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In the book

Aschenbrenner, Matthias; Friedl, Stefan; Wilton, Henry, 3-manifold groups, EMS Series of Lectures in Mathematics. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-154-5/pbk). xiv, 215 p. (2015). ZBL1326.57001.

they say that a $3$-manifold $M$ is (homotopically) atoroidal if any map $T^2\to M$ from a torus that induces an injection $\pi_1(T^2)\to\pi_1(M)$ is homotopic to a map $T^2\to\partial M$.

A related definition is that $M$ is (geometrically) atoroidal if any incompressible embedded $T^2\subset M$ is isotopic to a boundary component of $M$.

According to the book, these two notions agree except for "small Seifert fibered manifolds," where "a Seifert fibered manifold is called small if it is not Haken." And, "a $3$-manifold is Haken [...] if it is compact, orientable, irreducible, and has an embedded incompressible surface," where "embedded surface" means properly embedded and orientable.

Let $X$ be a pair of pants (a closed $3$-times punctured sphere) and let $M=S^1\times X$.

  • This is not homotopically atoroidal. If $a,b$ are generators of $\pi_1(X)$ with, say, counterclockwise orientation around the punctures, then let $\gamma$ represent the loop $ab^{-1}$, which requires a point of intersection. The immersed torus $S^1\times \gamma$ is $\pi_1$-injective but not boundary parallel. If it were boundary parallel, then this would give a homotopy of $\gamma$ to $\partial X$.

  • This is geometrically atoroidal. Essentially, every simple closed curve in $X$ is boundary parallel in the surface.

  • This is a Seifert-fibered space since it is an $S^1$ product bundle over the surface $X$.

  • This is Haken, so not a small Seifert-fibered space. $\{*\}\times X$ is a properly embedded orientable incompressible surface. (It is $\pi_1$-injective.)

In the arXiv version of the survey from a couple years before the book, they had previously written that "these two notions differ only for certain Seifert fibered $3$-manifolds where the base orbifold is a genus $0$ surface such that the number of boundary components together with the number of cone points equals three." $M$ is certainly such a Seifert fibered space, and I have been puzzled by the difference.

What is the correct condition for when the notions of a space being homotopically and geometrically atoroidal diverge? Or what have I misunderstood?

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Yes, the only class of compact 3-manifolds where you see the difference between the two notions consists of small Seifert manifolds.

Their book definition of small Seifert manifolds, as you correctly noticed, is wrong: The non-Haken condition is too strong. (It is correct though if you only consider oriented closed manifolds.) Instead they should have asked for "no essential closed incompressible surfaces." Here an incompressible surface is called essential if it is not boundary-parallel. However, this condition as stated is useless for your purposes since non-existence of essential incompressible tori is exactly what you are trying to characterize.

Their arxiv definition of a small Seifert manifold is correct as long as you restrict to the class of oriented 3-manifolds. If you allow nonorientable 3-manifolds then, for instance, the product of the Klein bottle with the interval would have to be excluded as well (and few more examples where the base is a non-orientable Euclidean orbifold). A better way to proceed is to strengthen the requirement of geometric atoroidality: In addition to tori, require that every incompressible 2-sided Klein bottle is inessential (i.e. is boundary-parallel).

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  • $\begingroup$ I suppose a motivation for my question was that I had a hard time finding what the definition of a small Seifert-fibered space was (for example: are the "special" Seifert manifolds on page 155 of Jaco's book the "small" ones?). With the pair of pants example I give, the pants form an essential incompressible surface, correct? So, to clarify, small SFS's are characterized by having no essential closed 2-sided incompressible surfaces? (I'm emphasizing the word that excludes the pants example.) $\endgroup$ Mar 4 '19 at 19:49
  • $\begingroup$ @KyleMiller Right. $\endgroup$ Mar 4 '19 at 22:56
  • $\begingroup$ Ok, thank you Prof. @MoisheCohen $\endgroup$ Mar 4 '19 at 23:40
  • $\begingroup$ @KyleMiller: A small correction: In the context of my answer "small" means "no essential closed 2-sided incompressible torus". There might be a closed horizontal incompressible surface of higher genus (if the Euler number of Seifert fibration is zero). $\endgroup$ Mar 5 '19 at 0:36

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