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In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It is also a fundamental tool in the field of operator ...
In mathematics, topological K-theory is a branch of algebraic topology. It was founded to study vector bundles on topological spaces, by means of ideas now recognised as (general) K-theory that were introduced by Alexander Grothendieck. The early work on topological K-theory is due to Michael Atiyah and Friedrich Hirzebruch.
Topological K-theory was one of the first examples of an extraordinary cohomology theory: It associates to each topological space X (satisfying some mild technical constraints) a sequence of groups K n (X) which satisfy all the Eilenberg–Steenrod axioms except the normalization axiom. The setting of algebraic varieties, however, is much more ...
This is precisely the usual construction of topological K-theory. Thus the gauge bundles on stacks of D9's and anti-D9's are classified by topological K-theory. If Sen's conjecture is right, all D-brane configurations in type IIB are then classified by K-theory. Petr Horava has extended this conjecture to type IIA using D8-branes.
1. The K-topology is strictly finer than the standard topology on R. Hence it is Hausdorff, but not compact. 2. The K-topology is not regular, because K is a closed set not containing , but the set and the point have no disjoint neighborhoods. And as a further consequence, the quotient space of the K-topology obtained by collapsing K to a point ...
K-theory. Topological K-theory; Adams operation; Algebraic K-theory; Whitehead torsion; Twisted K-theory; Cobordism; Thom space; Suspension functor; Stable homotopy theory; Spectrum (homotopy theory) Morava K-theory; Hodge conjecture; Weil conjectures; Directed algebraic topology
The boundary of a k-chain is a (k−1)-chain. Note that the boundary of a simplex is not a simplex, but a chain with coefficients 1 or −1 – thus chains are the closure of simplices under the boundary operator. Example 1: The boundary of a path is the formal difference of its endpoints: it is a telescoping sum.
The K-theory K(X) of a topological space X is a λ-ring, with the lambda operations induced by taking exterior powers of a vector bundle. Given a group G and a base field k, the representation ring R(G) is a λ-ring; the λ-operations are induced by the exterior powers of k-linear representations of the group G.