For a closed topological ▫$n$▫-manifold ▫$K$▫ and a map ▫$p :K \to B$▫ inducing an isomorphism ▫$\pi _{1}(K) \to \pi _{1}(B)$▫, there is a canonically defined morphism ▫$b: H_{n+1}(B,K,\mathbb{L})\to ...\mathcal{S}(K)$▫, where ▫$\mathbb{L}$▫ is the periodic simply connected surgery spectrum and ▫$\mathcal{S}(K)$▫ is the topological structure set. We construct a refinement ▫$a: H_{n+1}^{+}(B,K,\mathbb{L} ) \to \mathcal{S}_{\varepsilon, \delta }(K)$▫ in the case when ▫$p$ is $UV^{1}$▫, and we show that ▫$a$▫ is bijective if ▫$B$▫ is a finite-dimensional compact metric ANR. Here, ▫$H_{n+1}^{+}(B,K,\mathbb{L}) \subset H_{n+1}(B,K,\mathbb{L})$▫, and ▫$\mathcal{S}_{\varepsilon,\delta}(K)$▫ is the controlled structure set. We show that the Pedersen-Quinn-Ranicki controlled surgery sequence is equivalent to the exact ▫$\mathbb{L}$▫-homology sequence of the map ▫$p: K \to B$▫, i.e. that ▫$$ H_{n+1}(B,\mathbb{L} )\to H_{n+1}^{+}(B,K,\mathbb{L})\to H_{n}(K,\mathbb{L}^{+}) \to H_{n}(B,\mathbb{L}), \; \mathbb{L}^{+} \to \mathbb{L},$$▫ is the connected covering spectrum of ▫$\mathbb{L}$▫. By taking for ▫$B$▫ various stages of the Postnikov tower of ▫$K$▫, one obtains an interesting filtration of the controlled structure set.
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