Petters, AO, *Multiplane gravitational lensing. I. Morse theory and image counting*,
Journal of Mathematical Physics, vol. 36 no. 8
(1995),
pp. 4263-4275 .
**Abstract:**

*The image counting problem for gravitational lensing by general matter deflectors distributed over finitely many lens planes is considered. Counting formulas and lower bounds are found via Morse theory for the number of images of a point source not on a caustic. Images are counted within a compact region D not necessarily assumed to properly contain the deflector space. In addition, it is shown that Morse theory is applicable because multiplane time-delay maps Ty generically satisfy the Morse boundary conditions relative to D. All results obtained depend only on the topological properties induced in the lens planes by the deflector potentials and the behavior of grad Ty at boundary points of D. © 1995 American Institute of Physics.*