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

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

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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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Authors and Affiliations

  1. Dipartimento di Matematica “Federigo Enriques”, Università degli studi di Milano, 20133, Milan, Italy

    Friedrich Hegenbarth

  2. Faculty of Education, University of Ljubljana, 1000, Ljubljana, Slovenia

    Dušan Repovš

  3. Faculty of Mathematics and Physics, University of Ljubljana, 1000, Ljubljana, Slovenia

    Dušan Repovš

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  1. Friedrich Hegenbarth
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  2. Dušan Repovš
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Correspondence to Dušan Repovš.

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Dedicated to the memory of Professor Andrew Ranicki (1948–2018).

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Hegenbarth, F., Repovš, D. Controlled Surgery and \(\mathbb {L}\)-Homology. Mediterr. J. Math. 16, 79 (2019). https://doi.org/10.1007/s00009-019-1354-6

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  • DOI: https://doi.org/10.1007/s00009-019-1354-6

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