  We study complexities of 3-manifolds defined from triangulations, Heegaard
splittings, and surgery presentations. We show that these complexities are
related by linear inequalities, by presenting explicit geometric constructions.
We also show that our linear inequalities are asymptotically optimal. Our
results are used in [arXiv:1405.1805] to estimate Cheeger-Gromov $L^2$
$\rho$-invariants in terms of geometric group theoretic and knot theoretic
data.
