Topology of Sobolev mappings

This paper addresses some topological and analytical issues concerning Sobo lev mappings between compact Riemannian manifolds. Among the results we obt ained are unified proofs of various generalizations of results obtained in a recent work of Brezis and Li. In particular we solved two conjectures in [BL]. We also give a topological obstruction for the weak sequential densit y of smooth maps in a given Sobolev mapping space. Finally we show a necess ary and sufficient topological condition under which the smooth maps are st rongly dense in the Sobolev spaces. The earlier result, Theorem 1 of [B2], was shown to be not correct.