Assume that X is an oriented smooth (n+k)-manifold. Then the kernel of the forgetful map F considered in this work consists of immersions f: Mn → X nullbordant as a continuous map. Using an exact sequence of normal bordism groups previously given, we present a homological characterization of the kernel of the forgetful map F. Also, we prove that Ωi(X, εs - ηs and Hi(X,Z) are -isomorphic for i≤3 and C2-isomorphic for i≤2, where C2,3 (resp. C2 is the class of abelian groups whose elements have order 2p. 3q (resp. 2p), and ηs is an orientable stable vector bundle over X. © 2009 Pushpa Publishing House.