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 !


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.

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.