  We study exact Lagrangian immersions with one double point of a closed
orientable manifold K into n-complex-dimensional Euclidean space. Our main
result is that if the Maslov grading of the double point does not equal 1 then
K is homotopy equivalent to the sphere, and if, in addition, the Lagrangian
Gauss map of the immersion is stably homotopic to that of the Whitney
immersion, then K bounds a parallelizable (n+1)-manifold. The hypothesis on the
Gauss map always holds when n=2k or when n=8k-1. The argument studies a filling
of K obtained from solutions to perturbed Cauchy-Riemann equations with
boundary on the image f(K) of the immersion. This leads to a new and simplified
proof of some of the main results of arXiv:1111.5932, which treated Lagrangian
immersions in the case n=2k by applying similar techniques to a Lagrange
surgery of the immersion, as well as to an extension of these results to the
odd-dimensional case.
