# Arithmetic Morse theory?

Is there any analogue of Morse theory in Number theory? Naive idea arising in my head is that defining a Morse function on scheme and find etale cohomology using that function. Since I'm not an expert about algebraic geometry and Morse theory, I can't advance my thoughts.

Instead, Lefschetz considers a map from a smooth projective variety to $\mathbb P^1$. One demands a Morse-like condition that the critical points of this map are as simple as possible. Applying vanishing cycles theory one can relate the cohomology of the total space to the critical points but one also needs to know as a starting point the cohomology of a smooth fiber.