Assume that X and Y both have piecewise linear integral structures. Let f : X → Y be a piecewise linear homeomorphism such that


• V-1: f (t · x) = t · f (x), and


• V-2: x ∈ X is integral if and only if f (x) is integral in Y.


Then f is volume preserving.


how can i prove it? is it any classical proposition?



volume preserving homeomorphism between piecewise linear integral structures
To show that f is volume preserving, we only need to show that each of its linear pieces is volume preserving. So, it’s a property of linear maps.
If a linear map $f:R^n \to R^n$ sends integer points to integer points, it must have integer coefficients. Because of the “only if” part of the hypothesis, its inverse also has integer coefficients. Therefore, its determinant is equal to +1 or -1, and f is consequently volume preserving.


