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Consider the following constant associated to smooth maps $F: S^{2n-1} \to S^n$ for $n \geq 2$:

Let $\omega \in \Omega^n(S^n)$ be a volume form with $\int_{S^n} \omega = 1$. Then there exists $\eta \in \Omega^{n-1}\left( S^{2n-1} \right)$ with $d\eta = F^*\omega$; define $$ h(F) = \int_{S^{2n-1}} \eta \wedge d\eta. $$

On a recent exam, I was given this definition and asked to show it is independent of choices of $\omega$ and $\eta$, invariant under smooth homotopy, and satisfies $h(F) = 0$ for odd $n$.

Is there a name for this constant? Does it have any important properties or show up in any interesting places?

Thanks!

tyo
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    Note that $\mathrm{d}(\eta\wedge \eta) = \mathrm{d}\eta\wedge \eta + (-1)^{n-1}\eta\wedge\mathrm{d}\eta$. As $(n-1)n$ is always even, $\mathrm{d}\eta\wedge \eta = \eta \wedge \mathrm{d}\eta$. Look at Stokes theorem when $n$ is odd. – Didier May 18 '21 at 20:10
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    This should be an incarnation of the Hopf invariant, though off the top of my head I don't recall how to prove the equivalence with the usual definition. – Eric Wofsey May 18 '21 at 20:31
  • I may expand this into an answer later, but I recently learned that it also measures the 3-skeleton obstruction to homotopy between 2-plane fields over a 3-manifold. – tyo May 12 '22 at 20:39

1 Answers1

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As Eric says, this integral equals the Hopf Invariant. This equality was first proven by Whitehead in

Whitehead, J. H. C., An expression of Hopf’s invariant as an integral, Proc. Natl. Acad. Sci. USA 33, 117-123 (1947). ZBL0030.07902.

when $n=2$ and then by Whitney in full generality; see sections 31, 33 in Chapter IV of

Whitney, Hassler, Geometric integration theory, Princeton Mathematical Series. Princeton, N. J.: Princeton University Press; London: Oxford University Press. XV, 387 p. (1957). ZBL0083.28204.

Moishe Kohan
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    There is also an earlier proof in J.H.C. Whitehead, "An Expression of Hopf's Invariant as an Integral", PNAS, vol. 33, no. 5 (1947), for the special case when $n = 2$. – JHF May 19 '21 at 13:21
  • @JHF: Right, I should have mentioned this. – Moishe Kohan May 19 '21 at 13:27