Controlled Surgery and $$\mathbb {L}$$ L -Homology

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