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K-theory involves the construction of families of K-functors that map from topological spaces or schemes, or to be even more general: any object of a homotopy category to associated rings; these rings reflect some aspects of the structure of the original spaces or schemes.
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 ...
The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology or topological space might possess; for that, see List of general topology topics and Topological property.
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
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 ...
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 first were Michael Atiyah's Künneth theorem for complex K-theory and Pierre Conner and Edwin E. Floyd's result in cobordism. [ 3 ] [ 4 ] A general method of proof emerged, based upon a homotopical theory of modules over highly structured ring spectra .
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 ...