n-manifolds into Rn+k. Here a bordism between two immersions io: Mo Rn+k and

The bordism group of immersions of oriented n-manifolds into Rn+k is identified with the stable homotopy group IIn+k(MSO(k)). We study these groups for n-2 < k < n, and discuss the behaviour of double points and their relation with the corresponding bordism groups of embeddings. il: Ml 9 Rn+k iS an immersion of a compact oriented (n + l)-manifold j: Wv Rn+k X I such that dW = Mo U -Ml and jlM0 = io X {0} and jlM, = il X {1}. In the usual manner bordism defines an equivalence relation and bordism classes form an abelian group (under disjoint union) which is identified with the stable homotopy group sTn+k(MSO(k)), of the Thom space MSO(k) of the canonical oriented k-plane bundle over BSO(k). The object of this paper is to study the groups If2n k for n-2 s k < n and to discuss the behaviour of double points and the relation of these groups with the corresponding bordism groups of embeddings. Bordism groups of immersions were studied first by Wells (We) who determined the unoriented groups ISn n and IX4n4n_l These results were extended by Koschorke and OLk who completed the computations of I Sn k for n-2 < k < n (see (K, §10)). We shall make use of these computations.. This work is part of the author's thesis under the supervision of B. J. Sanderson at Warwick University. It is a pleasure to express thanks for his help and for helpful conversations with the topology staff at the Centro de Investigacion del IPN. 2. Some exact sequences involving bordism groups of immersions. We describe three exact sequences and compute some low-dimensional bordism groups appearing in them. The first sequence was obtained by Szucs (Sz) and Koschorke (K). The other two sequences are due to Salomonsen (Sa). We refer to these articles for a detailed description of the sequences. Given a subgroup G of the orthogonal group O(m) we will denote by &2J the bordism group of j-manifolds whose stable normal bundle admits a reduction to G. We will be mainly interested in the cases where G is