So a while ago, I was reading Hatcher’s notes on 3-manifolds. In there, he defines what it means for a manifold to be **prime **and states, casually, that the 3-sphere is prime. He later says that it follows immediately from **Alexander’s Theorem** as, and I quote: *Every 2-sphere in bounds a 3-ball*. And that’s it. Done.

Wait, what?!

Elsewhere, Hatcher expands his above statement: *…every 2-sphere in bounds a ball on each side…[and h]ence is prime*. Again, though, it isn’t accompanied by anything, and while this is clearly a trivial result, I just couldn’t *see it *for the longest time…I knew that it followed from a number of things, e.g. the fact that is the identity of the connected sum operation, that is **irreducible **(and that every irreducible manifold is prime), that one gets the trivial sum by splitting along a 2-sphere in which bounds a 3-ball in , etc. Even so, I didn’t want to leverage some enormous machinery to deduce the smallest of results and what I *really *wanted was for someone to tell me what I was missing. So I never stopped thinking about this, even after moving forward, until finally – it just clicked!

I figure other people who are as visualization-impaired as I may benefit from seeing this explained in greater depth, so in lieu of typing a blog post containing something new and attention-worthy, I figure I’d share this instead. Details after the break.

Throughout, I’m going to assume familiarity with the connected sum operation. From there, we begin with a definition:

**Definition 1: **A connected 3-manifold is said to be *prime *if writing implies that either or .

As Hatcher indicates, Alexander’s theorem will be a fundamental piece of the puzzle. I’ll state the version given in the first of the above-linked references here, deferring proof with the note that Hatcher’s proof is very readable (even if you’re totally unfamiliar with Morse theory).

**Alexander’s Theorem: **Every 2-sphere in bounds a 3-ball.

A 1924 version of this result (again attributed to Alexander) says that, moreover, the result holds when is replaced with and hence, every 2-sphere in bounds a 3-ball. This is the version we care about.

To prove that is prime, we’ll show that is only possible if and to do that, we’ll need the following observations/facts/lemmas/axioms/whatever…they’re the crux of the whole argument and they’re the things that totally escaped me.

**Fact 2:**

**Every 2-sphere in bounds a ball***on both sides*;**Gluing two closed 3-balls along their 2-sphere boundaries via the identity map yields a 3-sphere.**

To see the truth of the first fact, imagine as the union of with a point at infinity. As above, a 2-sphere in bounds a ball (in its interior) by Alexander’s theorem; moreover, the complement is homeomorphic to (since we can homeomorphically shrink to be arbitrarily small). Replacing with , we have that where is the open 3-ball.

The second fact can be seen in a couple different ways. In one dimension lower, note that a pair of closed 2-disks and glued along their boundary circles and by the identity map yield a 2-sphere . In three dimensions, the argument is the same, and this can be seen to generalize to arbitrary finite dimensions using cell decompositions: Take cell decompositions of closed -disks and consisting of one -cell, one -cell, and a 0-cell each, and glue them so that the boundary circles are identified, i.e. so that the resulting space consists of two -cells, one -cell, and one 0-cell. Appealing to Euler characteristic arguments, , and there’s no topology, and homology stuff, and yadda yadda yadda and…. Hence . 😆 😛 😉

Now, we formalize the point at hand.

**Proposition 3: is prime.**

*Proof*. Let be a 2-sphere in and let be the connected sum formed by splitting along (i.e., by removing a small open tubular neighborhood of from and by defining and to be the result of filling the boundary spheres corresponding to of the two components and of with a ball). The goal is to show that .

Now, bounds a ball in its interior (by Alexander) *and *in its exterior (by Fact 2.1 above) and so is the disjoint union of two open 3-balls and : . What’s more, the connected sum is formed by removing a tubular neighborhood from and by gluing closed 3-balls back in the resulting boundary components, i.e. and are both formed by gluing a closed 3-ball to another closed 3-ball along their 2-sphere boundaries: where denotes a closed 3-ball. By fact 2.2 above, it follows that both and are homeomorphic to : .

I’m not 100% sure *which* parts of this argument I struggled with the most but honestly, I never didn’t struggle with it. For me personally, it helped to draw the corresponding picture in one dimension lower:

- Take a 2-sphere , form by deleting an (open) annular region from and look at the pair of resulting manifolds and (in this case, each are surfaces of genus 0 with one boundary component).
- Rearranging and looking at how one would glue them to form the connected sum, it becomes obvious that (up to homeomorphism) it’s just a matter of capping the boundary of each of with 2-disks then gluing the boundary circles by the identity map.
- In particular, is being identified with the connected sum of two manifolds and which are each 2-spheres minus a (closed 2-)disk plus a (closed 2-)disk so that and so .

This is all very trivial, of course, but spending the extra time to ensure I wasn’t taking it for granted helped improved my ability to visualize things, I think. Even if not, maybe this (clumsy) (unpolished) (probably unsatisfactory) expounding will help someone fill in the gaps more quickly than I did!

**Note:** Though no exposition was entirely satisfactory for me, I found the proof of Corollary 2.2.3 of Schleimer’s lecture notes to be somewhat helpful, particularly in the realization of Fact 2.2.