Mapping the surgery exact sequence for topological manifolds to analysis. (English) Zbl 1408.57033
Summary: In this paper we prove the existence of a natural mapping from the surgery exact sequence for topological manifolds to the analytic surgery exact sequence of Higson and Roe. This generalizes the fundamental result of N. Higson and J. Roe [\(K\)-Theory 33, No. 4, 277–299 (2004; Zbl 1083.19002); \(K\)-Theory 33, No. 4, 301–324 (2004; Zbl 1083.19003); \(K\)-Theory 33, No. 4, 325–346 (2004; Zbl 1085.19002)], but in the treatment given by P. Piazza and T. Schick [Ann. K-Theory 1, No. 2, 109–154 (2016; Zbl 1335.46063)], from smooth manifolds to topological manifolds. Crucial to our treatment is the Lipschitz signature operator of Teleman.
We also give a generalization to the equivariant setting of the product defined by P. Siegel in his Ph.D. thesis [Homological calculations with analytic structure groups. Pennsylvania State University (2012)]. Geometric applications are given to stability results for rho classes. We also obtain a proof of the APS delocalized index theorem on odd dimensional manifolds, both for the spin Dirac operator and the signature operator, thus extending to odd dimensions the results of Piazza and Schick [loc. cit.]. Consequently, we are able to discuss the mapping of the surgery sequence in all dimensions.
We also give a generalization to the equivariant setting of the product defined by P. Siegel in his Ph.D. thesis [Homological calculations with analytic structure groups. Pennsylvania State University (2012)]. Geometric applications are given to stability results for rho classes. We also obtain a proof of the APS delocalized index theorem on odd dimensional manifolds, both for the spin Dirac operator and the signature operator, thus extending to odd dimensions the results of Piazza and Schick [loc. cit.]. Consequently, we are able to discuss the mapping of the surgery sequence in all dimensions.
MSC:
| 57R67 | Surgery obstructions, Wall groups |
| 19J25 | Surgery obstructions (\(K\)-theoretic aspects) |
| 19K35 | Kasparov theory (\(KK\)-theory) |
| 19K56 | Index theory |
| 58J22 | Exotic index theories on manifolds |
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