Epsilon-delta surgery over \(\mathbb Z\)
The surgery exact sequence is a main tool for computation of the structure set \({\mathcal S}^{\text{CAT}} (X)\) \((\text{CAT}= \text{Diff,PL,TOP})\) of a given Poincaré complex \(X\) [see \textit{C. T. C. Wall}, Surgery on compact manifolds. 2nd ed. Mathematical Surveys and Monographs. 69. Providence, RI: AMS (1999; Zbl 0935.57003)]. Controlled surgery is a development of surgery theory for spaces \(X\) equipped with a reference map \(p: X\to B\) to a metric space \(B\). A map \(f: Y\to X\) is an \(\varepsilon\)-equivalence over \(B\) if there exist a map \(g: X\to Y\) and homotopies \(h_t: f\circ g\sim \text{id}, k_t: g\circ f\sim \text{id}\) so that the tracks \(p\circ h_t(x)\) and \(p\circ f\circ k_t(y)\) have diameter \(\leq e\) for all \(x\in X\) and \(y\in Y\). Let \((M,\partial M)\) be a manifold with a reference map \(p: M\to B\). An \(\varepsilon\)-structure on \((M, \partial M)\) over \(B\) is an equivalence class of pairs \((N,f)\), where \(f: (N, \partial N)\to (M, \partial M)\) is an \(\varepsilon\)-equivalence over \(B\) which restricts to a homeomorphism of the boundaries. The pairs \((N,f)\) and \((N', f')\) are said to be \(\varepsilon\)-related if there is a homeomorphism \(\varphi: N\to N'\) so that \(f'\circ\varphi\) is \(\varepsilon\)-homotopic to \(f\) over \(B\) \(rel\) boundary. Denote by \({\mathcal S}'_{\varepsilon} \left(\overset{M}{\underset{B}\downarrow}\right)\) the collection of equivalence classes of \(\varepsilon\)-structures on \(M\) generated by this relation. The author provides a proof of the controlled surgery exact sequence, including a stability theorem, as used in the construction of exotic homology manifolds. Theorem 1.1 If \(M^n\) is a compact topological manifold, \(n\geq 6\), or \(n\geq 5\) when \(\partial M=\emptyset\), \(B\) is a finite polyhedron with the standard metric, and \(p: M\to B\) is a \(UV^1\) map, then there exist an \(\varepsilon_0> 0\) and a \(T> 0\) depending only on \(n\) and \(B\) so that for every \(\varepsilon\leq \varepsilon_0\) there is a surgery exact sequence \[ \dots H_{n+1}(B;\mathbb L) - - \to \mathcal S_{\varepsilon} \left(\overset{M}{\underset{B}\downarrow}\right) \longrightarrow [M,\partial M;G/TOP] \longrightarrow H_{n}(B;\mathbb L) \] where \(\mathbb L\) is the periodic \(L\)-spectrum of the trivial group and \[ {\mathcal S}_{\varepsilon} \left(\overset{M}{\underset{B}\downarrow}\right)= \text{im}\left( {\mathcal S}'_{\varepsilon} \left(\overset{M}{\underset{B}\downarrow}\right{)} {\mathcal S}'_{T\varepsilon} \left(\overset{M}{\underset{B}\downarrow}\right)\right). \] Moreover, for \(\varepsilon\leq \varepsilon_0\), \({\mathcal S}_{\varepsilon} \left(\overset{M}{\underset{B}\downarrow}\right)={\mathcal S}_{\varepsilon_0} \left(\overset{M}{\underset{B}\downarrow}\right)\). This surgery exact sequence is also valid in the smooth and PL categories where \([M, G/\text{TOP}]\) should be replaced by \([M, G/\text{CAT}]\). The approach is to show that this result is a formal consequence of the alpha-approximation theorem of \textit{T. A. Chapman} and \textit{S. Ferry} [Am. J. Math. 101, 583--607 (1979; Zbl 0426.57004)].
scientific article; zbMATH DE number 5806788