On bordism groups of immersions

Abstract

Abstract. The bordism group of immersions of oriented n-manifolds into R"+/t is identified with the stable homotopy group n^ + i(MSO(^)). We study these groups for n -2 *s k < n, and discuss the behaviour of double points and their relation with the corresponding bordism groups of embeddings.

Oriented bordism groups of immersions

“…Here a(n) is the number of ones in the binary expansion of n. As the first step of determining C n fc , Pastor calculated Q^*' for rn ^ 2 and k > 2 using a normal bordism sequence of Koschorke (see theorem 9 -3 of [4]). A close look at that exact sequence convinces us that Pastor's proposition 2-5 in fact gives the values of Q^' for m ^ 2 with no restriction on k. Now using Koschorke's sequence in Theorem 1 instead of (2-2) in [7], it is readily seen that theorems 3-1-3-3 of Pastor [7] remain valid in the range of n ^ 2k. Now return to the group extension problem.…”

“…The definitions of other homomorphisms could be found in [4]. Notice that the bordism group Q^f c) used in [7] is canonically isomorphic to Q m (RP°° xBSO k , A ®f k + y k ), hence to Q<£> if m < k. The exact sequence (2-2) in [7] used by Pastor, which was also developed independently by Sziics [9], is a weak form of the exact sequence above.…”

“…In the case of k ^ n -2, the first part has been done for n<2k-i by Pastor [7] using known results on immersing manifolds up to oriented bordism. For example, K nk = Q. n with a single exception when k = n -2 and a(n) = l; in that case K n k c Cl n is a subgroup of index 2 generated by elements [M] satisfying w 2 w n _ 2 (M) = 0.…”

“…ra (see [4]). A similar program has been carried out by Pastor [7] to determine the oriented bordism group ICl n k for k ^ n -2 except for certain group extension problems and some low dimensional cases.…”

Cobordisms of embeddings of oriented five-dimensional manifolds in Euclidean space

“…As shown by Pastor (see [2], Theorem 4.2), if n > 3 and neither n nor n + 1 are powers of two, then there is a short exact sequence 0 ~ Z2 -* Eft.,.-1 ---* f~. -* 0.…”