Say I have two smooth one-cycles $Z, Z'$ on a 3-manifold $M$, i.e. just formal linear combinations of (let us say disjoint) smooth circles. Assume they're homologous. 



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> Is there a map to a surface $\pi: M \to S$ and homologous zero-cycles $P, P'$ on the surface such that $\pi$ is a submersion over $P, P'$ and one has $\pi^{-1}(P) = Z$ and $\pi^{-1}(P') = Z'$?
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Homologous curves and maps to surfaces
I suspect the answer to this question is no in general. 


Suppose $Z$ has two components $Z\_1 \cup Z\_2$ with different multiplicities (as a formal linear combination of knots) and $Z'$ has one component. Then $P$ must have two points $P=\{p\_1,p\_2\}$ (ignoring multiplicity), and $P'$ one $P'=\{p'\}$. Then $\pi\_1(M-(Z\cup Z'))$ surjects a free group of rank at least two (the meridians of $Z\_i$ go to loops about $p\_i$, which must generate a subgroup of $\pi\_1(S-(P\cup P'))$ of rank at least two). 


The *corank* of a group is the largest rank of a free group that it surjects. It is known that there are 3-manifold groups with [arbitrarily large $b\_1$ and corank 1.](http://msp.org/gt/2002/6-1/p15.xhtml) What is needed to get a counterexample to your question is a manifold with 3 torus boundary components which has fundamental group of corank 1, and satisfying an appropriate homological condition. I feel like a construction of the sort that Shelly Harvey carries out ought to have the right properties. 


**Addendum:** I looked at a paper of [Leininger-Reid](http://msp.org/agt/2002/2-1/p04.xhtml), which gives an example of a compact 3-manifold $M$ with hyperbolic interior, 4 boundary components, and cut number 1. From this, one can get an example which answers your question in the negative. By half-lives, half-dies, there is an oriented surface $\Sigma\subset M$ which is homologically non-trivial in $H\_2(M, \partial M)$, and intersects each boundary component in consistently oriented parallel non-trivial curves (in particular, non-empty). Now, perform Dehn fillings on each boundary component so that the intersection number with $\partial \Sigma$ are different to get a manifold $N$. Partition the cores of the Dehn fillings into $Z$ and $Z'$, with multiplicity determined by the intersection number of $\partial \Sigma$ with the meridian of each Dehn filling, and orienting the components of $Z'$ opposite to the induced orientation of $\partial \Sigma$. This gives the desired pair of cycles which are homologous and which cannot be induced by pullback from 0-cycles on a surface because the link complement has cut number 1. 


