“Finite part of operator K-theory” IV

Continuing this series (earlier posts here, here and here) on the paper of Weinberger and Yu, I’m expecting to make two more posts: this one, which will say something about the class of groups for which they can prove their Finite Part Conjecture, and one more, which will say something about what can be done with the conjecture once one knows it. Continue reading

“Finite part of operator K-theory” III – repeat

(I posted this yesterday but it seems to have vanished into the ether – I am trying again.)

This series of posts addresses the preprint “Finite Part of Operator K-theory for Groups Finitely Embeddable into Hilbert Space and the Degree of Non-rigidity of Manifolds” (ArXiv e-print 1308.4744. http://arxiv.org/abs/1308.4744) by Guoliang Yu and Shmuel Weinberger.  In my previous post I gave the description of their main conjecture (let’s call it the Finite Part Conjecture) and showed how it would follow from the Baum-Connes conjecture (or, simply, from the statement that the Baum-Connes assembly map was an injection). Continue reading

Positive scalar curvature partial vanishing theorems and coarse indices

This paper, http://arxiv.org/abs/1210.6100, has been accepted by the Proceedings of the Edinburgh Mathematical Society.  I just sent off the copyright transfer form this evening, so everything is now set, I hope.

The paper is mostly paying an expository debt.  In my CBMS lecture notes I said that if one has the Dirac operator on a complete spin manifold \(M\), and if there is some subset \(N\subseteq M\) such that \(D\) has uniformly positive scalar curvature outside \(N\), then the index of \(D\) belongs to the K-theory of the ideal \(I_N \triangleleft C^*(M) \) associated to the subset \(N\).  A very special case of this is the observation of Gromov-Lawson that \(D\) is Fredholm if we have uniformly positive scalar curvature outside a compact set.  There are of course analogous results using thepositivity of the Weitzenbock curvature  term for other generalized Dirac operators.

Until now, I had not written up the proof of this assertion, but I felt last year that it was (past) time to do so.  This paper contains the proof and also that of the associated general form of the Gromov-Lawson relative index theorem which also appears in my CBMS notes. The latter proof uses some results from my paper with Paul Siegel on sheaf theory and Pashcke duality.

The submission to PEMS is in honor of a very pleasant sabbatical spent in Edinburgh in fall 2004.

 

“Finite part of operator K-theory” II

This is a sequel to an earlier post on the Weinberger-Yu paper referenced below.  Weinberger and Yu state their main conjecture as follows. Let \(G\) be a discrete group.

Conjecture 1.1. If \(\{g_1, · · · , g_n\}\) is a collection of non-identity elements in G with distinct
finite orders, then
(1) \(\{[p_{g_1}], · · · , [p_{g_n}]\}\) generates an abelian subgroup of \(K_0(C^*(G))\) having rank \(n\);
(2) any nonzero element in this abelian subgroup is not in the image of the assembly map \(\mu \colon  K^G_0 (EG) \to K_0(C^∗(G))\), where \(EG\) is the universal space for proper and free \(G\)-actions.

Recall that, for \(g\in G\) of finite order \(n\), \(p_g\) is the projection in the group algebra defined by averaging the powers of \(g\), that is \(p_g = \frac{1}{n}\sum_{k=0}^{n-1}g^k\).

The authors then  add: “In fact, we can state a stronger conjecture in terms of K-theory elements coming from finite subgroups and the number of conjugacy classes of nontrivial finite order elements. Such a stronger conjecture follows from the strong Novikov conjecture but would not survive inclusion into large groups.”  In this post I want to expound this perhaps slightly mysterious paragraph.

The Baum-Connes assembly map (for groups with torsion) runs from the equivariant K-homology of the space \(\underline{E}G\), the universal space for proper \(G\)-actions, to the K-theory of the group \(C^*\)-algebra of \(G\).  In low dimensions, it is well known that this map can be described by using a Chern character – see the papers of Baum-Connes and Matthey referenced below.   In particular, Matthey’s theorem 1.1 includes a diagram of assembly map which incorporates an injective homomorphism

\[ \beta_i^{(t)}\colon H_i(G;FG) \to K_i^G(\underline{E}G)\otimes{\mathbb C} \]

for \(i=0,1,2\), where \(FG\) is the collection of finitely supported complex-valued functions on the finite order elements of \(G\), on which \(G\) acts by conjugation.  In particular, \(H_0(G;FG) \) is simply the vector space spanned by the conjugacy classes of finite order elements.  From the Baum-Connes conjecture (in fact, from the injectivity of the Baum-Connes assembly map) it would therefore follow that \(K_0(C^*(G))\otimes{\mathbb C}\) contains a summand of rank equal to the number of conjugacy classes of finite order elements of \(G\).  This is, of course, at least equal to the number of distinct orders of finite order elements, since conjugate elements of \(G\) have the same order.  Thus we would obtain part (i) of the authors’ conjecture (in a strengthened form) form the injectivity of the BC assembly map (which is what they are referring to as the ‘strong Novikov conjecture’ above).  Their part (ii) would also follow from BC injectivity comparing the homological version of the LHS of the Baum-Connes assembly map with the corresponding homological version of the LHS of the ordinary assembly map (involving \(EG\) rather than \(\underline{E}G\) ).

Weinberger and Yu don’t stop at this point for two reasons, which I think are related.

(a) They want a conjecture which will “survive inclusion into large groups”.  What they mean by this is that if  \(G\) is a subgroup of some larger group \(G’\), they want a lower bound not just for the rank of \(K_0(C^*G)\) but also for \(K_0(C^*G’)\).  Now “the number of conjugacy classes of finite order elements” does not behave monotonically under inclusion of subgroups – non-conjugate elements in \(G\) can become conjugate in \(G’\) and in fact if \(g_1,g_2\in G\) have the same order then there will always be an HNN extension \(G’\) in which they become conjugate – but “the number of distinct orders of finite order elements” obviously does behave monotonically in this situation.

(b) Related to this is the method of the authors’ proof of the conjecture which appears to involve embedding finite subsets of \(G\) in larger groups or spaces.  The point is that by some kind of decomposition procedure, which incorporates the flexibility to increase the size of the group, one can prove Conjecture 1.1 even in some situations where the injectivity of the Baum-Connes map itself seems to be out of reach.
I’ll try to say more about the proof next time.

References

Baum, Paul, and Alain Connes. 1988. “Chern Character for Discrete Groups.” In A Fête of Topology, 163–232. Boston, MA: Academic Press.

Matthey, Michel. 2004. “The Baum–Connes Assembly Map, Delocalization and the Chern Character.” Advances in Mathematics 183 (2) (April 1): 316–379. doi:10.1016/S0001-8708(03)00090-2.

Weinberger, Shmuel, and Guoliang Yu. 2013. “Finite Part of Operator K-Theory for Groups Finitely Embeddable into Hilbert Space and the Degree of Non-Rigidity of Manifolds”. ArXiv e-print 1308.4744. http://arxiv.org/abs/1308.4744

 

Determinantal point processes 2

So, to try to understand these determinantal processes better, let’s take a look at the case when the set \(E\) contains only two points, say \(E=\{x,y\}\).  Then the kernel \(K\) that defines the process is a symmetric \(2\times 2\) matrix, say

\[ K = \left(\begin{array}{cc} a & b \\ b& d \end{array}\right) , \]

and the definition of a determinantal process gives us

\[ {\mathbb P}( x \in {\mathfrak S}) = a, \quad {\mathbb P}( y \in {\mathfrak S}) = d, \quad {\mathbb P}( x,y \in {\mathfrak S}) = ad-b^2 \]

for a random subset \(\mathfrak S\).   From these we can obtain by the inclusion-exclusion principle the exact probabilities of all four subsets of \(E\): the probability of \(E=\{x,y\}\) is \(ad-b^2\), that of \(\{x\}\) is \( a – (ad-b^2)\), that of \(\{y\}\) is \(d-(ad-b^2) \), and finally that of \(\emptyset\) is \( 1-a-d+(ad-b^2) = (1-a)(1-d)-b^2 \).

By construction these four quantities add up to 1, but in order to have a valid probability measure we also require that they all be  non-negative.  What condition will bring this about?

Lemma If the matrix \(K\) is positive and contractive (which is to say that its eigenvalues are \(\le 1\), or equivalently that \(\|K\|\le 1\) then the four quantities above are non-negative (and thus we have a determinantal probability measure).

Proof Let the eigenvalues be \(\lambda,\mu \in [0,1]\).  The probabilities defined above are, respectively, \(\lambda \mu, a-\lambda\mu, d-\lambda\mu, (1-\lambda)(1-\mu)\).  The first and fourth quantities are obviously non-negative, and the second and third are as well since \(a,d\) lie in the closed interval whose endpoints are \(\lambda\) and \(\mu\).

Conversely suppose that all the named quantities are strictly positive.  Since \(a,d>0\) are convex combinations of \(\lambda,\mu\), at least one eigenvalue is positive.  The determinant is positive, so they have the same sign, hence they are both positive.  Since also \((1-\lambda)(1-\mu)> 0 \) the eigenvalues are either both less than 1 or both greater than 1.  They can’t both be greater than one or else their sum \(a+d\) would be greater than 2 and therefore one of \(a,d\) would be greater than 1.

Thus at least in the \(2\times 2\) case we have established that determinantal measures are defined by positive contractions.  Of course this is very ad hoc.  To proceed more generally we need to use exterior algebra – topic for the next post.