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