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! 



Heegaard splitting of maps between 3-manifolds
*Waldhausen, Friedhelm*, On mappings of handlebodies and of Heegaard splittings, Topology of Manifolds, Proc. Univ. Georgia 1969, 205-211 (1971). [ZBL0282.57003](https://zbmath.org/?q=an:0282.57003). [Bielefeld](https://pub.uni-bielefeld.de/download/1782175/2313705).


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. 


