Appears in collection : Spectral theory of novel materials / Théorie spectrales des nouveaux matériaux

We propose a yet another definition of KR-groups, which combines those of Atiyah and Karoubi and gives a simple proof of the Bott periodicity. Using the new definition, we can formulate the bulk-edge correspondence for free fermion systems as the functoriality of the Gysin map. This is joint work with M. Furuta, S. Hayashi, M. Kotani, Y. Kubota, and K. Sato.