It is conjectured that any spin 4-manifold $X$ satisfies the equality $b_2(X)/\sigma(X) \geq 11/8$, where $b_2(X)$ is the second Betti number and $\sigma(X)$ is the signature of $X$. In this talk I would like to explain the equality could be improved when the intersection form on $H_1(X,{\bf Z})$ has some properties.

The main tool is the Seiberg-Witten equation. When the first Betti number $b_1(X)$ is positive, the equation is regarded as a ``proper-like" nonlinear map between two Banach bundles over the Jacobian torus. The formal difference of the two Banach bundles is an index of a family of elliptic operators with some symmetry. The outline of the argument is as follows: When the intersection form on $H_1(X,{\bf Z})$ is non-trivial, then the index becomes non-trivial. This non-triviality gives a restriction for the existence of the ``proper-like" map.