Complexities of 3-manifolds from triangulations, Heegaard splittings, and surgery presentations

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.