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In problem number 13 of Milnor's 'Topology from the Differentiable Viewpoint', the linking number for two compact boundary-less manifolds $M,N \subset \mathbb{R}^{k+1}$ of dimensions $m,n$ such that $m+n =k$ is defined as the degree of the map $\lambda\colon M \times N \to S^{k}$ given by $\lambda(x,y)=\frac{x-y}{\|x-y\|}$. The problem then asks to prove some properties of the linking number. The final question, where I need help, is

Define the linking number for disjoint manifolds of the sphere $S^{m+n+1}$.

I have no clue as to what the definition should be. I need this in order to proceed to the subsequent problems. Could somebody please give the definition, and a suitable reference ? Thanks !

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you can find three different definitions in bott and tu's book: differential forms in algebra topology page 229. The definitions by intersection number and differential forms are both in sphere.

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  • $\begingroup$ Your post would be more helpful if you included those definitions in your answer. $\endgroup$ – ekkilop Jul 31 '17 at 11:04

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