  We show that the correction terms in Heegaard Floer homology give a lower
bound to the the genus of one-sided Heegaard splittings and the $\mathbb
Z_2$--Thurston norm. Using a result of Jaco--Rubinstein--Tillmann, this gives a
lower bound to the complexity of certain closed $3$--manifolds. As an
application, we compute the $\mathbb Z_2$--Thurston norm of the double branched
cover of some closed 3--braids, and give upper and lower bounds for the
complexity of these manifolds.
