For any $n \geq 1$, let $\Sigma\_n$ denote the closed orientable surface of genus n. In <http://arxiv.org/abs/1202.6302>, the authors showed that for any $n$, there is a degree *two*, $\pi\_1$-surjective, map $ f: S^1 \times \Sigma\_n \to \#\_n S^1 \times S^2$ (the $n$-fold connected sum). 


Question: For any $n \geq 2$, is it possible to produce a possibly different closed orientable surface $B$ together with a degree *one* map $f': S^1 \times B \to \#\_n S^1 \times S^2 $ ? 


For $n=1$ of course it is possible to (for example) set $B=S^2$ and find a degree one map (the identity). 



Is there a degree one map from a product $B\times S^1 \to \#_n S^2 \times S^1$ for any n
You can find such a map which is degree one. The cut number $n$ of a closed 3-manifold $M$ is the maximal number of 2-sided disjointly embedded surfaces $\Sigma\_1,\ldots,\Sigma\_n$ which do not separate $M$. For each connected surface $\Sigma\_i$, we can make a degree one map to the 2-sphere by taking a spine of the surface (a embedded graph whose complement is a disk) and crushing it to a point. Do this for each component of the surfaces $\Sigma\_i$ in the 3-manifold mapping to $S^2\times \ast$ for each factor of $\#\_n S^2\times S^1$. The complement of these 2-spheres is a $2n$-punctured 3-sphere. There is no obstruction to extending the maps of $\cup\_n \Sigma\_i$ to $\cup\_n S^2\times \ast$ to a map $M-\Sigma\_i \to \#\_n S^2\times S^1 - \cup\_n S^2 \times \ast$, and get a degree one map from $M$ to $\#\_n S^2\times S^1$. Now, we note that $\Sigma\times S^1$ has cut number at least $g$, by taking $g$ disjoint non-separating curves on $\Sigma$ and crossing them with $S^1$. 


Remark on extending maps: Take a vertex in $M$, and connect it by $2n$ arcs to the surfaces $\Sigma\_i$, one for each side of $\Sigma\_i$ (which is possible by the non-separating assumption). Then the boundary of a tubular neighborhood of this graph and $\cup\_i \Sigma\_i$ gives a connected surface. We map do a similar thing in $\#\_n S^2\times S^1$ with the surfaces $\cup\_n S^2\times \ast$, where the boundary of the tubular neighborhood is a 2-sphere which bounds a ball. Extend the map on surfaces to a map on graphs, and then of a tubular neighborhood (getting a degree one map), then we may extend the map over the ball (just crush the degree one map of the surface to $S^2$ to a point to extend over the ball). 


