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A branch of mathematics which brings together ideas from algebraic geometry, Linear Algebra, and Number
Theory. 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 do
with Vector Bundles over Topological Spaces. Elements of a
-theory are Stable Equivalence classes of Vector Bundles over a Topological
Space. You can put a Ring structure on the collection of Stably Equivalent bundles
by defining Addition through the Whitney Sum, and Multiplication through the Tensor Product
of 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 Real
Vector Bundles, Complex Vector Bundles are used.
Topological
-theory is significant because it forms a generalized Cohomology theory, and it leads to a solution to
the 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 the
Category of suitably nice Topological Spaces (something like regular Hausdorff
Spaces) and C*-Algebra. The idea is to associate to every Space the
C*-Algebra of Continuous Maps from that Space
to the Reals.
A Vector Bundle over a Space has sections, and these sections can be
multiplied by Continuous Functions to the Reals. Under Swan's
correspondence, Vector Bundles correspond to modules over the C*-Algebra of
Continuous Functions, the Modules being the modules of sections of the
Vector Bundle. This study of Modules over C*-Algebra is the starting point
of algebraic -theory.
The Quillen-Lichtenbaum Conjecture connects algebraic -theory to Étale cohomology.
See also C*-Algebra
References
Srinivas, V. Algebraic
Swan, R. G. ``Vector Bundles and Projective Modules.'' Trans. Amer. Math. Soc. 105, 264-277, 1962.
-Theory, 2nd ed. Boston, MA: Birkhäuser, 1995.
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© 1996-9 Eric W. Weisstein