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I'm having some trouble understanding this picture from Guillemin and Pollack Chapter 3:

enter image description here

In G&P chapter 3, they define $\deg(f) = I(f, \{y\})$. In the above image, they claim that they're mapping $S^1$ to a curly circle and then projecting back to $S^1$ (I assume it's a projection along the radial direction). But it's not clear to me what the bold dots with arrows exactly represent. Yes, they're tangents but since $S^1 \to \text{curly circle} \overset{\text{projection}}{\to} S^1$ is a composition of two maps, I'm not sure that any of them represent regular values or regular points of the composite map. Could someone explain the figure to me?

Ted Shifrin
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S.D.
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They indicate oriented tangent vectors to the image (and then to the projected image). Yes, they are showing two regular values in $S^1$, as the tangent vector to the image is nonzero. The non-regular values would be those where the radial line is tangent to the "curvy circle" — at these points, the derivative of $f$ would be $0$. The orientation, of course, is needed to compute degree (not so with mod $2$ degree).

Ted Shifrin
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  • Thank you for your answer. I'm still a bit confused about a few things, but could you perhaps clarify which map $f$ represents in your answer? (Main confusion is how the tangent vectors on the curvy circle are being mapped to tangent vectors on $S^1$ upon projection.) – S.D. Nov 15 '22 at 21:52
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    $f$ is their map, the composite of wiggly map and the radial projection, so that the derivative is the composition of the derivative of the wiggly map with the derivative of radial projection. (Perhaps it would be helpful to trace out the mapping $f$ with a finger as the point in the domain moves with constant speed around that circle.) – Ted Shifrin Nov 15 '22 at 21:55
  • Aha, that's an excellent mental picture. Thank you once again for your help. – S.D. Nov 15 '22 at 22:04
  • P.S. If you don’t already have it, download the errata for G&P from my webpage linked in my profile. – Ted Shifrin Nov 15 '22 at 22:23