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I am reading Knows, Molecules, and the Universe: An Introduction to Topology. A definition of for shrinkable has been given (pg. 68, chapter 3): If a loop can be pulled in, we say the loop is shrinkable. Otherwise, we say the loop is unshrinkable Three examples are given: a curved torus with a loop around the hole, a knotted (trefoil, I think) torus that has a curve around the whole loop, and a loop crossing the left/right cut line of a flat torus. I understand the first two just fine. I don't understand the third. It seems that the loop can be pulled in easily to me.

mcr
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  • https://photos.app.goo.gl/n84cWqj36exRcS926 – mcr Dec 22 '19 at 21:27
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    None of these are loops. A loop is supposed to start and end at the same point. – Moishe Kohan Dec 22 '19 at 21:36
  • Yes, I agree with Moishe the figures are misleading as these are not loops. The last one can be thought as a torus by glueing the sides with same type of arrow. The top and bottom sides would create a "tube". Once you have glued those into a "tube" you need to glue the other two sides (left and right), that now are the top and bottom "circles" of the "tube". A bit rough as an explanation, but should give the intuition, I hope. – geguze Dec 23 '19 at 00:18

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I browsed the book that you are reading. It is written by a good professional topologist, but is totally non-rigorous to the point where definitions are utterly lacking and (even intuitive) proofs are impossible. For instance, the notion of a "loop" is undefined and where it is used in pictures, it is rather an "arc", which is inconsistent with the common notion of a loop in topology. (On many pictures in the book a "loop" is an "arc" connecting a square to a triangle.) The notion of a "shrinkable loop" is defined as a "loop which can be pulled in", which is also left undefined. To make any sense of this, one will have to redraw the pictures so that the square equals the triangle, and the latter equals a point on the surface. A shrinkable loop then is "loosely speaking" a loop which can be continuously deformed to a single point. (Defining a "continuously deformation" requires some work, but it is at least intuitively clear.)

If you want to continue reading this book, I suggest supplementing with a more rigorous basic algebraic topology book (parts related to fundamental groups) which will explain all this staff and will provide rigorous proofs. My personal favorite is

W.Massey, A Basic Course in Algebraic Topology, Chapters I and II.

Also:

J. Lee, Introduction to Topological Manifolds, Chapter 1-8, excluding the Category Theory and higher homotopy groups.

A.Hatcher, Algebraic Topology, Chapter 1. (Yes, there are many complaints about the book being too geometric, but, for most readers, this will be a plus.)

I would not recommend (to somebody who is not a pure mathematician) any of the other books mentioned here.

Moishe Kohan
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