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This is a follow-up question to my previous question on "winding numbers" of curves in higher-dimensional space. I thought it best to post a new question rather than changing the setup of the previous one from the ground... The comments and answers there suggest that in this case the point of view (pun intended) shouldn't be a point, but really a higher-dimensional object. So, what we are really looking for is $w(\gamma,X)$, the number of windings of a loop $\gamma$ around an appropriate topological subspace $X\subseteq\mathbb{R}^n$ (to be determined).

Example 1.: Let $L$ be a line and $X:=C:=\mathbb{D}^2\times L$ a solid cylinder in $\mathbb{R}^3$. Then intuitively $w(\gamma,C)$ should be determined by the homotopy class of $[\gamma]$ in $\pi_1(\mathbb{R}^3\setminus C)$. But I believe this example to be misleading.

Example 2: Let $X:=\mathbb{S}^2$. Surely we can count how many rounds/loops a yarn $\gamma$ makes around $X$, even though $\pi_1(X)=0$. It seems $w(\gamma,X)$ shouldn't always be homotopy-invariant.

Is there a reasonable way to formalize the intuitive notion of number of windings of a loop $\gamma$ around $X$, where $\gamma\cap X=\emptyset$, for a suitable class of topological spaces $X$?

For instance, in $\mathbb{R}^2$ the classical winding number $w(\gamma,a)$ would correspond to $w(\gamma,C_\varepsilon(a))$ for a small circle around $a$. (It's perhaps that second argument that should always be homotopy-invariant?)

*a cat: NORTH AMERICAN informal (especially among jazz enthusiasts) a man.

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    $\begingroup$ Re: the word "surely" in example 2. What should be the winding number of the seam on a baseball? $\endgroup$ – Willie Wong Apr 10 '18 at 16:31
  • $\begingroup$ @WillieWong: sorry, I am not really familiar with baseball. Could you give another example illustrating your point? $\endgroup$ – M.G. Apr 10 '18 at 16:33
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    $\begingroup$ (How can you know the word "cat" but not what a baseball look like?) Here's an animation of a baseball shutterstock.com/video/… Look at the closed curve corresponding to the red parts. $\endgroup$ – Willie Wong Apr 10 '18 at 16:41
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    $\begingroup$ If the winding number is always an integer but is not homotopy-invariant, then it must sometimes jump discontinuously when the curve is deformed continuously. It would be interesting to see where such jumps occur. $\endgroup$ – Andreas Blass Apr 10 '18 at 16:58
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    $\begingroup$ If you remove a point from a sphere you are back at the planar case. I suspect that is what we do when we count the loops of a yarn around a sphere: remove a pole and count the winding number around the other pole. Or do you have an other intuition? The "surely" doesn't really help... Example 1 can also be brought back to the usual case by projecting on a plane orthogonal to cylinder. $\endgroup$ – Maxime Lucas Apr 10 '18 at 19:44

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