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Abstract

The purpose of the present work is to derive a classification for topologically stable Fermi surfaces for translationally invariant systems with no electron-electron interactions. To derive such a classification we introduce the necessary concepts in condensed matter and electronic band theory as well as those in mathematics such as topological spaces, building up to topological K-theory and its connections with Fredholm operators. We further compute such classes when there is only translational invariance for dimensions d = 1, 2, 3 and discuss the inclusion of other symmetries.