The Whitney embedding theorem for tropical torsion modules: Classification of tropical modules

Abstract

We prove here a tropical version of the well-known Whitney embedding theorem [32] stating that a smooth connected m-dimensional compact differential manifold can be embedded into R 2 m + 1 . The tropical version of this theorem states that a tropical torsion module with m generators can always be embedded into the free tropical module R ̲ p , where p (equals to 2 for m = 2 , and 3 ⩽ p ⩽ m ( m - 1 ) otherwise) is the number of rows supporting the torsion, when the generators are given by the (independent) columns of a matrix of size n × m . As a corollary, we get that tropical m-dimensional torsion modules are classified by a ( m - 1 ) m ( m - 1 ) - 1 -parameter family.