Holes involving 1 are disjoint surfaces, eg either disjoint objects or objects with a hidden interior. These are not usually considered, so we have "rigid" holes that can participate in chains etc. In 3d, the rigid chain is a string of loops (22).
Central inversion, or "turning the surface inside out"
wendy wrote:A 23 hole, formed by a spheric prism wrapped into a circle
wendy wrote:Consider for example, a pipe with cylinder section, and length. Were you to connect it end to end like a hose, the thing becomes a 23 hole. If you roll it down like a sock, so the top is rolled down the side of the cylinder to meet the bottom, then the thing is a 3-sphere covered by a 2-circle, giving a 32 hole.
I have not figured out what H_0 and H_n might mean.
31 = Two disjoint spheres
13 = Two concentric spheres
31 can be transformed into 13 by turning it inside out as wendy said. There is no continuous transformation between a pair of disjoint spheres and a pair of concentric spheres, because the orientation matters
Correcting the orientation is not a continuous transformation.
Note that a space never looses its hole-ness, even should it loose its capacity to bound etc. A bi-circular torus is still a 22 hole, even in 15 dimensions. It's different to the knots, which in some large dimension unravels (eg a 22 knot, such as a piece of string forms, becomes a simple unconnected loop in 4d. You can still use 22 knots, as long as they are extended in prisms (ie 22P1), which is why you can tie hedra (planes) together in 4d.
Keiji wrote:One thing I just want to clarify, though:wendy wrote:A 23 hole, formed by a spheric prism wrapped into a circle
This implies a spherinder (spherical prism) joined end to end gives a toraspherinder and that a cubinder gives a toracubinder.
Therefore 23 is toraspherinder and 32 is toracubinder.wendy wrote:Consider for example, a pipe with cylinder section, and length. Were you to connect it end to end like a hose, the thing becomes a 23 hole. If you roll it down like a sock, so the top is rolled down the side of the cylinder to meet the bottom, then the thing is a 3-sphere covered by a 2-circle, giving a 32 hole.
This implies that a cubinder (cylindrical prism) joined end to end gives a toraspherinder and that a spherinder gives a toracubinder.
Therefore 32 is toraspherinder and 23 is toracubinder.
Which way around is it? I've believed the first case as the naming and toratopic notations make sense that way, but I recall some argument that it's actually the opposite.
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