This is the final one of a series of posts about the manuscript “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 previous posts (most recently this one) I’ve described their main result about the assembly map, what I call the Finite Part Conjecture, and explained some of the methodology of the proof for the large class of groups that they call “finitely embeddable in Hilbert space”.  Now I want to explain some of the consequences of the Finite Part Conjecture.Recall the statement of the Finite Part Conjecture

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 \(\mu\colon K^G_0(\underline{E}G)→K_0(C^∗(G))\), where \(\underline{E}G\) is the universal space for proper and free \(G\)-actions.

Let \(M\) be a compact manifold of dimension \(4k-1\ge 7\). Recall that the structure set \({\mathscr S}(M)\) classifies (topological) manifold structures within the homotopy type of \(M\). It is an abelian group fitting into the surgery exact sequence

\[ \to L_{4k}({\mathbb Z}G) \to {\mathscr S}(M) \to H_{4k-1}(M;{\mathbb L}(e)) \to L_{4k-1}({\mathbb Z}G) \]

where the arrow from homology to L-theory is the assembly map and \(G=\pi_1(M)\). Here is one of the main results from the paper.

Theorem 3.4 In the above situation, if \(\pi_1(M) \) contains elements of \(n\) distinct finite orders, and is finitely embeddable in Hilbert space (or otherwise satisfies Conjecture 1.1), then the abelian group \({\mathscr S}(M)\) has rank at least \(n\).

Proof. We will use the results of the Higson-Roe “Surgery to analysis” papers.  This is a bit of a cheat, as those papers dealt with smooth manifolds and surgery, whereas WY consider topological manifolds. However, it seems likely to me that a topological manifold version of everything in HR exists (see the PRW paper below for some evidence of this): if not, one should take the remarks below as a compressed version of some diagram-chasing with assembly maps which can be formulated, for topological manifolds, without developing the whole HR machinery.

The main result of HR is a big diagram which relates the rationalized version of the topological surgery exact sequence with a rationalized version of the analytic surgery exact sequence, in which the assembly map is the analytic one of Kasparov-Mischenko and Baum-Connes.  In our case the relevant part of this diagram is the following
\[  \begin{array}{ccccc} H_{4k}(M;{\mathbb L}(e)){\otimes\mathbb Q} & \to & L_{4k}({\mathbb Z}G) {\otimes\mathbb Q} &\to& {\mathscr S}(M){\otimes\mathbb Q} \\ \downarrow &&\downarrow && \downarrow\\ K_{4k}(M) {\otimes\mathbb Q} &\to & K_{4k}(C^*G){\otimes\mathbb Q} &\to& {\mathscr S}^{an}(M){\otimes\mathbb Q} \end{array} \]

Everything in this picture is a rational vector space: the top row is the (rationalized) surgery exact sequence, the bottom row its analytic counterpart.  Suppose that \(G\) satisfies conjecture 1.1   Then  the middle space in the bottom row has a subspace \(W\) of dimension \(n\) which meets the image of the assembly map only at zero.   Thus (by exactness) the rationalized analytic structure space \({\mathscr S}^{an}(M){\otimes\mathbb Q}\) has dimension at least \(n\).  If one knows that \(W\) lies in the image of the vertical map (from surgery to analysis) it will follow that the corresponding topological structure space has dimension at least \(n\) too.  But since the elements of \(W\) are constructed explicitly in terms of (selfadjoint) projections in the group algebra, this is easy to check.

There are similar applications to metrics of positive scalar curvature.  It seems plausible that these could also be formulated in the same way as above, using the natural transformation from the Stolz exact sequence of positive scalar curvature metrics to the analytic surgery sequence (as formulated by Piazza-Schick), but I did not check the details of this.

 References

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.

Piazza, Paolo, and Thomas Schick. “Rho-Classes, Index Theory and Stolz’ Positive Scalar Curvature Sequence.” arXiv:1210.6892 (October 25, 2012). http://arxiv.org/abs/1210.6892.

Pedersen, Erik K., John Roe, and Shmuel Weinberger. “On the Homotopy Invariance of the Boundedly Controlled Analytic Signature of a Manifold over an Open Cone.” In Novikov Conjectures, Index Theorems and Rigidity, Vol. 2 (Oberwolfach, 1993), 227:285–300. London Math. Soc. Lecture Note Ser. Cambridge: Cambridge Univ. Press, 1995.