# What are the implications of the simple loop conjecture?

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?

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.

Folklore theorem: Every continuous map from $\Sigma$ to a graph $\Gamma$ kills an essential simple closed curve.

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.

• The "fairly complete understanding" would mean something like: every map of a surface is obtained (up to homotopy) by a map of a $\pi_1$-injective surface with some handles attached along maps of arcs in the manifold with endpoints on the surface. – Ian Agol Nov 30 '15 at 21:58
• Right. I only wanted to describe the understanding as 'fairly complete' because the SLC wouldn't say anything about $\pi_1$-injective surfaces. Somehow, the Makanin--Razborov framework only classifies the kernel, not the image. – HJRW Dec 1 '15 at 9:27
• There appears to be a proof. See Vlad Markovic Harmonic surfaces in 3-manifolds and the Simple Loop Theorem, at its.caltech.edu/~markovic/preprints.html – Joel Hass Sep 14 '19 at 2:32
• @JoelHass: Indeed. I haven't yet encountered anyone who understands the proof, though (and I haven't spoken to Vlad himself about it). – HJRW Sep 16 '19 at 10:28

The simple loop conjecture can be viewed as a statement about how to construct all surfaces in a 3-manifold.

Fix any orientable 3-manifold M. There are two well known constructions that produce oriented surfaces in M.

1. Start with a sphere that bounds a ball, and successively add a finite number of 1-handles (allowing self-intersections).

2. Start with a surface subgroup of π1(M), or equivalently with an immersed π1-injective surface, and add a finite of 1-handles, (again allowing self-intersections.)

Does this produce all immersed surfaces in M?
Yes if and only if the simple loop conjecture is true.

I've come across this conjecture in questions about Dehn fillings. Suppose one has a hyperbolic 3-manifold with torus boundary, and one performs a Dehn filling to get a 3-manifold which is not hyperbolic. Then there is an essential surface in the 3-manifold of genus $0$ or $1$ (an essential torus or sphere, or a planar surface bounding some multiple of the core of the Dehn filling which is contractible). One may pull this surface back to the 3-manifold with torus boundary up to homotopy so that it becomes a punctured genus $\leq 1$ surface which is simple-loop injective (notice this is for surfaces with boundary, but it reduces to the closed case by doubling). Now we'd like to know that this surface is $\pi_1$-injective in order to make certain geometric arguments.