The fourth oriented cobordism group \(\Omega_4\) is isomorphic to \(\mathbb Z\) (Q1404788)

Using generic singular maps, the author proves the following theorem of \textit{V. A. Rokhlin} [Dokl. Akad. Nauk SSSR, n. Ser. 84, 221--224 (1952; Zbl 0046.40702)]: Theorem 1. The fourth oriented cobordism group \(\Omega_4\) is isomorphic to the group of integers; in short, \(\Omega_4\approx\mathbb{Z}\). Corollary 2 (signature formula of Rokhlin). For any smooth, closed \(4\)-manifold \(M\), we have \[ \sigma(M)= {\langle p_1(M),[M]\rangle\over 3}. \] For the proof the author considers an invariant under any generic cobordism of a generic map \(g\) of a \(4\)-manifold \(M^4\) in \(\mathbb{R}^6\). This invariant is a difference \(t(g)- l(g)\), with \(t(g)\) the algebraic number of triple points of \(g\) and the number \(l(g)\in \mathbb{Z}\) is defined as follows: Let \(n(g)\) be a short outward pointing normal vector field of the \(1\)-manifold of the Whitney umbrella points \(\Sigma(g)\) in the double-point surface \(\Delta(g)\) immersed in \(\mathbb{R}^6\). Let \(\widetilde\Sigma\) be the curve formed by endpoints of \(n(g)\) and then \(l(g)\) is the linking number of \(\widetilde\Sigma\) with the image of \(g\). In this way a homomorphism \(\Phi: \Omega_4(\mathbb{R}^6)= \Omega_4\to \mathbb{Z}\) is obtained with \(\Phi(g)= t(g)- l(g)\). It is proved that this homomorphism is nontrivial and mono. The proof of these facts and of the fact that \(\Phi(g)\) is invariant under any generic cobordism are very interesting. For example, as a spin-off it is proved that the cobordism class of \(M^4\) contains a manifold \(\widetilde M\) that can be embedded in \(\mathbb{R}^7\) and various other results on generic maps are given.