Controlled \(L\)-theory. (Preliminary announcement). (English) Zbl 0905.57021

Ranicki, A.; Yamasaki, M.

Sci. Bull. Josai Univ. Spec. Iss., No. 2, 119-136 (1997).

This article contains a preliminary announcement of a controlled algebraic surgery theory following a proposal of Quinn. The authors have already defined controlled \(K\)-theory in [Topology Appl. 61, No. 1, 1-59 (1995; Zbl 0835.57013)] and introduce and analyse in this paper its \(L\)-theoretic analogue, the \(\varepsilon\)-controlled \(L\)-groups \(L_n(X, p_X, \varepsilon)\). A normal map \((f,b): K\to L\) from a closed \(n\)-dimensional manifold to a \(\delta \)-controlled Poincaré complex determines its controlled surgery obstruction \(\sigma_*^\delta (f,b)\in L_n (X,1_X, 100 \delta)\). If \((f,b)\) can be made into a \(\delta\)-controlled homotopy equivalence by \(\delta\)-controlled surgery, then \(\sigma_*^\delta (f,b)= 0\in L_n (X,1_X, 100 \delta)\). Conversely, if \(n\geq 5\) and \((f,b)\) satisfies \(\sigma^\delta_* (f,b)= 0\in L_n (X,1_X, 100 \delta)\), then \((f,b)\) can be made into an \(\varepsilon\)-controlled homotopy equivalence by \(\varepsilon\)-controlled surgery, where \(\varepsilon =C_n \cdot 100 \delta\) for a certain constant \(C_n>1\) that depends on \(n\). The limit of the controlled \(L\)-groups \[ L^c_n (X;1_X) =\varprojlim_\varepsilon \varprojlim_\delta \text{im} \bigl(L_n (X, 1_X, \delta)\to L_n (X,1_X, \varepsilon) \bigr) \] is the obstruction group for controlled surgery to \(\varepsilon\)-controlled homotopy equivalence for all \(\varepsilon>0\). Given a compact polyhedron \(X\) and a nonnegative integer \(n\), there exist numbers \(\varepsilon_0 >0\) and \(0<\mu_0\leq 1\) such that \(L^c_n (X;1_X) =\text{im} (L_n(X,1_X, \delta) \to L_n (X,1_X, \varepsilon))\) for every \(\varepsilon \leq\varepsilon_0\) and every \(\delta\leq \mu_0 \varepsilon\).

Reviewer: W.Lück (Münster)

MSC:

57R67

Surgery obstructions, Wall groups

Keywords:

controlled algebraic surgery theory; controlled \(L\)-groups; controlled surgery obstruction

Citations:

Zbl 0835.57013