The String Coffee Table

So this are the main concepts, informally:

Pick some topological space $X$. Consider all complex vector bundles over X. Including their homomorphisms these form a category $\mathrm{Vect}(X)$.

There is sort of a semi-ring structure on that category in that vector bundles can be added, using the direct sum, and multiplied, using the direct product. When we take equivalence classes in $\mathrm{Vect}(X)$ the result is in fact an ordinary semi-ring. There is a standard way, called the Grothendieck group completion, to throw in additive inverses such as to obtain an honest ring. This ring is denoted $K_0(X)$ and is called the K-theory of $X$.

Instead of $X$, one can consider the suspension $SX$ of $X$. This is sort of a ‘sphere’ which at every latitude looks like $X$, if you wish. More precisely, it is the space obtained by taking the product of $X$ with the unit interval and identifying all points of $X$ attached to $0\in [0,1]$ and all those attached to $1\in [0,1]$:

The $n$-th iteration of taking the suspension is written $S^n X$ and one defines

Bott periodicity says that $K_n(X)$ is isomorphic to $K_{n+2}(X)$:

(In the case of K-theory for real vector bundles we’d have periodicity 8 instead of 2).

Since K-theory is something algebraic, it is natural to do away with topological spaces and replace them by their algebraic equivalent, $C^*$-algebras.

Given the $C^*$-algebra $A = C(X) = \{f : X \to \mathbb{C}\}$ of continuous complex-valued functions on $X$, a vector bundle over $X$ can equivalently be characterized as a projection in $M_n(A)$, i.e. as an $n\times n$-matrix $P$ with values in $A$ that satisfies $P = P^* = P^2$.

So an alternative way to define $K_0$, but now generalized to arbitrary $C^*$-algebras $A$, is as the abelian group $K_0(A)$ which has one generator $[p]$ for every projector in $M_n(A)$ for all $n$ subject to the identification of projections which can be continuously connected and to the relation

This formulation makes it easy to define sort of a dual K-theory.

For every $C^*$-algebra $A$ there is are dual $C^*$-algebras $D_\rho(A)$ defined as the commutant-up-to-compact-operators of any rep of $A$. So given $A$, choose any rep $\rho : A \to B(H)$ of $A$ in terms of bounded operators on some seperable Hilbert space $H$. Then

where $T_1 \sim T_2$ means that $T_1 -T_2 \in K(H)$ is a compact operator.

It may happen that $A$ didn’t have a unit. Let the result of making it unital by adjoining a unit be denoted by $\tilde A$.

The K-theory of $D(\tilde A) := D_\rho(\tilde A)$ (for any $\rho$) is now called K-homology. One writes

The position of the indices here denotes the covariant or contravariant nature of these maps, when these are regarded as functors from the category of $C^*$-algebras to that of abelian groups.

The reason for calling the map $A \mapsto K^p(A)$ a homology is that in the case where $A = C(X)$ is the $C^*$-algebra of functions on $X$, this map can be shown to define what is called a generalized homology theory.

This is any theory which associates a list of groups to any topological space such that a couple of crucial properties copied from ordinary (simplicial, say) homology, are satisfied.

So one might wonder what the cohomology theory corresponding to $K^p$ might be. Turns out that this is nothing but the K-theory that we started with (which is another way to understand the position of the indices):

But at the level of the above definitions this duality is neither straightforward to see nor really nice. It becomes much neater once a cool result is established which says that $K^p(C(X))$ is nothing but equivalence classes of generalized Dirac operators on rank-$p$ vector bundles over $X$, loosely speaking.

More precisely, this works as follows:

A Fredholm operator is a bounded operator $F$ which admits a notion of index:

A Fredholm module is like a spectral triple with a Fredholm operator instead of a Dirac operator, namely its a triple $(H,\rho(A),F)$ of a Hilbert space $H$ on which $F$ and the $C^*$-algebra $A$ is represented by $\rho : A \to B(H)$ and such that $F$ satisfies three technical conditions.

If in addition a Clifford algebra $C_p$ is represented on $H$ which commutes with $F$ this is called a $p$-multigraded Fredholm module.

Now, Kasparov went ahead and defined the Kasparov K-homology group $KK^{-p}(A)$ to be the abelian group of equivalence classes of Fredholm modules.

More precisely, $KK^{-p}(A)$ is the abelian group generated by $p$-multigraded Fredholm modules $(H,\rho(A),F)$ up to unitary equivalence, up to modules which can be continuously connected, and up to the relation

where $a,b$ are Fredholm modules, $[a],[b]$ their images in $KK^{-1}(X)$ and $\oplus$ denotes the rather obvious direct sum of such modules.

So, this definition looks rather similar in spirit to the one above. And indeed, it turns out that Kasparov’s groups are isomorphic to the ordinary K-homology groups like this:

This is nice, among other things because it gives a nice interpretation for the pairing between K-homology and K-theory:

First of all, any Dirac operator on $X$ gives rise to a Fredholm module (that’s pretty much the motivation for the definition of Fredholm module, I guess). More generally, every symmetric elliptic first-order differential operator $D$ on some vector bundle over $X$ gives a Fredholm module and hence an element in $KK^{-\cdot}(C(X))$ and hence an element in K-homology $K^\cdot(C(X))$.

Now suppose in addition an element in K-theory $K_\cdot(C(X))$ itself is given, i.e. the class of a further vector bundle $V$ over $X$. We can in natural way get the operator $D_V$ obtained by tensoring with $V$ and then compute it’s index. This gives a map

So to coin a slogan summarizing this situation:

K-theory consists of classes of vector bundles, K-homology consists of classes of Dirac operators and under the index map Dirac operators are dual to vector spaces.

Before quitting I should admit that most of what I said here comes from the very nice textbook

N. Higson & J. Roe
Analytic K-Homology
Oxford Univ. Press (2000)