# Most common small 3-manifolds in the 4-sphere

I've been enumerating vertex-normal 3-manifolds in triangulated homotopy 4-spheres recently. I thought I'd put a list of the most commonly found 3-manifolds somewhere. This seems like a good a place as any.

In my search, what I do is run through all homotopy 4-spheres that can be triangulated (in the unordered delta complex / semisimplicial / Thurston sense) with 6 or less 4-dimensional simplices. There's millions of those, but from a sampling (of about 1000) of those I do a more or less random walk in Pachner space and generate a list of roughly 120,000 triangulated homotopy 4-spheres that are Pachner-equivalent -- roughly it's a little neighbourhood of the original 4-sphere in the Pachner graph. Typically I explore triangulations up to about 14 or 16 pentachora.

The 3-sphere is by far the most commonly found 3-manifold, since the link of any vertex is a normal 3-manifold. Next most common is connect-sums of S2 x S1. Roughly 8 or less summands is fairly typical when searching triangulations with 14 or 16 pentachora. S3/Q8 where Q8 is the quaternionic group Q8 = \pm { 1,i,j,k } is fairly common as well, as well as that direct-sum up to three copies of S1 x S2. The oriented S^1 bundle over a Klein bottle is fairly common. L3,1 connect sum its mirror image is fairly common as well, this one sometimes appears connect sum up to about 4 copies of S1 x S2. The Seifert-fibred manifolds over a torus with one singular fiber of order n/1 occurs quite often as well for n <= 3. These manifolds also tend to come with a few S1 x S2 summands.

Relatively rare but occuring manifolds tend to be Seifert-fibered manifolds over S2 with three and four singular fibers. There are also a few hyperbolic manifolds that occur frequently, primarily the hyperbolic manifold that's the 0-surgery on the 2-component link 7a6, giving in my Embeddings of 3-manifolds...' preprint (with Ben Burton now, who wrote much of the code I'm using for these computations).

More rare still are manifolds with slightly more complicated JSJ-decompositions, things like the figure-8 knot complement with an identification on the torus cusp that turns it into a Klein bottle.

These manifolds with complicated JSJ-decompositions are what I'm most curious in at present. These manifolds tend to have fairly complicated surgery descriptions so traditional handle type constructions tend to be difficult to use for these manifolds.

One experimentally re-occuring theme that comes up is that if a manifold M # (S1 x S2) embeds in a homotopy 4-sphere, then it is nearby' an embedding of M. Likely one is just the other with a 1-handle attachment but my software does not allow for checking this. I think it's reasonable to assume that this is fairly typical behaviour but it's interesting so far that there's been no deviations from it. The 4-manifolds literature does not have a theorem that would predict this -- some kind of generalized Dehn Lemma for 2-spheres in 3-manifolds embedded in 4-manifolds.

Next up I'll try exploring larger homotopy 4-spheres.