3
$\begingroup$

What would you call the product of an annulus and $S^1$ (a 'thickened' torus like 3-manifold)?

More generally, is there an archive or list online of names assigned to various (non-standard) manifolds by people? Or a set convention by which to name them?

$\endgroup$
6
  • 1
    $\begingroup$ A tubular neighborhood of the torus. $\endgroup$ Jan 10, 2010 at 16:25
  • 1
    $\begingroup$ @Steve That only makes sense if it's embedded in an ambient polyhedron. $\endgroup$ Jan 11, 2010 at 0:44
  • 1
    $\begingroup$ Sorry to reretag, but "names" is more apt that "notation". $\endgroup$ Jan 11, 2010 at 2:20
  • 1
    $\begingroup$ A common way to name manifolds is via cobordism + surgery. Thom classified manifolds up to cobordism, and you get between any two cobordant manifolds via surgery. So that's a common (if highly ambiguous) naming convention. Moskovich's response falls under the verbiage of fibre bundle terminology, which is far less ambiguous. $\endgroup$ Jan 11, 2010 at 2:35
  • $\begingroup$ I also added the tag "3-manifolds". $\endgroup$ Jan 11, 2010 at 4:29

3 Answers 3

6
$\begingroup$

I would call it a thickened torus. I don't know how standard that is, but it is quite normal to speak of thickened manifolds, where one means that manifold times a closed interval.
I have long felt that there should be a mathematical dictionary- not an encyclopaedia, by a dictionary- in order to fix and record standard usage.

$\endgroup$
2
  • 2
    $\begingroup$ A trivial $I$-bundle over a torus would be another -- since there are non-orientable $I$-bundles over the torus this is a tiny bit more specific. $\endgroup$ Jan 11, 2010 at 2:19
  • $\begingroup$ @Ryan: $M\ddot{o}\times S^1$ is the non-orientable twisted $I$-bundle over the torus. $\endgroup$
    – janmarqz
    Oct 12, 2012 at 1:48
1
$\begingroup$

that corresponds to the complement of a trivial (but essencial) torus knot in a open solid torus. For those -Fico had mention- they are called cable spaces and have nice foliation into circles. Its name is CS(1,0). Can you see what is CS(2,1)?

Edit at: utc-6 = 11:50 approx

you could also say the trivial I-bundle over the torus

$\endgroup$
0
$\begingroup$

May I {humbly} suggest "inner tube", as in "Floating down the river in an ... " ? (Strike while the terminological iron is hot.)

Gerhard "Ask Me About System Design" Paseman, 2010.02.09

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.