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**Controlled surgery with trivial local fundamental groups.**
*(English)*
Zbl 1050.57025

Farrell, F. T. (ed.) et al., High-dimensional manifold topology. Proceedings of the school, ICTP, Trieste, Italy, May 21–June 8, 2001. River Edge, NJ: World Scientific (ISBN 981-238-223-2/hbk). 421-426 (2003).

The controlled surgery exact sequence, used by Bryant, Ferry, Mio and Weinberger in the construction of exotic homology manifolds, is a refinement of the exact sequences developed in [S. C. Ferry and E. K. Pedersen, Lond. Math. Soc. Lect. Note Ser. 227, 167–226 (1995; Zbl 0956.57020)], and depends on a stability property of the limit process. Stability is a key element in the construction of exotic homology manifolds but the proof of the refinement was postponed to a planned project that was never completed.

The aim of this fine paper is to fill this gap. The authors provide a proof of the controlled surgery sequence, including stability, in the special case that the local fundamental groups are trivial. More precisely, if \(B\) is a finite \(n\)-dimensional , \(n\geq 4\), compact metric ANR there is a stability threshold \(\varepsilon_0 >0\) so that for any \(\varepsilon_0> \varepsilon>0\) there is \(\delta >0\) with the following property: if \(f:N\to B\) is \((\delta,1)\)-connected and \(N\) is a compact \(n\)-manifold, then there is a controlled surgery exact sequence \[ H_{n+1}(B;\mathbb L) \to S_{\varepsilon, \delta}(N,f) \to [N, \partial N;G/ \text{TOP}, *] \to H_n(B;\mathbb L). \] Here \(S_{\varepsilon, \delta}(N,f)\) is the controlled structure set, i.e., it is the set of equivalence classes of \((M,g)\), where \(M\) is a topological manifold and \(g:M\to N\) is a homeomorphism on boundaries and a \(\delta\) homotopy equivalence rel boundary. The equivalence relation is given by: \((M,g)\sim (M',g')\) if there is a homeomorphism \(h:M\to M'\) whose restriction to the boundary commutes with \(g\) and \(g'\), and which \(\varepsilon\) homotopy commutes on all of \(M\). \([N, \partial N;G/ \text{TOP}]\) is the set of homotopy classes of maps of \(N\), rel boundary, to the classifying space \(G/ \text{TOP}\); \(H_{*}(B;\mathbb L)\) is homology with coefficients in the 4-periodic simply-connected surgery spectrum \(\mathbb L\), with \(\pi_{*}(\mathbb L)=L_{*}(\mathbb Z)\).

The proof of this result is based on work of M. Yamasaki [Invent. Math. 88, 571–602 (1987; Zbl 0622.57022)].

For the entire collection see [Zbl 1029.00029].

The aim of this fine paper is to fill this gap. The authors provide a proof of the controlled surgery sequence, including stability, in the special case that the local fundamental groups are trivial. More precisely, if \(B\) is a finite \(n\)-dimensional , \(n\geq 4\), compact metric ANR there is a stability threshold \(\varepsilon_0 >0\) so that for any \(\varepsilon_0> \varepsilon>0\) there is \(\delta >0\) with the following property: if \(f:N\to B\) is \((\delta,1)\)-connected and \(N\) is a compact \(n\)-manifold, then there is a controlled surgery exact sequence \[ H_{n+1}(B;\mathbb L) \to S_{\varepsilon, \delta}(N,f) \to [N, \partial N;G/ \text{TOP}, *] \to H_n(B;\mathbb L). \] Here \(S_{\varepsilon, \delta}(N,f)\) is the controlled structure set, i.e., it is the set of equivalence classes of \((M,g)\), where \(M\) is a topological manifold and \(g:M\to N\) is a homeomorphism on boundaries and a \(\delta\) homotopy equivalence rel boundary. The equivalence relation is given by: \((M,g)\sim (M',g')\) if there is a homeomorphism \(h:M\to M'\) whose restriction to the boundary commutes with \(g\) and \(g'\), and which \(\varepsilon\) homotopy commutes on all of \(M\). \([N, \partial N;G/ \text{TOP}]\) is the set of homotopy classes of maps of \(N\), rel boundary, to the classifying space \(G/ \text{TOP}\); \(H_{*}(B;\mathbb L)\) is homology with coefficients in the 4-periodic simply-connected surgery spectrum \(\mathbb L\), with \(\pi_{*}(\mathbb L)=L_{*}(\mathbb Z)\).

The proof of this result is based on work of M. Yamasaki [Invent. Math. 88, 571–602 (1987; Zbl 0622.57022)].

For the entire collection see [Zbl 1029.00029].

Reviewer: Fulvia Spaggiari (Modena)

### MSC:

57R67 | Surgery obstructions, Wall groups |