Drew Zemke recently posted a preprint on arXiv proving the Simple Loop Conjecture for 3-manifolds modeled on Sol.
Simple Loop Conjecture: Consider a 2-sided immersion $F\colon\, \Sigma\rightarrow M$ of a closed orientable surface $\Sigma$ into a closed 3-manifold $M$. If $F_\ast \colon\, \pi_1 \Sigma \rightarrow \pi_1 M$ is not injective then there is an essential simple closed curve in $\Sigma$ that represents an element in the kernel of $F_\ast$.
If $F$ were an embedding then this would follow from Papakyriakopoulos's Loop Theorem. "To be an embedding" is not an algebraic property, so the Simple Loop Conjecture is more of a `$ \pi_1$ to 3-manifolds' statement than the loop theorem. It allows us to replace non-$\pi_1$-injective immersions by immersions of lower genus surfaces by surgery paralleling passage to a normal subgroup; So it does translate from algebra to topology.
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 (not Sol).
In Kirby's problem list it also states that the Simple Loop Conjecture arises in trying to characterize 3-manifold groups among Poincaré duality groups, but I'm not sure what Kirby means. I also don't know what else it implies.
Question: What wonderful things would follow from the Simple Loop Conjecture if it were true? Beyond it being a "natural question" and beyond the abstract considerations brought above, what is the significance of this conjecture?
Best Answer
I would motivate the simple loop conjecture as follows. (I'm fairly idiosyncratic about this; I fear I'm going to turn off many 3-manifold topologists.)
As well as understanding spaces, we want to understand the maps between them. One instance of this is that it would be extremely useful to have some sort of 'classification' of the set of all maps from your favourite (closed, say) surface $\Sigma$ to any 3-manifolds
What might such an understanding look like? If we replace 3-manifolds by graphs, then the answer is provided by a folklore theorem, often attributed to either Stallings or Zieschang.
This fits into Sela's framework of Makanin--Razborov diagrams (over free groups), where it implies that the natural homomorphism $\pi_1\Sigma\to F$ induced by including $\Sigma$ in the boundary of a handlebody forms the Makanin--Razborov diagram for $\pi_1\Sigma$.
So the simple loop conjecture is the analogous statement over 3-manifolds. Basically, it would take us from having a relatively hazy understanding of what the set of all maps from $\Sigma$ to a 3-manifold might look like, to a fairly complete understanding.
You might say "Fine, but what's so special about surfaces?" In fact, surfaces play a distinguished role, because of the way they arise naturally in JSJ theory. For this reason, they are one of the key cases; if we can understand maps from surfaces to 3-manifolds, we have a chance of understanding maps from arbitrary aspherical spaces to 3-manifolds.