  We establish a lower bound for the number of sign changing solutions with
precisely two nodal domains to the singularly perturbed nonlinear elliptic
equation -{\epsilon}^{2}{\Delta}_{g}u+u=|u|^{p-2}u on an n-dimensional
Riemannian manifold M, p in (2,2n/(n-2)), in terms of the cup-length of the
configuration space of M. We give a precise description of the asymptotic
profile of these solutions as {\epsilon} goes to 0. We also provide new
estimates for the cup-length of the configuration space of M.
