Topological insulators are materials that are insulating in their interior but can support the flow of electrons on their surface.

 The underlying cause for this phenomenon is time-reversal symmetry. Magnetic materials on the other hand break time-reversal symmetry,

but some of them may also be classified as topological insulators, namely, insulating in the but with conduction or transport properties on the surfaces or hinges.

These materials are called “axion insulators”.

The underlying cause for the phenomenon is the existence of a symmetry which is a combination of time-reversal or space inversion with a rotation or a translation.

To understand physically and mathematically this phenomenon several approaches have been proposed. One of the approaches shows that the integral

of Chern-Simons form on the occupied states precisely detects the topological invariant which protects the topological insulating property.

In this talk I will present some partial results on determining relations between the Chern-Simons invariant and other topological invariants associated to K-theory.