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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 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 .
In topology, a topological space is called a compactly generated space or k-space if its topology is determined by compact spaces in a manner made precise below. There is in fact no commonly agreed upon definition for such spaces, as different authors use variations of the definition that are not exactly equivalent to each other.
Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called K-groups. These are groups in the sense of abstract algebra.
KO-theory is a version of this construction which considers real vector bundles. K-theory with compact supports can also be defined, as well as higher K-theory groups. The famous periodicity theorem of Raoul Bott asserts that the K-theory of any space X is isomorphic to that of the S 2 X, the double suspension of X.
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 ...
The direct limit of this system is the general linear group of K, written as GL(K). An element of GL(K) can be thought of as an infinite invertible matrix that differs from the infinite identity matrix in only finitely many entries. The group GL(K) is of vital importance in algebraic K-theory. Let p be a prime number.
In algebraic K-theory and homotopy theory there are several notions of categories equipped with some specified classes of morphisms. If C has a structure of an exact category, then by defining we(C) to be isomorphisms, co(C) to be admissible monomorphisms, one obtains a structure of a Waldhausen category on C.