Given a connected, compact, totally geodesic submanifold Y^m of noncompacttype inside a compact locally symmetric space of noncompact type X^n, weprovide a sufficient condition that ensures that [Y^m] is nonzero in H_m(X^n;R); in low dimensions, our condition is also necessary. We provide conditionsunder which there exist a tangential map of pairs from a finite cover(X-bar,Y-bar) to the nonnegatively curved duals (X_u,Y_u).
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