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.
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