Topological K-theory is a branch of mathematics that studies vector bundles over topological spaces through the lens of homotopy theory. It arises in both algebraic topology and functional analysis and is a fundamental concept in modern mathematics, bridging several areas, including geometry, representation theory, and mathematical physics. The main idea behind K-theory is to classify vector bundles (or more generally, modules over topological spaces) up to stable isomorphism.
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Complex topological K-theory is the generalized cohomology theory built from stable equivalence classes of complex vector bundles. Its degree-zero group is the Grothendieck group , and its reduced theory is denoted .