  The purpose of this paper is to establish analogues of the classical
Lefschetz Section Theorem for smooth tropical varieties. More precisely, we
prove tropical analogues of the section theorems of Lefschetz,
Andreotti-Frankel, Bott-Milnor-Thom, Hamm-L\^e and Kodaira-Spencer, and the
vanishing theorems of Andreotti-Frankel and Akizuki-Kodaira-Nakano.
  We start the paper by resolving a conjecture of Mikhalkin and Ziegler (2008)
concerning the homotopy types of certain filtrations of geometric lattices,
generalizing several known properties of full geometric lattices. This
translates to a crucial index estimate for the stratified Morse data at
critical points of the tropical variety; it can also by itself be interpreted
as a Lefschetz-type theorem for matroids.
