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In Milnor's book Topology from the Differentiable Viewpoint, there is a problem concerning the definition of the Hopf invariant. Let $y\neq z$ be regular values of a smooth map $f:S^{2p-1}\to S^p$, then we want to show that the linking number $\ell(f^{-1}(y),f^{-1}(z))$ is locally constant as a function of $y$. The Hopf invariant of $f$ is then defined as $\ell(f^{-1}(y),f^{-1}(z))$, after several more parts of this exercise showing that this quantity only depends on the homotopy class of $f$.

Here, the linking number $\ell(M,N)$ for compact, oriented, boundaryless submanifolds $M^m,N^n$ of $S^{m+n+1}$ is defined by picking some $p\in S^{m+n+1}\setminus(M\cup N)$, identifying $S^{m+n+1}$ with $\mathbb R^{m+n+1}$. The linking number is then defined by the degree of the map $\lambda:M\times N\to S^{m+n}$ given by $$\lambda(x,y)=\frac{x-y}{\|x-y\|}.$$

My idea was to use the framed cobordism theory outlined in $\S7$. For if we choose a neighborhood $U$ of $y$ consisting of regular values of $f$ with $z\in U$, and if $y_0\in U$, then $f^{-1}(y)$ is framed cobordant to $f^{-1}(y_0)$. But I don't know where to go from here.

Any hints about how I should proceed would be greatly appreciated.

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  • $\begingroup$ You don't need anything so fancy. Just think about what you know about degrees of homotopic maps. $\endgroup$ – Ted Shifrin Feb 20 '18 at 3:52
  • $\begingroup$ @TedShifrin I'm afraid I still don't have anything. If $y,y_0\in S^p$ are sufficiently close, will there be a diffeomorphism on $S^{2p-1}$ homotopic to the identity, taking $f^{-1}(y)$ to $f^{-1}(y_0)$? $\endgroup$ – Aweygan Feb 21 '18 at 0:39
  • $\begingroup$ @Aweygan I am stuck on the same problem. Have you found an answer since posting? And if so would you be kind enough to share it? $\endgroup$ – D. Brogan May 20 '18 at 19:32
  • $\begingroup$ @D.Brogan I Haven't thought about this in a while, but if I come up with anything, I'll post it. Please feel free to do the same :) $\endgroup$ – Aweygan May 20 '18 at 19:35
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Here's a shot at this problem:

We know that if $y$ is a regular value, then there exists an open neighborhood $U\subset S^{p}$ containing $y$ that only contains regular values. Further, we know that for any $y'\in U$ and bases $b,b'$ of $T(S^p)_y,T(S^p)_{y'}$ respectively, the manifolds $(f^{-1}(y),f^*b),(f^{-1}(y'),f^*b')$ are framed cobordant, meaning that if we stereographically project these manifolds from $S^{2p-1}$ into $\mathbb{R}^{2p-1}$ we see that $(h^+(f^{-1}(y)),h^{+*}(f^*b))$ and $(h^+(f^{-1}(y')),h^{+*}(f^*b'))$ are framed cobordant in $\mathbb{R}^{2p-1}$ (where $h^+$ is the stereographic projection) . This implies that if we pick bases $\mathcal{B}$ of $f^{-1}(y)\times f^{-1}(z)$ and $\mathcal{B}'$ of $f^{-1}(y')\times f^{-1}(z)$, we see that

$((h^+(f^{-1}(y))\times h^+(f^{-1}(z)),h^+\mathcal{B})\textit{ is framed cobordant to }(h^+(f^{-1}(y'))\times h^+(f^{-1}(z)),h^{+*}\mathcal{B'})$

Giving us that their respective $\lambda$ maps (as defined in the previous problem) have the same degree.

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