The graded cobordism group of codimension-one immersions (Q1811213)

For a differentiable manifold \(M\), let \(N(M)\) denote the cobordism group of codimension-one immersions into \(M\). This group was shown to be isomorphic to \([M^+, \Omega^{\infty}S^{\infty} \mathbb{R} P^{\infty}]\), e.g., by \textit{P. Vogel} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 7, 317--357 (1974; Zbl 0302.57011)], where \(M^+\) means the one-point compactification of \(M\). For \(M\) a three-manifold, the group \(N(M)\) has been studied by, e.g., \textit{R. Benedetti} and \textit{R. Silhol} [Topology 34, 651--678 (1995; Zbl 0996.57519)] and \textit{R. Gini} [Manuscr. Math. 104, 49--69 (2001; Zbl 0971.57039)]. In this article, the authors study the cobordism group \(N(M)\) for an \(n\)-manifold \(M\) equipped with a cubulation, \(M_0\subset M_1 \subset \cdots \subset M_n=M\). Let \(F^k\) be the subgroup of cobordism classes represented by codimension-one immersions into \(M\), which do not intersect \(M_{k-1}\). Then \(N(M)\) has a filtration \(0\subset F^n\subset F^{n-1}\subset\cdots \subset F^1\subset F^0=N(M)\). The authors investigate the group \(F^k/F^{k+1}\) for \(k\leq n\), associated with this filtration, and its graded group \(gr(N(M))\), which is isomorphic to \(N(M)\) as a set. They intend to describe \(F^k/F^{k+1}\) by using the cohomology group \(H^k(M;P_k)\), where \(P_k=N(\mathbb{R}^k)\). They succeed in doing so for \(n\leq 7\) under the condition \(\text{Ext}(H_3(M; \mathbb{Z}),\mathbb{Z}_8)=0\) when \(n=5,6,7\). The precise description of \(\text{gr}(N(M))\) is given in Theorem 1.1. The group \(N(M)\) for an orientable 4-manifold \(M\) is further studied and a short exact sequence of abelian groups containing \(N(M)\) is given in proposition 7.1. As a corollary to this, \(N(\mathbb{C} P^2)=\mathbb{Z}_2\) follows. Manifolds treated in this article are supposed to be closed.