Topological computation of some Stokes phenomena on the affine line

Let M be a holonomic algebraic D-module on the affine line, regular everywhere including at infinity. Malgrange gave a complete description of the Fourier-Laplace transform Mc, including its Stokes multipliers at infinity, in terms of the quiver of M. Let F be the perverse sheaf of holomorphic solutions to M. By the irregular Riemann-Hilbert correspondence, Mc is determined by the enhanced Fourier-Sato transform Ff of F. Our aim here is to recover Malgrange's result in a purely topological way, by computing Ff using Borel-Moore cycles. In this paper, we also consider some irregular M's, like in the case of the Airy equation, where our cycles are related to steepest descent paths.