Topological vs. Algebraic $K$-Theory

Suppose I can calculate the extraordinary cohomology encoded in topological $K$-groups of a topological space $X$ with CW structure. What information does this give me about $C^{*}$-algebras associated with $X$? What is the algebraic analogue of topological suspension or the algebraic version of Bott Periodicity?

Bigger Question: More generally, what is the deep connection between topological $K$-theory and algebraic $K$-theory?

Motivation: I've computed the reduced topological $K$-groups of a wedge sum of $m$ $n$-spheres. They are given by \begin{eqnarray} \tilde{K}^{p}_{\text{top}} \left(\bigvee^{m} S^{n} \right) \cong
\mathbb{Z}^{m} \quad \text{if} \quad p + n \quad \text{is even} \end{eqnarray} and trivial otherwise.

Question: Is the positive integer $m$ the dimension of an algebra associated with the wedge sum? If so, is there an intuitive description of this algebra?

Thanks!

Solution 1:

I cannot fully answer your questions, but hopefully I can help orient you. I know a little about C*-algebra K-theory, and less about topological K-theory, so take this with a grain of salt.

The $K_0$ group of a (complex) C*-algebra $A$ is defined as the Grothendieck group of a commutative monoid of equivalence classes of projections in the matrix algebras over $A$. The $K_1$ group of $A$ can be defined as a group of equivalence classes of unitary elements of matrix algebras over $A$. (The subscripts are used instead of superscripts because the corresponding functors are covariant rather than contravariant.) To define higher $K_n$ groups, "suspensions" are used. In this setting, the suspension $SA$ of $A$ is defined to be $C_0(0,1)\otimes A\cong C_0((0,1),A)$, which can be thought of as the C*-algebra of continuous $A$-valued functions on the unit interval that vanish at $0$ and $1$. When $n$ is greater than $1$, $K_n(A)$ is defined to be $K_{n-1}(SA)$. It turns out that this is consistent with the $n=1$ case, i.e., $K_1(A)$ is isomorphic to $K_0(SA)$. Bott periodicity says that $K_{n+2}(A)$ is isomorphic to $K_n(A)$ for all $n$.

In the commutative case, there is a locally compact Hausdorff space $X$ such that $A$ is isomorphic to $C_0(X)$, the algebra of complex-valued continuous functions on $X$ vanishing at infinity. The homeomorphism class of $X$ is determined by $A$, and the isomorphism class of $A$ is determined by $X$. One can consider either the operator K-theory of $C_0(X)$ or the topological K-theory of $X$. It turns out that the groups are isomorphic: $K_n(C_0(X))\cong K^n(X)$.

The K-groups don't tell you anything literally about the dimension of the C*-algebra. For example, the infinite dimensional C*-algebra $C(S^2)$ of continuous complex-valued functions on the $2$-sphere has K-groups $K_0(C(S^2))=\mathbb{Z}^2$ and $K_1(C(S^2))=0$. These are the same as the K-groups of the algebra $\mathbb{C}\oplus\mathbb{C}$ (which is the algebra of continuous complex-valued functions on a $2$-point discrete space).

As a terminological aside, usually a distinction is made between algebraic K-theory and K-theory of operator algebras.

Here are some references. For topological K-theory, see Atiyah and Karoubi. For operator K-theory, in increasing order of difficulty and assumed background, see Wegge-Olsen, Rørdam et al., and Blackadar. Blackadar's book is a wonderful reference and the most comprehensive, while the other two serve as good introductions. (Rørdam et al. is the one I've studied the most.)

Some insight on the relationship between the two theories can be gained in light of Swan's theorem on the equivalence between finitely generated projective modules over $C(X)$ and vector bundles over $X$. Roughly speaking, since projective modules are direct summands of free modules, they correspond to idempotents in the endomorphism rings of free modules, and in the C*-algebra case this corresponds to the projections in matrix algebras mentioned above. Each of the operator K-theory books mentioned above includes at least some material on the relationship to topological K-theory.