[Math] Most manifolds are hyperbolic

Question

I heard the claim as in the title for a long time, but can not find the precise reference for this claim, what's the reference with proof for this claim? Thanks for the help.
To be more precise, is there a canonical topology structure on the space $\Omega$ of all compact $n$-dim smooth manifolds, such that for any compact smooth $n$-dim manifold $M^n$, any neighborhood $U$ of $M^n$ in $\Omega$, there is a $n$-dim smooth manifold $N^n\in U$ such that $N^n$ admits a Riemannian metric with curvature equal to $-1$. Everything in my mind is just Riemannian hyperbolic, no complex structure involved.

Answer 1

The quotes are from Thurston's survey paper Three dimensional manifolds, kleinian groups and hyperbolic geometry page 362:

2.6. THEOREM [Th 1]. Suppose $L \subset M^3$ is a link such that $M — L$ has a hyperbolic structure. Then most manifolds obtained from $M$ by Dehn surgery along $L$ have hyperbolic structures. In fact, if we exclude, for each component of $L$, a finite set of choices of identification maps (up to the appropriate equivalence relation as mentioned above), all the remaining Dehn surgeries yield hyperbolic manifolds.

 

Every closed 3-manifold is obtained from the three-sphere $S^3$ by Dehn surgery along some link whose complement is hyperbolic, so in some sense Theorem 2.6 says that most 3-manifolds are hyperbolic.