Hi, I am trying to show that for M a 4-manifold, and [a]_2 a class in H_2(M,Z) , there is always a surface that represents [a]_2 , i.e., there exists a surface S , and an embedding i of S into M , with [ioS]_2 =[a]_2. (Equiv.: there exists S, and an embedding i of S of M , so that a triangulation of S induces the class [a]_2) ** What I have ** If M is simply-connected, so that Pi_1(M)=0 (Notation: Pi_1:=Fund. Grp.) Then, by the Hurewicz Theorem (Hip, Hip Hurewicz!) Pi_2(M) is actually Isomorphic to H_2(M;Z) , so that every class in H_2(M;Z) can be represented as an embedded sphere S^2 (possibly with self-intersections, which can be smoothed away ). **BUT** I can't think of what can be done if M is not simply-connected. Any Ideas.?