  This article reviews the foundations of the theory of the Bott index of a
pair of unitary matrices in the context of condensed matter theory, as
developed by Hastings and Loring in J. Math. Phys. (51), 015214, (2010) and
Annals of Physics 326 (2011) 1699-1759, providing a novel proof of the equality
with the Chern number. The Bott index is defined for a pair of unitary
matrices, then extended to a pair of invertible matrices and homotopic
invariance of the index is proven. An insulator defined on a lattice on a
2-torus, that is a rectangular lattice with periodic boundary conditions, is
considered and a pair of quasi-unitary matrices associated to this physical
system are introduced. It is shown that their Bott index is well defined and
the connection with the transverse conductance, the Chern number, is
established proving the equality of the two quantities.
