Piazza and Schick on the analytic surgery sequence

In my last post I mentioned a paper by Deeley and Goffeng whose aim is to construct a geometric counterpart of the Higson-Roe analytic surgery sequence.  This week, there appeared on the arXiv a new paper by Piazza and Schick which gives a new construction of the natural transformation from the original DIFF surgery exact sequence of Browder-Novikov-Sullivan-Wall to our analytic surgery sequence.  This is a counterpart to a slightly earlier paper by the same authors in which they carry out the same project for the Stolz exact sequence for positive scalar curvature metrics.

In our original papers, Nigel and I made extensive use of Poincaré spaces – the key facts being that the “higher signatures” can be defined for such spaces, and that the mapping cylinder of a homotopy equivalence between manifolds is an example of a Poincaré space (with boundary).  In fact, these observations can be used to prove the homotopy invariance of the higher signatures – this argument is the one that appears in the 1970s papers of Kasparov and Mischenko, essentially – and the natural transformation from geometric to analytic surgery should be thought of as a “quantification” of this homotopy invariance argument.

Now there is a different argument for homotopy invariance, due to Hilsum and Skandalis, that has a more analytical feel.  The point of the new Piazza-Schick paper is to “quantify” this argument in the same way that we did the Poincaré complex argument.  This should lead to the same maps (or at least, to maps having the same properties – then one is faced with a secondary version of the “comparing assembly maps” question) in perhaps a more direct way.

References

Hilsum, Michel, and Georges Skandalis. “Invariance Par Homotopie de La Signature à Coefficients Dans Un Fibré Presque Plat.” Journal Fur Die Reine Und Angewandte Mathematik 423 (1992): 73–99. doi:10.1515/crll.1992.423.73.

Kasparov, G.G. “K-theory, Group C*-algebras, and Higher Signatures (Conspectus).” In Proceedings of the 1993 Oberwolfach Conference on the Novikov Conjecture, edited by S. Ferry, A. Ranicki, and J. Rosenberg, 226:101–146. LMS Lecture Notes. Cambridge University Press, Cambridge, 1995.

Mischenko, A.S. “Infinite Dimensional Representations of Discrete Groups and Higher Signatures.” Mathematics of the USSR — Izvestija 8 (1974): 85–111.

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

———. The Surgery Exact Sequence, K-theory and the Signature Operator. ArXiv e-print, September 17, 2013. http://arxiv.org/abs/1309.4370

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.