Ktheory

K-theory is a branch of mathematics that provides algebraic invariants to classify vector bundles and related objects on spaces and rings. It arose from the need to understand how bundles can be assembled and compared, and it uses Grothendieck groups to convert geometric information into algebraic data.

Topological K-theory assigns groups to topological spaces. For a compact space X, K^0(X) is the Grothendieck group of isomorphism classes of complex vector bundles over X, with the operation given by direct sum. Reduced K-theory, tilde K^0(X), is defined by factoring out the class of the trivial line bundle; higher K-groups can be defined for more general spaces.

Higher groups K^n(X) are defined via Bott periodicity, giving a 2-periodic theory: K^{n+2}(X) ≅ K^n(X). This framework extends to pairs, suspensions, and maps, and connects with characteristic classes through the Chern character.

Algebraic K-theory, introduced by Quillen, assigns groups K_i(R) to rings or exact categories and encodes projective modules and their automorphisms. It has deep connections to algebraic geometry, number theory, and motivic homotopy theory, with key results such as localization and the fundamental theorems of Bass and Quillen.

Operator K-theory, or K-theory of C*-algebras, is a variant developed to study noncommutative algebras. The groups K_0(A) and K_1(A) are built from projections and unitary elements, and play central roles in index theory, extension problems, and the classification of C*-algebras.

Applications span topology, geometry, and mathematical physics. K-theory provides stable invariants of vector bundles, informs index theorems such as the Atiyah-Singer index theorem, and interacts with cohomology via the Chern character.

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