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

  • Friedrich Hegenbarth
  • Dušan RepovšEmail author


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 \(H_n (B, \mathbb {L}) \rightarrow H_n (B, L_0)\).


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 



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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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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