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It's problem 13 on the book. For disjoint compact manifold $M, N$ with no boundary in $R^{k+1}, m+n=k$, define their linking number $l(M,N)$ by the degree of mapping$\lambda (x,y)=\frac{x-y}{||x-y||}$. If M is the boundary of an oriented manifold $\Sigma$ disjoint to $N$, prove that $l(M,N)=0$.

I thought it might be divided into two cases.

  1. When $\Sigma$ is unbounded, it seems possible to find an $\widetilde{M}$ homotopic to $M$ which is far away from $N$, thus $\lambda \simeq 0$.
  2. When $\Sigma$ is bounded, $M$ seems to be retracted to a single point.

I want to make the illusion to be rigorous if possible. Any help is appreciated.

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HINT: Have you considered applying the powerful Lemma 1 on p. 28 of Milnor? (Guillemin and Pollack, appropriately, call this the Boundary Theorem.)

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  • $\begingroup$ I see. Thanks so much. I didn't notice that the map could be extended. $\endgroup$ – Kirby Lee Mar 27 '17 at 11:20

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