The Disc Embedding Theorem is a recent book carefully explicating the work of Mike Freedman in proving the topological 4D Poincare Conjecture. I am very excited about this book. I have always wanted to understand the proof of the theorem, but prior to this book, there has not been a feasible accessible approach to it. A dream of mine is to be able to actually see one of the crazy topological disks that the proof creates, for example bounding a topologically slice knot.
The book is based on a series of lectures that Freedman gave, though saying that tends to obscure the amount of work the contributors of the book put into it! One nice feature of the book is its reflection of the structure of Freedman's original presentation, introducing important techniques through historical examples.
One such example is the "shrinking" proof of the Schönflies Theorem. Not only is this a cool and foundational result in topology, the argument introduces the concept of shrinking, which figures into the 4D proof.
I'd actually like to go through this proof of Schönflies. It is super elegant and really showcases the power of point set topology as found in Munkres's standard text that so many of us learned from as undergraduates!
Definition: A subset $X$ of an $n$-manifold $M$ is said to be cellular if it is a nested intersection of countably many closed $n$-cells. More precisely, $X=\cap_{n\geq 1} C_n,$ where $C_n$ is homeomorphic to the closed $n$ dimensional ball $D^n$, and $C_{n+1}\subset \operatorname{int}C_n.$
I invite the reader to come up with examples of cellular sets. The one given in the book is a literal letter $X$, which you can imagine a nested series of disks approaching. You can also imagine more exotic examples like a truncated topologist's sine curve, or simply connected sets with fractal like filaments.
Of course $X$ has to be compact. It seems obvious that $X$ has to be connected and simply connected, though I don't even see a proof of these "easy" facts. In fact, I would bet that $X$ is contractible, though I would have to consider further why that is the case.
Update: After thinking about it, aside from connectivity, these "obvious" facts are wrong. First, the topologist's sine curve in the plane, defined by $Y=\{(x,\sin(1/x)\,|\,0<x\leq 1\}\cup \{0\}\times[-1,1],$ is a cellular set. I learned this by watching a video of Arunima Ray. You can see it is a cellular set by having each successive closed cell trace out more of the oscillations of the sine curve. Hence, cellular sets do not have to be path connected. Indeed, one can modify this example by considering the topologist's sine curve of revolution in $\mathbb R^3$, formed by rotating $Y$ around the $y$-axis. This space is not simply connected.
As for connectivity, suppose $X$ is separated by sets $U$ and $V$ which are open in the ambient manifold. Then $C_i\subset U\cup V$ for sufficiently large $i$, but then $U\cup V$ would separate $C_i$ which is a contradiction.
Cellular sets seem like they can get pretty pathological, but in fact we have the following theorem.
Theorem: If $X$ is a cellular set in a compact manifold $M$, then the quotient map $\pi\colon M\to M/X$ is a homeomorphism.
The proof of this fact is a beautiful function space argument. One constructs homeomorphisms $h_\epsilon\colon M\to M$ which collapse $X$ to small radius and then argue that the sequence $h_\epsilon$ converges to a limit homeomorphism in the uniform topology on the appropriate function space.
In a follow-up post, I'll sketch the proof of the topological Schönflies theorem.
