TITLE: Minimal genus, adjunction inequality
QUESTION [5 upvotes]: Let's consider closed simply-connected 4-manifold $M$ and some $a\in H^2(M)$. It is very natural question to estimate minimal $g$ that $a$ can be presented as embedded surface of genus $g$.
As I know there is the adjunction inequality for estimation of minimal genus via Seiberg-Witten theory.
Question 1: Does there exist other methods to estimate minimal genus ?
I heard that there are homeomorphic but not diffeomorphic 4-manifolds $M,N$ such that for some $a\in H^2$, $a$ has different minimal genus in $M,N$.
Question 2: Could you give me such examples? As I understand it should be some manipulations with Seiberg-Witten invariants...

REPLY [5 votes]: To add a bit to Andrey's answer, you can improve the method of Hsiang and Szczarba (also Kotschick and Matic deserve credit here) a bit using Furuta's approach to Seiberg-Witten theory. The method can give genus bounds on four manifolds where the adjunction formula doesn't apply (for example on $\mathbb{CP}^2 \# \mathbb{CP}^2$). I wrote a short paper on this: Math. Res. Lett. 5 (1998), no. 1-2, 165–183. The arXiv version is here: http://arxiv.org/pdf/dg-ga/9704010v1. Theorem 1.6 gives a typical genus bound from this method.
The basic idea is the following. If your class has certain divisibility properties and it is represented by an embedded surface, then the four manifold obtained by taking a cyclic cover branched along the surface will be a new four manifold with certain properties which can be used to constrain the genus of the embedded surface. Under appropriate hypotheses, the cover will be a spin four manifold with a cyclic group action. Refining Furuta's proof of the 10/8ths theorem for spin four manifolds to include the presence of a cyclic action, one obtains a bound on the size of the signature in terms of the second Betti number which then translates into a genus bound on the original manifold.