Is there a complete set of topological invariants for smooth 3-manifolds? Does there exist a complete set of topological invariants for smooth 3-manifolds? (If not for all smooth 3-manifolds, maybe for a certain class?)
By equipping the manifold with a metric, can some invariants be expressed in terms of the metric? For example, in 2d, the genus can be expressed as the integral of the Riemann curvature, due to the Gauss-Bonnet theorem.
 A: There exist various ways of classifying $3$-manifolds, but the result is nowhere near as simple as the one for closed $2$-surfaces, where orientability and genus (or, equivalently, $H^1$) give a full classification. Like $2$-manifolds, though, the smooth, continuous, and PL categories are all equivalent in dimension $3$. I'm also going to be considering only the oriented, closed case below.

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*A (closed, oriented) $3$-manifold $M\not = S^3$ is prime if $M = N \# N'$ forces $N = S^3$ or  $N' = S^3$. Then $X$ is a connected sum $X = M_1 \# \cdots \# M_r$ for some prime $M_i\not = S^3$, and the $M_i$ are unique up to rearrangement.

*If $X \not = S^1 \times S^2$ is prime, then it admits a unique minimal toral decomposition. It takes a bit of work to define exactly what a "toral decomposition" is, but the key concept is that of a Seifert manifold, which is basically a circle bundle over a torus with a finite number of bad fibers. The idea is that such a decomposition leaves you with a bunch of Seifert manifolds, which are much easier to classify (as well as another kind of manifold that's also much easier to classify, but $3$-manifold geometry is complicated).

*Given a $3$-manifold $M$ and a link $L\subset M$ (i.e., a finite set of knotted circles in $M$), you can obtain a new manifold by cutting out a neigborhood of each component of $L$, twisting it by a certain simple homeomorphism, and pasting it back into $M$. This technique is called Dehn surgery, and it turns out to produce all the (closed, oriented) $3$-manifolds, along with some results about how manifolds created by related Dehn surgeries are themselves related.

*Along the lines of the toral decomposition above, Thurston's geometrization conjecture states that every (closed, oriented) $3$-manifold can be decomposed by cutting along tori into pieces that are each one of eight well-known, finite-volume types. Furthermore, the geometry of these types can more or less be read off from the fundamental group. (Despite the name, Thurston's geometrization conjecture was proved by Perelman in the early 2000s. Among other things, it implies the Poincare conjecture in dimension $3$.)

These are some of the main types of classification, but the subject of $3$-manifolds gets technical very quickly and relies on many nontrivial results from both algebraic and geometric topology. If you're new to the subject and want something that's interesting but doesn't require as long of a wind-up to get to the results, take a look at Mostow rigidity for hyperbolic manifolds. It has the same flavor of "metric invariants are secretly topological invariants" as Gauss-Bonnet, and it often pops up in $3$-manifold geometry.
