Controlled homotopy equivalences and structure sets of manifolds

Abstract

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.