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 canonicaly defined morphism $b:H_{ n+1}(B,K,\mathbb{ L})\to \mathbb{ S} (K)$, where $\mathbb{ L}$ is the periodic simply-connected surgery spectrum and $\mathbb{ S} (K)$ is the topological structure set. We construct a refinement $a:H_{ n+1}^{ +}(B,K,\mathbb{ L} )\to \mathbb{ S}_{ \varepsilon ,\delta }(K)$ 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 $\mathbb{ S}_{ \varepsilon ,\delta }(K)$ 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.
Categories: math.GT math.AT