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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 ...
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 ...
Examples of different knots including the trivial knot (top left) and the trefoil knot (below it) A knot diagram of the trefoil knot, the simplest non-trivial knot. In topology, knot theory is the study of mathematical knots.
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.
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 ...
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
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 ...