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 $a:H_{n+1}^{+}(B,K,\mathbb{L})\to {\mathbb{S}}_{\epsilon ,\delta }(K)$ in the case when $p$ is $U{V}^1$, 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 ${\mathbb{S}}_{\epsilon ,\delta }(K)$ 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.

Cited by in corpus (3)

• On Steenrod $\mathbb{L}$-homology, generalized manifolds, and surgery
• Controlled surgery and $\mathbb{L}$-homology
• Limits of manifolds in the Gromov-Hausdorff metric space