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Tensor product of bundles gives K-theory a commutative ring structure. Without subscripts, () usually denotes complex K-theory whereas real K-theory is sometimes written as (). The remaining discussion is focused on complex K-theory. As a first example, note that the K-theory of a point is the integers. This is because vector bundles over a ...
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 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.
The K-theory group, K(X), of a compact Hausdorff topological space is defined as the abelian group generated by isomorphism classes [E] of complex vector bundles modulo the relation that, whenever we have an exact sequence, then [] = [] + [] in topological K-theory.
A ring spectrum is a spectrum X such that the diagrams that describe ring axioms in terms of smash products commute "up to homotopy" (corresponds to the identity.) For example, the spectrum of topological K-theory is a ring spectrum. A module spectrum may be defined analogously.
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 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.
One can define the algebraic K-theory of a stable ∞-category or dg-category C, giving a sequence of abelian groups () for integers i. The group () has a simple description in terms of the triangulated category associated to C. But an example shows that the higher K-groups of a dg-category are not always determined by the associated ...