Stable Homotopy theory
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The K-theory spectrum (for complex K-theory) or (for orthogonal K-theory) in the strict sense is the spectrum that represents the generalized (Eilenberg-Steenrod) cohomology theory topological K-theory. For complex topological K-theory this is periodic with period 2 (reflect Bott periodicity) of the form
More generally, to every stable (infinity,1)-category is associated a K-theory space which in good cases, such as when the category is presented by a Waldhausen category is the degree 0 piece of a corresponding algebraic K-theory spectrum. The detailed construction is known as the Waldhausen S-construction.
As a symmetric spectrum: Schwede 12, Example I.2.10
is a 2-periodic ring spectrum. This is the original Bott periodicity.
As a localization of an -group -ring
Snaith's theorem asserts that the K-theory spectrum for periodic complex K-theory is the ∞-group ∞-ring of the circle 2-group localized away from the Bott element :
Relation between , and -
as homotopy-fixed points of
Complex conjugation on complex vector bundles induces on the complex K-theory spectrum an involutive automorphism. KR-theory is the corresponding -equivariant cohomology theory.
In particular, the homotopy fixed point of KU under this automorphism is KO
and this way where in complex K-theory one has KU-modules (∞-modules), so in KR-theory one has -modules.
a proof in terms of moduli stacks is given in Mathew 13, section 3
Relation to Clifford algebras
There are close relations between K-theory and Clifford algebras. One conceptual statement relating them is this:
KO-theory is the first Weiss-derivative (in orthogonal calculus) of the K-theory of Clifford algebras. (Charles Rezk, MO comment, Sept ‘13)
cohomology theories of string theory fields on orientifolds
The weak homotopy eqivalence is discussed in detail in
- Marcelo Aguilar, Samuel Gitler, Carlos Prieto, appendix B.2 of Algebraic topology from a homotopical viewpoint, Springer (2002) (toc pdf)
Discussion of KU as a symmetric spectrum is in
Discussion of in analogy to the construction of tmf is in
with a summary in