Drew Zemke, who is a grad student of Jason Manning, posted a proof of the Simple Loop Conjecture for 3-manifolds modeled on Sol last week.
The Simple Loop Conjecture fits into that family of statements such as Dehn’s Lemma and the Sphere Theorem which translate statements about fundamental groups into statements about 3-manifolds. Such theorems allow us to trade 3-manifolds for their fundamental groups (which are much simpler mathematical objects).
Consider a 2-sided immersion of a closed orientable surface into a closed 3-manifold . The Simple Loop Conjecture states that if is not injective then there is an essential simple closed curve in that represents an element in the kernel of . If were an embedding then this would follow from Papakyriakopoulos’s loop theorem. `To be an embedding’ doesn’t translate to an algebraic property, so the Simple Loop Conjecture is more of a ` to 3-manifolds’ statement than the loop theorem. It allows us to replace non--injective immersions by immersions of lower genus surfaces by surgery paralleling passage to a normal subgroup; So it really does translate between algebra and topology. I asked an MO question regarding applications.
One of the nice things about the simple loop theorem is that it really does seem that the target seems to be a 3-manifold group or something similar. There have been several attempts to generalize, for instance to consider representations into , but they have all failed.
Joel Hass proved the conjecture for Seifert-fibered spaces using geometrical techniques in 1987, and Hyam Rubinstein and Shicheng Wang proved the conjecture in 1998 for non-trivial graph manifolds, which for them meant the graph manifold was also not Sol. Mayer Landau tells me that Rubinstein-Wang’s technique might work also for Sol manifolds, but he is not sure… Anyway, Zemke does something different.
The most interesting case for the Simple Loop Conjecture is of course for hyperbolic manifolds. There, it’s known to be false in higher dimensions.
Thanks to Mayer Landau for drawing my attention to this preprint and for explaining its significance.