K-theory

Topological K-theory is a generalised cohomology theory on the categoryof 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 $ℂ$ and denote by ${\mathrm{M}}_{\mathrm{\infty }}(A)$ the algebraic direct limit of matrix algebras ${\mathrm{M}}_n(A)$ under the embeddings.Identify the completion of ${\mathrm{M}}_{\mathrm{\infty }}(A)$ with the stable algebra $A\otimes 𝕂$ (where $𝕂$ 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.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 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 casesof $A$ being complex (over $ℂ$), real (over $ℝ$) or Real(with a given real structure).

$K_i(C(X,ℂ))$$=$${\mathrm{𝐾𝑈}}^{-i}(X)\mathit{\hspace{1em}}\text{(complex/unitary)},$(7)

$K_i(C(X,ℝ))$$=$${\mathrm{𝐾𝑂}}^{-i}(X)\mathit{\hspace{1em}}\text{(real/orthogonal)},$(8)

${\mathrm{𝐾𝑅}}_i(C(X),J)$$=$${\mathrm{𝐾𝑅}}^{-i}(X,J)\mathit{\hspace{1em}}\text{(Real)}.$(9)

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