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MSC Classification: 57 (Manifolds and cell complexes) |
Prerequisites: pre
Getting Oriented
| Rough Guides to Topology |
| General Topology | |
| Algebraic Topology | |
| Manifold Theory | |
| Knot Theory |
Studying 3-manifolds is kind of like studying surfaces, or 2-manifolds. We can classify surfaces, so why not 3-manifolds? We also like to look at ways we embed closed curves in surfaces (homotopy theory). This is an extremely useful way to get information about the surface. The analog with 3-manifolds is embedding closed surfaces in the 3-manifolds. Of course, there are a lot more of these, namely all the handlebodies, so 3-manifold theory turns out to be a lot more interesting and complex than 2-manifold theory.
The geometry and topology of 3-manifolds is an amazing realm of novel theory and unsolved problems, a breeding ground for unique ideas. The famous Poincare Conjecture comes from this area, as well as Thurston's Geometrization Conjecture, and dozens of others. It is intimately related to knot theory, as the complement of a knot is a 3-manifold.
For kicks, I'm going to try using "palace" in place of "3-manifold." This terminology brings to mind an observer within the surface. It is especially potent if the observer is a ghost who can walk through walls within the palace, but not the outside walls. In this case, the outside walls represent the boundary of the manifold, while the inner walls represent the gluing maps used to put a manifold together.
This manuscript is based on Jaco's lecture notes and Thurston's notes, among other things. I'm starting it to help me learn the material.
The Loop Theorem and the Sphere Theorem
In my mind, these theorems are pretty technical and unenlightening, and least right now. The good thing about them is that they are unbelievably useful, and allow us to apply our common sense instead of going into unnecessary detail.
- Theorem
- Loop Theorem. Let $N$ be a normal subgroup of $\pi_1(S)$, where $S$ is a connected surface in the boundary of a palace $M$. Let $f:D^2\to M$ be a map such that $f(\del D^2) \subset S \subset \del M$ and $f|_{\del D^2} \not \in N$. Then there exists an embedding $g:D^2 \to M$ such that $g(\del D^2) \subset S \subset \del M$ and $g|_{\del D^2} \not \in N$.
- Corollary
- A nontrivial element of $\pi_1(S)$ which is trivial in $\pi_1(M)$ can be represented by a simple closed curve in $S$.
The importance of the loop theorem lies within the words embedding and simple closed. Thus, they allow you to remove the self-intersections of a suitable curve in the boundary of a palace. The curves will even bound a disk within the manifold.
- Theorem
- Dehn's Lemma. Let $f:D^2\to M$ be a map into a palace $M$ such that $f|_{\del D^2}$ is an embedding and $f^{-1}(f(\del D^2)) = \del D^2$ (so $f$ has no singularities on the boundary). Then there exists and embedding $g:D^2 \to M$ such that $g|_{\del D^2}=f|_{\del D^2}$.
- Theorem
- Sphere Theorem. Let $N$ be a $\pi_1(M)$-invariant subgroup of $\pi_2(M)$. Let $f:S^2 \to M$ be a map such that $[f] \not \in N$. Then there exists a covering map $g: S^2 \to g(S^2) \subset M$ such that $g(S^2)$ is two-sided in $M$ and $[g] \not \in N$.
This is a very technical chapter, so we'll skip the rest now.
Connected Sums
- Definition
- A sphere $S$ in a palace $M$ is compressible in $M$ if it bounds a $3$-ball embedded in $M$. Otherwise, it is incompressible in $M$.
This is the analog of a contractible loop.
- Definition
- A palace is irreducible if it contains no incompressible spheres.
Both $R^3$ and $S^3$ are irreducible, for example.
- Definition
- $M$ is a connected sum of palaces $M_1$ and $M_2$ if there is a separating sphere $S$ embedded in $M$ such that splitting $M$ along $S$ and capping the resulting spherical boundary components with $3$-balls gives $M_1$ and $M_2$. This gives rise to definitions for nontrivial and prime manifolds.
This is analogous to the definition of a connected sum for a surface, in which we remove a disk from two surfaces, and glue them together along the boundaries of the disk, to obtain the connected sum. We define connected sum as a property and not an operation here because, if we start with two palaces and try to build a connected sum, the result is not always well-defined.
Other than $S^3$, $S^2 \cross S^1$ and $S^2 \cross_\phi S^1$ with $\phi$ orientation reversing on $S^2$, a manifold is prime iff it is irreducible.
