  We show a non-existence result for some class of equivariant maps between
sphere bundles over tori. The notion of equivariant KO-degree is used in the
proof. As an application to Seiberg-Witten theory, for a connected closed
oriented spin 4-manifold with indefinite intersection form, we have a new bound
of the second Betti number by the signature and a non-negative integer
determined by the quadruple cup product on the first cohomology group and some
maps in KO-theory induced from the Albanese map.
