TITLE: Heegaard splitting of maps between 3-manifolds
QUESTION [8 upvotes]: Let $M$ and $M'$ be closed oriented connected 3-manfolds and let $f : M \to M'$ be a continuous map.  Do there exist Heegaard splittings $M = H_1 \cup H_2$ and $M' = H_1' \cup H_2'$ and a map $f'$ homotopic to $f$ so that $f'$ preserves the Heegaard splittings in the sense that $f'(H_1) \subset H_1'$ and $f'(H_2) \subset H_2'$?
Let $\Sigma = H_1 \cap H_2$ and $\Sigma' = H_1' \cap H_2'$.  Note that by naturality of Mayer-Vietoris, the degree of $f'$ will have to equal the degree of $f'|_\Sigma : \Sigma \to \Sigma'$ which gives some restriction on the possible Heegaard splittings.  
I believe this should not be true for $\deg(f) >1$ but I can not seem to find an obstruction.  
Thanks!

REPLY [4 votes]: Waldhausen, Friedhelm, On mappings of handlebodies and of Heegaard splittings, Topology of Manifolds, Proc. Univ. Georgia 1969, 205-211 (1971). ZBL0282.57003.  Bielefeld.
See this paper of Waldhausen, which discusses the
case of degree 1 maps. However, in the proof of Theorem 2.1 in his paper, he sketches a proof that a map between 3-manifolds may be homotoped to preserve Heegaard splittings (the details are not given, but it seems clear that the hypothesis of degree 1 is not used in this part of the theorem). He seems to think this is "well-known", but doesn't give a reference.