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We solve a connection problem for the equation of rank four in Simpson's even family, which we call Even four. We use an integral representation of solutions, and consider the twisted homology group associated with it. The rank of the homology group is five. Its four dimensional subspace, consisting of regularizable cycles, corresponds to the solution space of Even four. The connection problem is reduced to solving the linear relations among twisted cycles. There are only few equations for which connection problems are solved in this way.

Connection matrices associated with the generalized hypergeometric function 3F2 are determined. Our approach is based on the integral representation, and the theory of twisted homology groups (homology with coefficients in local system). We also consider the related monodromy representation and the monodromy-invariant Hermitian form.