Oriented bordism groups of immersions

Abstract

Let IΩn, k denote the bordism group of immersions of closed oriented n-manifolds into (n + k)-space. The object of this paper is to study certain group extension problems arising from Pastor's calculations of IΩn, k.The bordism group of immersions was first studied by Wells [12] who calculated the unoriented bordism group I Rn, k for k = n and k = n − 1 ≡ 3(4). Later these unoriented bordism groups were completely determined by Koschorke and Olk for k ≥ n − 2 with the help of an exact sequence measuring the difference between IRn, k and Rn (see [4]). A similar program has been carried out by Pastor [7] to determine the oriented bordism group I Ωn, k for k ≥ n − 2 except for certain group extension problems and some low dimensional cases.

Analogues of the Atiyah--Wall exact sequences for cobordism groups of singular maps

“…Moreover, if k + 1 is not a power of 2 and k = 1, then by [Ko,theorem 10.8] In the following we shall always use the first diagram above lemma 9.7 which yields (again by the 5-lemma) that α int imm is epi in all cases. We will also use that by proposition 9.1 the endomorphism ϕ 2k is 0 on the direct complement of Ω k in Imm SO (k, k) and we obtain the forms of Imm SO (k + 1, k) and Imm O (k + 1, k) from [Li,theorem 6] and [Ko,theorem 10.8] respectively.…”

“…Proof of (2). If k ≡ 2 (4) and k + 2 is not a power of 2, then we have Imm SO (k + 1, k) ∼ = Ω k+1 ⊕ Z 4 and by [Li,p. 472] the involution ι in theorem I is the identity, hence the cokernel of ϕ 2k+1 is Ω k+1 ⊕ Z 2 .…”