Abstract
First we introduce the ten symmetry classes of free fermionic systems, among which eight are called real classes with either time-reversal symmetry (TRS) or particle-hole symmetry (PHS). One-particle gapped Hamiltonians in the eight real classes are bijectively mapped to all the eight independent embeddings of real Clifford algebras, corresponding to the complete set of classifying spaces in KO-theory. Moving to dimensional free fermionic systems, the complete topological classifications are presented for both Fermi surfaces and strong topological insulators/superconductors (TIs/TSCs) in the framework of KR-theory. We show that TRS and PHS have topologically different actions on TIs/TSCs and FSs through both topological invariants and the involutions in KR-theory, leading to the dimension shift from the classification of FSs to that of strong TIs/TSCs. The physical interpretations of the two classifications are also discussed in detail. Applying our theory of topological FSs on the boundary of strong TIs/TSCs, a general index theory is conjectured describing a faithful boundary-bulk correspondence of TIs/TSCs, motivated by the dimension shift. Then we construct all kinds of TIs/TSCs and Fermi surfaces by Dirac matrices, which provides us a rigorous proof of the general index theorem.