# “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

# A geometric version of the analytic surgery sequence?

In our Mapping surgery to analysis papers, Nigel and I proposed an analytic counterpart of the surgery exact sequence which summarizes the main results of the (Browder, Novikov, Sullivan, Wall) theory of high-dimensional manifolds.  This exact sequence identifies the set of manifold structures within a given homotopy type $$X$$ (the structure set) as the fiber of an assembly map

$H_*(X; {\mathbb L}(e)) \to L_*({\mathbb Z}\pi_1(X))$

which abstracts the ides of obtaining “signature obstructions” from a “surgery problem”.

Analogously, we constructed an analytic structure set (actually the K-theory of a certain C*-algebra) as the fiber of a Baum-Connes type assembly map, and showed that index theory provides a natural transformation from the topological surgery exact sequence to our analytic surgery exact sequence.

Our structure set is defined in purely analytic terms.  However, in a subsequent paper where we related our exact sequence to the theory of $$\eta$$ invariants, it became useful to have a more geometrical approach to the structure set also.  (The relation between the “more geometrical” and “more analytical” approaches is roughly the same as that between the Baum-Douglas and Kasparov models of K-homology.)  Our paper didn’t give a geometric definition of the structure set – just a geometric approach to certain elements.

A recent arXiv paper by Deeley and Goffeng proposes to take this idea to its logical conclusion by constructing a Baum-Douglas type model for the whole analytic structure set. The basic idea is this: An element of the structure set should be “an elliptic operator together with a reason that its index vanishes”.   The cobordism invariance of the index shows that one example of such a “reason” is that our elliptic operator is actually defined on the boundary of some manifold (and that our operator is a boundary operator).  Therefore a first approximation to a Baum-Douglas model of the structure set should have as cycles spin-c manifolds with boundary $$(M,\partial M)$$ together with maps $$\partial M \to X$$.

But of course this (cobordism) is not the only known reason for the vanishing of an index (e.g., as I understand it, the fundamental question about positive scalar curvature metrics is whether positive scalar curvature implies some bordism condition).  So suppose you have an elliptic operator whose index vanishes for some “positive scalar curvature type” reason.  How are you to build a structure class?  It seems to me that Deeley-Goffeng deal with this by incorporating quite a lot of analysis into their geometric cycles – as well as the bordism that I have described, there are also projective module bundles over the group algebra, etc… this makes the desired exactness true, but perhaps at the cost of making the groups less geometrical; they are a “geometry-analysis hybrid”.  And that is inevitable in this problem.

I should mention that several other applications of the analytic surgery sequence depend on constructing an appropriate ncie model for the structure set: e.g. Siegel, Xie-Yu (see below).  I’m not sure whether our original model is “nice” for anybody!

#### References

Deeley, Robin, and Magnus Goffeng. Realizing the Analytic Surgery Group of Higson and Roe Geometrically, Part I: The Geometric Model. ArXiv e-print, August 27, 2013. http://arxiv.org/abs/1308.5990.

Higson, Nigel, and John Roe. “Mapping Surgery to Analysis. I. Analytic Signatures.” K-Theory. An Interdisciplinary Journal for the Development, Application, and Influence of K-Theory in the Mathematical Sciences 33, no. 4 (2005): 277–299. doi:10.1007/s10977-005-1561-8.

———. “Mapping Surgery to Analysis. II. Geometric Signatures.” K-Theory. An Interdisciplinary Journal for the Development, Application, and Influence of K-Theory in the Mathematical Sciences 33, no. 4 (2005): 301–324. doi:10.1007/s10977-005-1559-2.

———. “Mapping Surgery to Analysis. III. Exact Sequences.” K-Theory. An Interdisciplinary Journal for the Development, Application, and Influence of K-Theory in the Mathematical Sciences 33, no. 4 (2005): 325–346. doi:10.1007/s10977-005-1554-7.

Higson, Nigel, and John Roe. “$$K$$-homology, Assembly and Rigidity Theorems for Relative Eta Invariants.” Pure and Applied Mathematics Quarterly 6, no. 2, Special Issue: In honor of Michael Atiyah and Isadore Singer (2010): 555–601.

Siegel, Paul. “The Mayer-Vietoris Sequence for the Analytic Structure Group.” arXiv:1212.0241 (December 2, 2012). http://arxiv.org/abs/1212.0241.

Siegel, Paul. “Homological Calculations with the Analytic Structure Group.” PhD Thesis, Penn State, 2012. https://etda.libraries.psu.edu/paper/16113/.

Xie, Zhizhang, and Guoliang Yu. “A Relative Higher Index Theorem, Diffeomorphisms and Positive Scalar Curvature.” arXiv:1204.3664 (April 16, 2012). http://arxiv.org/abs/1204.3664.

Xie, Zhizhang, and Guoliang Yu. “Positive Scalar Curvature, Higher Rho Invariants and Localization Algebras.” arXiv:1302.4418 (February 18, 2013). http://arxiv.org/abs/1302.4418.