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A complex projective manifold is rationally connected, resp. rationallysimply connected, if finite subsets are connected by a rational curve, resp.the spaces parameterizing these connecting rational curves are themselvesrationally connected. We prove that a projective scheme over a global functionfield with vanishing "elementary obstruction" has a rational point if itdeforms to a rationally simply connected variety in characteristic 0. Thisgives new, uniform proofs over these fields of the Period-Index Theorem, thequasi-split case of Serre's "Conjecture II", and Lang's $C_2$ property.

Rational points of rationally simply connected varieties over global function fields