# K-theory

Topological K-theory is a generalised cohomology^{} theory on the category^{}
of compact^{} Hausdorff spaces.
It classifies the vector bundles^{} over a space $X$ up to stable equivalences.
Equivalently, via the Serre-Swan theorem, it classifies the finitely generated projective modules over the ${C}^{*}$-algebra^{} $C(X)$.

Let $A$ be a unital ${C}^{*}$-algebra over $\u2102$ and denote by ${\mathrm{M}}_{\mathrm{\infty}}(A)$ the algebraic direct limit^{} of matrix algebras ${\mathrm{M}}_{n}(A)$ under the embeddings^{}
${\mathrm{M}}_{n}(A)\to {\mathrm{M}}_{n+1}(A):a\mapsto \left(\begin{array}{cc}\hfill a\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)$.
Identify the completion of ${\mathrm{M}}_{\mathrm{\infty}}(A)$ with the stable algebra $A\otimes \mathbb{K}$ (where $\mathbb{K}$ is the compact operators^{} on ${l}_{2}(\mathbb{N})$),
which we will continue to denote by ${\mathrm{M}}_{\mathrm{\infty}}(A)$.
The ${K}_{0}(A)$ group is the Grothendieck group (abelian group^{} of formal differences^{}) of the homotopy classes of the projections in ${\mathrm{M}}_{\mathrm{\infty}}(A)$.
Two projections $p$ and $q$ are homotopic^{} if there exists a norm continuous^{} path of projections from $p$ to $q$.
Let $p\in {\mathrm{M}}_{m}(A)$ and $q\in {\mathrm{M}}_{n}(A)$ be two projections.
The sum of their homotopy classes $[p]$ and $[q]$ is the homotopy class of their direct sum^{}:
$[p]+[q]=[p\oplus q]$ where $p\oplus q=\mathrm{diag}(p,q)\in {\mathrm{M}}_{m+n}(A)$.
Alternatively, one can consider equivalence classes^{} of projections up to unitary transformations.
Unitary equivalence coincides with homotopy equivalence^{} in ${\mathrm{M}}_{\mathrm{\infty}}(A)$ (or ${\mathrm{M}}_{n}(A)$ for $n$ large enough).

Denote by ${\mathrm{U}}_{\mathrm{\infty}}(A)$ the direct limit of unitary groups ${\mathrm{U}}_{n}(A)$ under the embeddings
${\mathrm{U}}_{n}(A)\to {\mathrm{U}}_{n+1}(A):u\mapsto \left(\begin{array}{cc}\hfill u\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)$.
Give ${\mathrm{U}}_{\mathrm{\infty}}(A)$ the direct limit topology^{}, i.e. a subset $U$ of ${\mathrm{U}}_{\mathrm{\infty}}(A)$ is open if and only if
$U\cap {\mathrm{U}}_{n}(A)$ is an open subset of ${\mathrm{U}}_{n}(A)$, for all $n$.
The ${K}_{1}(A)$ group is the Grothendieck group (abelian group of formal differences) of the homotopy classes of the unitaries in ${\mathrm{U}}_{\mathrm{\infty}}(A)$.
Two unitaries $u$ and $v$ are homotopic if there exists a norm continuous path of unitaries from $u$ to $v$.
Let $u\in {\mathrm{U}}_{m}(A)$ and $v\in {\mathrm{U}}_{n}(A)$ be two unitaries.
The sum of their homotopy classes $[u]$ and $[v]$ is the homotopy class of their direct sum:
$[u]+[v]=[u\oplus v]$ where $u\oplus v=\mathrm{diag}(u,v)\in {\mathrm{U}}_{m+n}(A)$.
Equivalently, one can work with invertibles in ${\mathrm{GL}}_{\mathrm{\infty}}(A)$
(an invertible $g$ is connected to the unitary $u=g{|g|}^{-1}$ via the homotopy^{} $t\to g{|g|}^{-t}$).

Higher K-groups can be defined through repeated suspensions^{},

$${K}_{n}(A)={K}_{0}({S}^{n}A).$$ | (1) |

But, the Bott periodicity theorem means that

$${K}_{1}(SA)\cong {K}_{0}(A).$$ | (2) |

The main properties of ${K}_{i}$ are:

${K}_{i}(A\oplus B)$ | $=$ | ${K}_{i}(A)\oplus {K}_{i}(B),$ | (3) | ||

${K}_{i}({\mathrm{M}}_{n}(A))$ | $=$ | ${K}_{i}(A)\mathit{\hspace{1em}}\text{(Morita invariance)},$ | (4) | ||

${K}_{i}(A\otimes \mathbb{K})$ | $=$ | ${K}_{i}(A)\mathit{\hspace{1em}}\text{(stability)},$ | (5) | ||

${K}_{i+2}(A)$ | $=$ | ${K}_{i}(A)\mathit{\hspace{1em}}\text{(Bott periodicity)}.$ | (6) |

There are three flavours of topological K-theory to handle the cases of $A$ being complex (over $\u2102$), real (over $\mathbb{R}$) or Real (with a given real structure).

${K}_{i}(C(X,\u2102))$ | $=$ | ${\mathrm{\mathit{K}\mathit{U}}}^{-i}(X)\mathit{\hspace{1em}}\text{(complex/unitary)},$ | (7) | ||

${K}_{i}(C(X,\mathbb{R}))$ | $=$ | ${\mathrm{\mathit{K}\mathit{O}}}^{-i}(X)\mathit{\hspace{1em}}\text{(real/orthogonal)},$ | (8) | ||

${\mathrm{\mathit{K}\mathit{R}}}_{i}(C(X),J)$ | $=$ | ${\mathrm{\mathit{K}\mathit{R}}}^{-i}(X,J)\mathit{\hspace{1em}}\text{(Real)}.$ | (9) |

Real K-theory has a Bott period of 8, rather than 2.

## References

- 1 N. E. Wegge-Olsen, K-theory and ${C}^{\mathrm{*}}$-algebras. Oxford science publications. Oxford University Press, 1993.
- 2 B. Blackadar, K-Theory for Operator Algebras. Cambridge University Press, 2nd ed., 1998.
- 3 M. Rørdam, F. Larsen and N. J. Laustsen, An Introduction to K-Theory for ${C}^{\mathrm{*}}$-Algebras. Cambridge University Press, 2000.

Title | K-theory |
---|---|

Canonical name | Ktheory |

Date of creation | 2013-03-22 12:58:06 |

Last modified on | 2013-03-22 12:58:06 |

Owner | mhale (572) |

Last modified by | mhale (572) |

Numerical id | 17 |

Author | mhale (572) |

Entry type | Topic |

Classification | msc 19-00 |

Synonym | Topological K-theory |

Related topic | KHomology |

Related topic | AlgebraicKTheory |

Related topic | GrothendieckGroup |