# Exact sequence between real and complex bivariant K theories and application to the Z2 pairing

## TL;DR

This paper establishes formulas for the ZZ pairing in KO theory by linking real and complex bivariant K theories through a long exact sequence, with applications to topological phases in physics.

## Contribution

It introduces a new framework connecting real and complex K theories via exact sequences, with implications for topological phases protected by symmetries.

## Key findings

- Formulas for ZZ pairing in KO theory derived
- Connection between real and complex K theories established
- Application to topological phases with time reversal symmetry

## Abstract

We give some formulas for the ZZ pairing in KO theory using a long exact sequence for bivariant K theory which links real and complex theories. This is discussed under the framework of real structures given by antilinear operators verifying some symmetries. Topological phases protected by time reversal symmetry from condensed matter physics will be discussed.