  We show that a partition function of topological twisted N=4 Yang-Mills
theory is given by Seiberg-Witten invariants on a Riemannian four manifolds
under the condition that the sum of Euler number and signature of the four
manifolds vanish. The partition function is the sum of Euler number of
instanton moduli space when it is possible to apply the vanishing theorem. And
we get a relation of Euler number labeled by the instanton number $k$ with
Seiberg-Witten invariants, too. All calculation in this paper is done without
assuming duality.
