3 ms·

Lost me at the end, but, don't inner-tubes have 2 holes (genus 2), topologically: one for inflation and one for the wheel to fit in. This makes them distinct fr

Lost me at the end, but, don't inner-tubes have 2 holes (genus 2), topologically: one for inflation and one for the wheel to fit in. This makes them distinct from a torus (genus 1) and no homotopy exists between them. Clearly IANAM.

superhuzza 3y ago
There is a distinction between a torus and a solid torus A torus is like an inner tube - an inner void and a big hole in the middle. A solid torus just has a big hole in the middle, like a donut. https://en.wikipedia.org/wiki/Solid_torus https://en.wikipedia.org/wiki/Solid_torus
- toth 3y ago
  You probably are aware, but just to make it clear for others: topologically a torus has one hole (genus 1). The inner void is not considered a hole. Similarly, a 2-sphere (surface of solid sphere in 3d) has 0 holes (the inner void is not considered).
  - adgjlsfhk1 3y ago
    to be slightly more technical what you're defining as "voids" are often considered 2-dimensional holes (as formalized by homology)
toth 3y ago
You are right, but I am pretty sure they just meant the inner tube ignoring the puncture for inflation. Usually people just use donut/bagel for an illustration of a torus. Not sure why they used inner tube here - maybe to make it clear it is just the surface?
- shrx 3y ago
  A hollow inner tube with the puncture for inflation is topologically equivalent to a solid torus, if I understood correctly.
  - pbhjpbhj 3y ago
    My understanding is they're non-homeomorphic. You can thread a string in through the inflation hole (valve hole) on an inner tube, around the great circle inside the tube, and then out the valve hole. You can also, of course go through the centre as with a donut/bagel/torus. But you can't do the former threading with a donut, so the torus and whatever the inner-tube are not typologically equivalent. Another way to consider it is if you have a circle around the valve hole, you can't stretch that circle to be one of the two possible circles you could draw on a torus (around the 'trunk' and around centred on the donut-hole). Anyway, I think they just chose a bad example, this was just where my mind went.

superhuzza 3y ago

There is a distinction between a torus and a solid torus A torus is like an inner tube - an inner void and a big hole in the middle. A solid torus just has a big hole in the middle, like a donut. https://en.wikipedia.org/wiki/Solid_torus https://en.wikipedia.org/wiki/Solid_torus

toth 3y ago
You probably are aware, but just to make it clear for others: topologically a torus has one hole (genus 1). The inner void is not considered a hole. Similarly, a 2-sphere (surface of solid sphere in 3d) has 0 holes (the inner void is not considered).
- adgjlsfhk1 3y ago
  to be slightly more technical what you're defining as "voids" are often considered 2-dimensional holes (as formalized by homology)

toth 3y ago

You are right, but I am pretty sure they just meant the inner tube ignoring the puncture for inflation. Usually people just use donut/bagel for an illustration of a torus. Not sure why they used inner tube here - maybe to make it clear it is just the surface?

shrx 3y ago
A hollow inner tube with the puncture for inflation is topologically equivalent to a solid torus, if I understood correctly.
- pbhjpbhj 3y ago
  My understanding is they're non-homeomorphic. You can thread a string in through the inflation hole (valve hole) on an inner tube, around the great circle inside the tube, and then out the valve hole. You can also, of course go through the centre as with a donut/bagel/torus. But you can't do the former threading with a donut, so the torus and whatever the inner-tube are not typologically equivalent. Another way to consider it is if you have a circle around the valve hole, you can't stretch that circle to be one of the two possible circles you could draw on a torus (around the 'trunk' and around centred on the donut-hole). Anyway, I think they just chose a bad example, this was just where my mind went.