Prescribing invariants for integral surfaces in the Grassmann bundle of 2-planes in 4-space (Q1910436)

A smoothly immersed integral surface \(M\) in the Grassmann bundle \(M@>s>>G_2(\mathbb{R}^4)@>\pi >>\mathbb{R}^4\) defines a possibly nonimmersed surface \(\pi\circ s(M)\) in \(\mathbb{R}^4\). Moreover, a surface in \(\mathbb{R}^4\) whose nonimmersive singularities are sufficiently well-behaved, lifts to an integral surface in \(G_2(\mathbb{R}^4)\). Thus a compact integral surface in \(G_2(\mathbb{R}^4)\) can be thought of as a resolution or lift of a nonimmersed surface in \(\mathbb{R}^4\) which is well-defined up to smooth diffeomorphisms of \(\mathbb{R}^4\) and \(M\). In this paper, the author defines five differential topological invariants for compact orientable rank 1 transversal integral surfaces in \(G_2(\mathbb{R}^4)\) and shows that these invariants are independent and can be freely prescribed. As a corollary, any 2-plane bundle can be realized as the pullback bundle \(s^*(\varepsilon)\) over a rank 1 transversal integral surface, where \(\varepsilon\) is the canonical bundle over \(G_2(\mathbb{R}^4)\).