# What do you call the product of a circle and an annulus?

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?

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A tubular neighborhood of the torus. –  Steve Huntsman Jan 10 '10 at 16:25
@Steve That only makes sense if it's embedded in an ambient polyhedron. –  Daniel Moskovich Jan 11 '10 at 0:44
Sorry to reretag, but "names" is more apt that "notation". –  Jonas Meyer Jan 11 '10 at 2:20
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. –  Ryan Budney Jan 11 '10 at 2:35
I also added the tag "3-manifolds". –  Daniel Moskovich Jan 11 '10 at 4:29

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.

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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. –  Ryan Budney Jan 11 '10 at 2:19
@Ryan: $M\ddot{o}\times S^1$ is the non-orientable twisted $I$-bundle over the torus. –  janmarqz Oct 12 '12 at 1:48

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

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