minimal genus of surface representing a homology class Can you give an example of a 4-manifold with embedded surfaces of different genera representing the same homology class? 
(If $i:\Sigma\to X$ is an embedding of a closed surface in a closed smooth 4-manifold $X$, then $\Sigma$ represents $a\in H_2(X)$ if $i_*[\Sigma]=a$ ($[\Sigma]$ is the fundamental class of $\Sigma$)). 
 A: You can just add handles to any surface inside the 4-manifold.
To be more explicit, for an embedded surface in a 4-manifold, take a chart in which the surface locally looks like $R^2$ in $R^4$, cut out a disk and glue in an embedded punctured torus (inside the 4-dimensional chart). 
You can repeat this procedure to get surfaces of arbitrarily high genus.
A: As user39082 said, the genus of a representing surface is very non-unique. You could add handles without changing the homology class (but changing genus) or (since you didn't ask for connectedness) you could just take the disjoint union with nullhomologous tori.
Using this non-uniqueness in 3-manifold theory you develop a norm called the Thurston norm. It measures the minimal complexity of a properly embedded surface representing a homology class in $H_2(M,\partial M)$. It turns out to be a nice map and extends to $H_2(M,\partial M;\mathbb R)$. The Thurston norm pops out in knot theory (before it was named after Thurston) called the "genus of a knot", which is just the genus of a minimal Seifert surface (which is basically a represantative of a generator of $H_2(M,\partial M) = H^1(M) =\mathbb Z$).
Hence to answer your question in a pretty intuitive way, look at Seifert surfaces of knots and the complement $M$ gives you a 4-manifold by $M\times S^1$ which gives you examples you might already know.
