I am working on the Gromov norm of submanifolds in the total space E of surface bundles over surfaces. So I am interested in knowing the minimal genus of a surface representing a given homology class in $H_2(E, \mathbb{Z})$.

It is known that every such homology class can be represented by an embedded surface, but this might not be the representant of minimal genus.

Is there in this context some result analogous to the one of D. Gabai for $3$-manifolds (see Corollary 6.18 of Foliations and the Topology of $3$-manifolds) stating that the immersed norm is the same as the embedded one?


In full generality, this is false.

I'll construct an example in the blow up $X$ of $\mathbb{CP}^2$ (which is, in fact, an $S^2$-bundle over $S^2$).

Kronheimer and Mrowka proved that every symplectic surface is genus-minimising in its homology class, and Severi proved that there are two irreducible plane complex curves $C_1, C_2\subset\mathbb{CP}^2$ of degree 4 with one and three nodes respectively. Notice that $g(C_1) = 2$, while $C_2$ is rational (i.e. it's an immersed sphere).

If you blow them up at one of their nodes, you get two symplectic surfaces in (a copy of) $X$, in the same homology class (namely, $4h-2e$, where $h$ is the line class and $e$ is the class of the exceptional divisor): the proper transform of $C_1$ is embedded and symplectic, so it's also genus-minimising in its homology class; on the other hand, the proper transform of $C_2$ is an immersed sphere (with two double points), so it has lower genus.

I would assume that similar counterexamples can be found for most (all?) surface bundles over surfaces, but I wouldn't swear.

In general, the minimal genus problem is a quite hard one, and the only answers I can think of come from gauge theory (Seiberg-Witten or Heegaard Floer).


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