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Let a $\pi$-manifold be a manifold with the property that its normal bundle is trivial if it is embedded into $\mathbb R^n$ for large enough $n$.

Homotopy spheres are $\pi$-manifolds.

Here it is stated that the only spheres that admit a framing are $S^1, S^3, S^7$. Here it is stated that a framed manifold is a manifold with trivial normal bundle (therefore the same as a $\pi$-manifold).

These statements contradict each other. Which of the statements is wrong?

snailspace
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  • It seems like "framing" is used in a few related but not identical senses. – Qiaochu Yuan Jun 29 '13 at 10:12
  • The Wikipedia page does not say that a framed manifold is a manifold admitting a framing though. :) The notion discussed on ncat is the same as what Wikipedia calls parallelizability. The ncat page seems to have forgotten $S^0$ though. – fuglede Jun 29 '13 at 10:37

1 Answers1

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«Framing» means a trivialization of some vector bundle (tangent / stable tangent / normal for some embedding / stable normal). Naturally, existence of a framing depends on what bundle we're talking about.

Pages you're linking to explicitly state that

In one sense of the term, a framing of a manifold is a choice of trivialization of its tangent bundle

and

The term framed manifold (...) is most usually applied to an embedded manifold with a given trivialisation of the normal bundle

Grigory M
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