k-Theory

A branch of mathematics which brings together ideas from algebraic geometry, Linear Algebra, and NumberTheory. In general, there are two main types of -theory: topological and algebraic.

Topological -theory is the ``true'' -theory in the sense that it came first. Topological -theory has to dowith Vector Bundles over Topological Spaces. Elements of a-theory are Stable Equivalence classes of Vector Bundles over a TopologicalSpace. You can put a Ring structure on the collection of Stably Equivalent bundlesby defining Addition through the Whitney Sum, and Multiplication through the Tensor Productof Vector Bundles. This defines ``the reduced real topological -theory of a space.''

``The reduced -theory of a space'' refers to the same construction, but instead of RealVector Bundles, Complex Vector Bundles are used.Topological -theory is significant because it forms a generalized Cohomology theory, and it leads to a solution tothe vector fields on spheres problem, as well as to an understanding of the -homeomorphism of Homotopy Theory.

Algebraic -theory is somewhat more involved. Swan (1962) noticed that there is a correspondence between theCategory of suitably nice Topological Spaces (something like regular HausdorffSpaces) and C*-Algebra. The idea is to associate to every Space theC*-Algebra of Continuous Maps from that Spaceto the Reals.

A Vector Bundle over a Space has sections, and these sections can bemultiplied by Continuous Functions to the Reals. Under Swan'scorrespondence, Vector Bundles correspond to modules over the C*-Algebra ofContinuous Functions, the Modules being the modules of sections of theVector Bundle. This study of Modules over C*-Algebra is the starting pointof algebraic -theory.

The Quillen-Lichtenbaum Conjecture connects algebraic -theory to Étale cohomology.

References

Srinivas, V. Algebraic -Theory, 2nd ed. Boston, MA: Birkhäuser, 1995.

Swan, R. G. ``Vector Bundles and Projective Modules.'' Trans. Amer. Math. Soc. 105, 264-277, 1962.

• Kolmogorov Constant
• Kolmogorov Entropy
• Kolmogorov-Sinai Entropy
• Kolmogorov-Smirnov Test
• Korselt's Criterion
• Kovalevskaya Exponent
• Kozyrev-Grinberg Theory
• k-Partite Graph
• Kramers Rate
• Krawtchouk Polynomial
• Kreisel Conjecture
• Kronecker Decomposition Theorem
• Kronecker Delta
• Kronecker Product
• Kronecker's Polynomial Theorem
• Kronecker Symbol
• Krull Dimension
• Kruskal's Algorithm
• Kruskal's Tree Theorem
• KS Entropy
• k-Statistic
• k-Subset
• k-Theory
• k-Tuple Conjecture
• Kuen Surface