Talk:Atoroidal

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I've never run across a situation where (geometrically) atoroidal also includes acylindrical. I think that should be removed. --C S 23:45, Dec 7, 2004 (UTC)

Fine: I've fixed it per your suggestion. But I do think discussion of anannular should stay here, (1) as the concepts are so closely related and (2) to avoid an article that's pure dict.def. —msh210 02:53, 8 Dec 2004 (UTC)

I think it's a good idea to have these definitions on one page also. As long as everything redirect here, it's not a problem. Nice pictures, BTW (in boundary parallel. --C S 04:16, Dec 8, 2004 (UTC)

Actually, Kirby's Problems has a definition of geometrically atoroidal that includes anannular. I quote from his introduction to chapter 3:

$M^{3}$ is called atoroidal if

• (geometric definition) M contains no essential, properly imbedded, nonperipheral annulus or torus;
• (algebraic definition) each $Z\oplus Z$ in $\pi _{1}(M)$ is conjugate to a subgroup in $\pi _{1}(\partial M).$

—msh210 19:37, 13 Dec 2004 (UTC)

Well, I never said nobody made this definition or that it doesn't make sense (it does). I've never seen it outside of your reference though, although this claim is suspect as I'm familiar with Kirby's list, so my memory has been shown fallible. --C S 07:21, Dec 14, 2004 (UTC)

Anyway, I will restore your deleted comment. --C S 08:55, Dec 14, 2004 (UTC)

Relations between notions of atoroidality

I modified "Any algebraically atoroidal 3-manifold is geometrically atoroidal and acylindrical" to omit acylindrical as it is not true. For example, take a torus cross an interval. There is an incompressible annulus that has each boundary component on a different boundary component and thus is not boundary parallel. I took a look at the reference from above (Kirby's list), which is where I presume the original statement came from. Kirby seems to be saying (after proper reinterpretation to avoid this and similar examples) that it ought to be true as long as we avoid certain "small" Seifert fiber spaces, e.g. base space is a disc and there are two exceptional fibers. --C S (Talk) 17:16, 1 March 2006 (UTC)

Actually, to my chagrin I note that my last statement about those SFSs is incorrect as they are not even algebraically atoroidal (there are immersed essential tori). Nonetheless my first example is still a counterexample, and I suppose one can even take surface x I to get more examples. So it's not totally clear what Kirby means, but things cannot be as he states.

So, it might be good to bring up that I believe geometrically atoroidal plus acylindrical does imply algebraically atoroidal excepting certain special cases (such as SFSs with base space a sphere and three exceptional fibers). That would be one reason for taking the defn of geom. atoroidal to include acylindrical. In any case, I'm confident of the current statement of alg. atoroidal implies geom. atoroidal but not nec. the converse. --C S (Talk) 19:54, 3 March 2006 (UTC)