  In this paper we give some results on the topology of manifolds with
$\infty$-Bakry-\'Emery Ricci tensor bounded below, and in particular of steady
and expanding gradient Ricci solitons. To this aim we clarify and further
develop the theory of f-harmonic maps from non-compact manifolds into
non-positively curved manifolds. Notably, we prove existence and vanishing
results which generalize to the weighted setting part of Schoen and Yau's
theory of harmonic maps.
