It’s the last week of classes here at UWO, and things have been wrapping up. There have also been a whole series of interesting talks, as both Doug Ravenel and Paul Baum have been visiting members of the department. Doug Ravenel gave a colloquium explaining work by himself, and collaborators Mike Hopkins and Mike Hill, solving the “Kervaire Invariant One” problem – basically, showing that certain kinds of framed manifolds – and, closely related, certain kinds of maps between spectra – don’t exist (namely, those where the Kervaire invariant is nonzero). This was an interesting and very engaging talk, but as a colloqium it necessarily had to skip past some of the subtleties of stable homotopy theory involved, and since my understanding of this subject is limited, I don’t really know if I could do it justice.

In any case, I have my work cut out for me with what I am going to try to do (taking blame for any mistakes or imprecisions I introduce in here, BTW, since I may not be able to do this justice either). This is to discussing the first two of four talks which Paul Baum gave here last week, starting with an introduction to K-theory, and ending up with some discussion of the Baum-Connes Conjecture. This is a famous conjecture in noncommutative geometry which Baum and Alain Connes proposed in 1982 (and which Baum now seems to be fairly convinced is probably not true, though nobody knows a counterexample at the moment).

It’s a statement about (locally compact, Hausdorff, topological) groups $G$; it relates K-theory for a $C^{\star}$-algebra associated to $G$, with the equivariant K-homology of a space associated to $G$ (in fact, it asserts a certain map $\mu$, which always exists, is furthermore always an isomorphism). It implies a great many things about any case where it IS true, which includes a good many cases, such as when $G$ is commutative, or a compact Lie group. But to backtrack, we need to define those terms:

K-Theory

The basic point of K-theory, which like a great many things began with Alexandre Grothendieck, is that it defines some invariants – which happen to be abelian groups – for various entities. There is a topological and an algebraic version, so the “entities” in question are, in the first case, topological spaces, and in the second, algebras (and more classically, algebraic varieties). Part of Paul Baum’s point in his talk was to describe the underlying unity of these two – essentially, both correspond to particular kinds of algebras. Taking this point of view has the added advantage that it lets you generalize K-theory to “noncommutative spaces” quite trivially. That is: the category of locally compact topological spaces is equivalent to the opposite category of commutative $C^{\star}$-algebras – so taking the opposite of the category of ALL $C^{\star}$ algebras gives a noncommutative generalization of “space”. Defining K-theory in terms of $C^{\star}$ algebras extends the invariant to this new sort of space, and also somewhat unifies topological and algebraic K-theory.

Classically, anyway, Atiyah and Hirzebruch’s definition for K-theory (adapted to the topological case by Adams) gives an abelian group from a (topological or algebraic) space $X$, using the category of (respectively, topological or algebraic) vector bundles over $X$. The point is, from this category one naturally gets a set of isomorphism classes of bundles, with a commutative addition (namely, direct sum) – this is an abelian semigroup. One can turn any abelian semigroup $J$ (with or without zero) into an abelian group, by taking pairs – $J \oplus J$, and taking the quotient by the relation $(x,y) \sim (x',y')$ which holds when there is $z \in J$ with $x + y' + z = x' + y + z$. This is like taking “formal differences” (and any $(x,x)$ becomes zero, even if there was no zero originally). In fact, it does a little more, since if $x$ and $x'$ are not equal, but become equal upon adding some $z$, they’re forced to be equal (so an equivalence relation is being imposed on bundles as well as allowing formal inverses).

In fact, a definition equivalent to Atiyah and Hirzebruch’s (in terms of bundles) can be given in terms of the coordinate ring of a variety $X$, or ring of continuous complex-valued functions on a (compact, Hausdorff) topological space $X$. Given a ring $\Lambda$, one defines $J(\Lambda)$ to be the abelian semigroup of all idempotents (i.e. projections) in the rings of matrices $M_n(\Lambda)$ up to STABLE similarity. Two idempotent matrices $\alpha$ and $\beta$ are equivalent if they become similar – that is, conjugate matrices – possibly after adjoining some zeros by the direct sum $\oplus$. (In particular, this means we needn’t assume $\alpha$ and $\beta$ were the same size). Then $K_0^{alg}(\Lambda)$ comes from this $J(\Lambda)$ by the completion to a group as just described.

A class of idempotents (projections) in a matrix algebra over $\mathbb{C}$ is characterized by the image, up to similarity (so, really, the dimension). Since these are matrices over a ring of functions on a space, we’re then secretly talking about vector bundles over that space. However, defining things in terms of the ring $\Lambda$ is what allows the generalization to noncommutative spaces (where there is no literal space, and the “coordinate ring” is no longer commutative, but this construction still makes sense).

Now, there’s quite a bit more to say about this – it was originally used to prove the Hirzebruch-Riemann-Roch theorem, which for nice projective varieties $M$ defines an invariant from the alternating sum of dimensions of some sheaf-cohomology groups – roughly, cohomology where we look at sections of the aforementioned vector bundles over $M$ rather than functions on $M$. The point is that the actual cohomology dimensions depend sensitively on how you turn an underlying topological space into an algebraic variety, but the HRR invariant doesn’t. Paul Baum also talked a bit about some work by J.F. Adams using K-theory to prove some results about vector fields on spheres.

For the Baum-Connes conjecture, we’re looking at the K-theory of a certain $C^{\star}$-algebra. In general, given such an algebra $A$, the (level-j) K-theory $K_j(A)$ can be defined to be the $(j-1)$ homotopy group of $GL(A)$ – the direct limit of all the finite matrix algebras $GL_n(A)$, which have a chain of inclusions under extensions where $GL_n(A) \rightarrow GL_{n+1}(A)$ by direct sum with the 1-by-1 identity. This looks a little different from the algebraic case above, but they are closely connected – in particular, under this definition $K_0(A)$ is just the same as $K^{alg}_0(A)$ as defined above (so the norm and involution on $A$ can be ignored for the level-0 K-theory of a $C^{\star}$-algebra, though not for level-1).

You might also notice this appears to define $K_0(A)$ in terms of negative-one-dimensional homotopy groups. One point of framing the definition this way is that it reveals that there are only two levels which matter – namely the even and the odd – so $K_0(A) = K_2(A) = K_4(A) \dots$, and $K_1(A) = K_3(A) = \dots$, and this detail turns out not to matter. This is a result of Bott periodicity. Changing the level of homotopy groups amounts to the same thing as taking loop spaces. Specifically, the functor $\Omega$ that takes the space of loops $\Omega(X)$ of a space $X$ is right-adjoint to the suspension functor $S$ – and since $S(S^{n-1}) = S^n$, this means that $\pi_{j+1}(X) = [S(S^n),X] \cong [S^n,\Omega(X)]$. (Note that $[S^n,X]$ is the group of homotopy classes of maps from the $n$-sphere into $X$). On the other hand, Bott periodicity says that $\Omega^2(GL(A)) \sim GL(A)$ – taking the loop-space twice gives something homotopic to the original $GL(A)$. So the tower of homotopy groups repeats every two dimensions. (So, in particular, one may as well take that $j-1$ to be $j+1$, and just find $K_2$ for $K_0$).

Now, to get the other side of the map in the Baum-Connes conjecture, we need a different part of K-theory.

K-Homology

Now, as with homology and cohomology, there are two related functors in the world of K-theory from spaces (of whatever kind) into abelian groups. The one described above is contravariant (for “spaces”, not algebras – don’t forget this duality!). Thus, maps $f : X \rightarrow Y$ give maps $K^0(f) : K^0(Y) \rightarrow K^0(X)$, which is like cohomology. There is also a covariant functor $K_0$ (so $f$ gives $K_0(f) : K_0(X) \rightarrow K_0(Y)$), appropriately called K-homology. If the K-theory is described in terms of vector bundles on $X$, K-homology – in the case of algebraic varieties, anyway – is about coherent sheaves of vector spaces on $X$ – concretely, you can think of these as resembling vector bundles, without a local triviality condition (one thinks, for instance, of the “skyscraper sheaf” which assigns a fixed vector space $V$ to any open set containing a given point $x \in X$, and $0$ to any other, which is like a “bundle” having fibre $V$ at $x$, and $0$ everywhere else – generalizations putting a given fibre on a fixed subvariety – and of course one can add such examples. This image explains why any vector bundle can be interpreted as a coherent sheaf – so there is a map $K^0 \rightarrow K_0$. When the variety $X$ is not singular, this turns out to be an isomorphism (the groups one ends up constructing after all the identifications involved turn out the same, even though sheaves in general form a bigger category to begin with).

But to take $K_0$ into the topological setting, this description doesn’t work anymore. There are different ways to describe $K_0$, but the one Baum chose – because it extends nicely to the NCG world where our “space” is a (not necessarily commutative) $C^{\star}$-algebra $A$ – is in terms of generalized elliptic operators. This is to say, triples $(H, \psi, T)$, where $H$ is a (separable) Hilbert space, $\psi$ is a representation of $A$ in terms of bounded operators on $H$, and $T$ is some bounded operator on $H$ with some nice properties. Namely, $T$ is selfadjoint, and for any $a \in A$, both its commutator with $\psi(a)$ and $\psi(a)(I - T^2)$ land in $\mathcal{K}(H)$, the ideal of compact operators. (This is the only norm-closed ideal in $\mathcal{L}(H)$, the bounded operators – the idea being that for this purpose, operators in this ideal are “almost” zero).

These are “abstract” elliptic operators – but many interesting examples are concrete ones – that is, $H = L^2(S)$ for some space $S$, and $T$ is describing some actual elliptic operator on functions on $S$. (He gave the case where $S$ is the circle, and $T$ is a version of the Dirac operator $-i \partial/\partial \theta$ – normalized so all its nonzero eigenvalues are $\pm 1$ – then we’d be doing K-homology for the circle.)

Then there’s a notion of homotopy between these operators (which I’ll elide), and the collection of these things up to homotopy forms an abelian group, which is called $K^1(A)$. This is the ODD case – that is, there’s a tower of groups $K^j(A)$, but due to Bott periodicity they repeat with period 2, so we only need to give $K^0(A)$ and $K^1(A)$. The definition for $K^0(A)$ is similar to the one for $K^1(A)$, except that we drop the “self-adjoint” condition on $T$, which necessitates expanding the other two conditions – there’s a commutator for both $T$ and $T^*$, and the condition for $T^2$ becomes two conditions, for $TT^*$ and $T^* T$.  Now, all these $K^j(A)$ should be seen as the K-homology groups $K_j(X)$ of spaces $X$ (the sub/super script is denoting co/contra-variance).

Now, for the Baum-Connes conjecture, which is about groups, one actually needs to have an equivariant version of all this – that is, we want to deal with categories of $G$-spaces (i.e. spaces with a $G$-action, and maps compatible with the $G$-action). This generalizes to noncommutative spaces perfectly well – there are $G$$C^{\star}$-algebras with suitable abstract elliptic operators (one needs a unitary representation of $G$ on the Hilbert space $H$ in the triple to define the compatibility – given by a conjugation action), $G$-homotopies, and so forth, and then there’s an equivariant K-homology group, $K^G_j(X)$ for a $G$-space $X$.  (Actually, for these purposes, one cares about proper $G$-actions – ones where $X$ and the quotient space are suitably nice).

Baum-Connes Conjecture

Now, suppose we have a (locally compact, Hausdorff) group $G$. The Baum-Connes conjecture asserts that a map $\mu$, which always exists, between two particular abelian groups found from K-theory, is always an isomorphism. In fact, this is supposed to be true for the whole complex of groups, but by Bott periodicity, we only need the even and the odd case. For simplicity, let’s just think about one of $j=0,1$ at a time.

So then the first abelian group associated to $G$ comes from the equivariant K-homology for $G$-spaces.  In particular, there is a classifying space $\underline{E}G$ – this is the terminal object in a category of (“proper”) $G$-spaces (that is, any other $G$-space has a $G$-map into $\underline{E}G$). The group we want is the equivariant K-homology of this space: $K_j^G(\underline{E}G)$.  Since $\underline{E}G$ is a terminal object among $G$-spaces, and $K_j$ is covariant, it makes sense that this group is a limit over $G$-spaces (with some caveats), so another way to define it is $K^G_j(\underline{E}G) = lim K^G_j(X)$, where the limit is over all ($G$-compact) $G$-spaces.  Now, being defined in this abstract way makes this a tricky thing to deal with computationally (which is presumably one reason the conjecture has resisted proof).  Not so for the second group:

The second group is $K_j(C^{\star}_r(G))$ the reduced $C^{\star}$-algebra of a (locally compact, Hausdorff topological) group $G$. To get this, you take the compactly supported continuous functions on $G$, with the convolution product, and then, thinking of these as acting on $L^2(G)$ by multiplication, take the completion in the algebra of all such operators. This is still closed under the convolution product. Then one takes the K-theory for this algebra at level $j$.

So then there is always a particular map $\mu : K^G_j(\underline{E}G) \rightarrow K_j(C^{\star}_r(G)$, which is defined in terms of index theory.  The conjecture is that this is always an isomorphism (which, if true, would make the equivariant K-homology much more tractable).  There aren’t any known counterexamples, and in fact this is known to be true for all finite groups, and compact Lie groups – but for infinite discrete groups, there’s no proof known.  Indeed, it’s not even known whether it’s true for some specific, not very complicated groups, notably $SL(3,\mathbb{Z})$ – the 3-by-3 integer matrices of determinant 1.

In fact, Paul Baum seemed to be pretty confident that the conjecture is wrong (that there is a counterexample $G$) – essentially because it implies so many things (the Kadison-Kaplansky conjecture, that groups with no torsion have group rings with no idempotents; the Novikov conjecture, that certain manifold invariants coming from $G$ are homotopy invariants; and many more) that it would be too good to be true.  However, it does imply all these things about each particular group it holds for.

Now, I’ve not learned much about K-theory in the past, but Paul Baum’s talks clarified a lot of things about it for me.  One thing I realized is that some invariants I’ve thought more about, in the context of Extended TQFT – which do have to do with equivariant coherent sheaves of vector spaces – are nevertheless not the same invariants as in K-theory (at least in general).  I’ve been asked this question several times, and on my limited understanding, I thought it was true – for finite groups, they’re closely related (the 2-vector spaces that appear in ETQFT are abelian categories, but you can easily get abelian groups out of them, and it looks to me like they’re the K-homology groups).  But in the topological case, K-theory can’t readily be described in these terms, and furthermore the ETQFT invariants don’t seem to have all the identifications you find in K-theory – so it seems in general they’re not the same, though there are some concepts in common. But it does inspire me to learn more about K-theory.

Coming up: more reporting on talks from our seminar on Stacks and Groupoids, by Tom Prince and Jose Malagon-Lopez, who were talking about stacks in terms of homotopical algebra and category theory.