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?

Improve this question
$\endgroup$
6
  • 1
    $\begingroup$ A tubular neighborhood of the torus. $\endgroup$
    – Steve Huntsman
    Commented Jan 10, 2010 at 16:25
  • 1
    $\begingroup$ @Steve That only makes sense if it's embedded in an ambient polyhedron. $\endgroup$
    – Daniel Moskovich
    Commented Jan 11, 2010 at 0:44
  • 1
    $\begingroup$ Sorry to reretag, but "names" is more apt that "notation". $\endgroup$
    – Jonas Meyer
    Commented 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. $\endgroup$
    – Ryan Budney
    Commented Jan 11, 2010 at 2:35
  • $\begingroup$ I also added the tag "3-manifolds". $\endgroup$
    – Daniel Moskovich
    Commented Jan 11, 2010 at 4:29

3 Answers 3

Reset to default
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.

Improve this answer
$\endgroup$
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. $\endgroup$
    – Ryan Budney
    Commented 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
    Commented 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

Improve this answer
$\endgroup$
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

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .