K Is for K-Theory
18 hours ago
- #Mathematical History
- #K-theory
- #Algebraic Topology
- K-theory originated with Alexander Grothendieck in the late 1950s, using 'K' for 'Klasse' (class) in German.
- Topological K-theory, introduced by Atiyah and Hirzebruch, relates to homology and contributed to solutions like Adams' vector fields on spheres and the Atiyah-Singer Index Theorem.
- Grothendieck's approach involved categories and isomorphism classes, leading to K-groups that formally allow addition and subtraction, capturing deeper category information.
- Higher K-theory groups retain information on how isomorphic objects are identified, unlike just counting isomorphism classes.
- Bott Periodicity Theorem connects higher invariants to lower K-groups in topological K-theory, but not in algebraic K-theory.
- Algebraic K-theory extends to rings, forming a basis for non-commutative geometry, with Dan Quillen solving the problem of higher invariants in 1973.
- K-theory has deep links to number theory, as seen in conjectures like Milnor, Quillen-Lichtenbaum, and Bloch-Kato, and has been expanded by Waldhausen to ring spectra.