# Homotopically vs geometrically atoroidal

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?