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Controlled Surgery and \(\mathbb {L}\)-Homology

  • Friedrich Hegenbarth
  • Dušan RepovšEmail author
Article

Abstract

This paper presents an alternative approach to controlled surgery obstructions. The obstruction for a degree one normal map \((f,b): M^n \rightarrow X^n\) with control map \(q: X^n \rightarrow B\) to complete controlled surgery is an element \(\sigma ^c (f, b) \in H_n (B, \mathbb {L})\), where \(M^n, X^n\) are topological manifolds of dimension \(n \ge 5\). Our proof uses essentially the geometrically defined \(\mathbb {L}\)-spectrum as described by Nicas (going back to Quinn) and some well-known homotopy theory. We also outline the construction of the algebraically defined obstruction, and we explicitly describe the assembly map \(H_n (B, \mathbb {L}) \rightarrow L_n (\pi _1 (B))\) in terms of forms in the case \(n \equiv 0 (4)\). Finally, we explicitly determine the canonical map \(H_n (B, \mathbb {L}) \rightarrow H_n (B, L_0)\).

Keywords

Generalized manifold resolution obstruction controlled surgery controlled structure set \(\mathbb {L}_q\)-surgery Wall obstruction 

Mathematics Subject Classification

Primary 57R67 57P10 57R65 Secondary 55N20 55M05 

Notes

Acknowledgements

This research was supported by the Slovenian Research Agency Grants P1-0292, J1-7025, J1-8131, N1-0064, and N1-0083. We thank K. Zupanc for her technical assistance with the preparation of the manuscript. We acknowledge the referee for comments and suggestions.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Dipartimento di Matematica “Federigo Enriques”Università degli studi di MilanoMilanItaly
  2. 2.Faculty of EducationUniversity of LjubljanaLjubljanaSlovenia
  3. 3.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia

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