J. Math. Volume 36, Number 3 (1971), 615-621] but this is a now standard surgery-theoretic argument.

It is proved in [Kaminker, J., Proc. Amer. Math. Soc. 41 (1973), 305–308] that the tangent sphere bundle of a connected, closed smooth H-manifold is fibre homotopy trivial. On other other hand, surgery theory allows to construct H-manifolds with non-trivial rational Pontrjagin classes, see e.g. [Victor Belfi, Pacific J. Math. Volume 36, Number 3 (1971), 615-621] but this is a now standard surgery-theoretic argument.

It is proved in [Kaminker, J., Proc. Amer. Math. Soc. 41 (1973), 305–308] that the tangent sphere bundle of a closed smooth H-manifold is (unstably) fibre homotopy trivial. On other other hand, surgery theory allows to construct H-manifolds with non-trivial rational Pontrjagin classes, see e.g. [Victor Belfi, Pacific J. Math. Volume 36, Number 3 (1971), 615-621] but this is a standard surgery-theoretic argument.

Source Link
Igor Belegradek
Igor Belegradek
  • 29.1k
  • 2
  • 80
  • 176

It is proved in [Kaminker, J., Proc. Amer. Math. Soc. 41 (1973), 305–308] that the tangent sphere bundle of a connected, closed smooth H-manifold is fibre homotopy trivial. On other other hand, surgery theory allows to construct H-manifolds with non-trivial rational Pontrjagin classes, see e.g. [Victor Belfi, Pacific J. Math. Volume 36, Number 3 (1971), 615-621] but this is a now standard surgery-theoretic argument.