TITLE: How to get convinced that there are a lot of 3-manifolds?
QUESTION [25 upvotes]: My question is rather philosophical : without using advanced tools as Perlman-Thurston's geometrisation, how can we get convinced that the class of closed oriented $3$-manifolds is large and that simple invariants as Betti number are not even close to classify ?
For example i would start with :

If $S_g$ is the closed oriented surface of genus $g$, the family $S_g \times S^1$ gives an infinite number of non pairwise homeomorphic $3$-manifolds.
Mapping tori of fiber $S_g$ gives as much as non-diffeomorphic $3$-manifolds as conjugacy classes in the mapping class group of $S_g$ which can be shown to be large using the symplectic representation for instance.

I think that I would like also say that Heegaard splittings give rise to a lot of different $3$-manifolds which are essentially different, but I don't know any way to do this. 
So if you know a nice construction which would help understanding the combinatorial complexity of three manifolds, please share it :)

REPLY [5 votes]: Read Chapter 4 of Thurston's notes http://library.msri.org/books/gt3m/. He produces infinitely many closed hyperbolic 3-manifold of different volumes, and hence non-homeomorphic, just by doing Dehn surgery on the figure 8 knot in $S^3$. This does not used advanced geometrisation theorems, just some basic (but clever) hyperbolic geometry, combined with the Mostow rigidity theorem.