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**Controlled \(L\)-theory.**
*(English)*
Zbl 1127.57014

Quinn, Frank (ed.) et al., Exotic homology manifolds. Proceedings of the mini-workshop, Oberwolfach, Germany, June 29–July 5, 2003. Coventry: Geometry & Topology Publications. Geometry and Topology Monographs 9, 105-153 (2006).

In this nice paper the authors introduce and develop a controlled algebraic \(L\)-theory, extending their previous work on controlled algebraic \(K\)-theory [see A. Ranicki and M. Yamasahi, Topology Appl. 61, 1–59 (1995; Zbl 0835.57013)]. \(L\)-theory is based on duals. Fix the control map \(p_X:M \to X\) from a space \(M\) to a metric space \(X\), and let \(R\) be a ring with involution. The dual \(f^*\) of a geometric morphism \(f=\sum_ {\lambda} a_{\lambda} \rho_{\lambda}\) is defined by \(f^*=\sum_ {\lambda}\overline a_{\lambda} \overline \rho_{\lambda}\), where \(\overline a_{\lambda} \in R\) is the image of \(a_{\lambda}\) by the involution of \(R\) and \(\overline{\rho}_{\lambda}\) is the path obtained from the path \(\rho_{\lambda} \) in \(M\) by reversing the orientation. If \(f\) has radius \(\varepsilon\), then so does its dual \(f ^*\), and \(f \sim_{\varepsilon} g\) implies \(f^* \sim_{\varepsilon} g^*\). Let \(C\) be a free \(R\)-module chain complex on \(p_X :M \to X\). An \(n\)-dimensional \(\varepsilon\)-quadratic structure \(\psi\) on \(C\) is a collection \(\{\psi_s \mid s\geq 0\}\) of geometric morphisms \(\psi_s:C^{n-r-s}=(C_{n-r-s})^* \to C_r\) \((r \in \mathbb Z)\) of radius \(\varepsilon\) such that
\[
d \psi_s +(-)^r \psi_s d^* +(-)^{n-s-1}(\psi_{s+1} +(-)^{s+1}T\psi_{s+1}) \sim_{3\varepsilon} 0:C^{n-r-s-1} \to C_r,
\]
for \(s \geq 0\). An \(n\)-dimensional free chain complex \(C\) on \(p_X\) equipped with an \(n\)-dimensional \(\varepsilon\)-quadratic structure is called an \(n\)-dimensional \(\varepsilon\)-quadratic \(R\)-module complex on \(p_X\). An \(n\)-dimensional \(\varepsilon\)-quadratic structure \(\psi\) on \(C\) is \(\varepsilon\) Poincaré (over a subset \(W\) of \(X\)) if the algebraic mapping cone of the duality \(3\varepsilon\) chain map \(\mathcal D_{\psi}=(1+T)\psi_0:C^{n-*} \to C\) is \(4 \varepsilon\)-contractible (over \(W\)). A quadratic complex \((C, \psi)\) is Poincaré (over \(W\)) if \(\psi\) is \(\varepsilon\)-Poincaré (over \(W\)). Let \(Y\) be a subset of \(X\). For any integer \(n \geq 0\), for any pair of non-negative numbers \(\delta \geq \varepsilon \geq 0\), the \(\varepsilon\)-controlled \(L\)-groups \(L_n^{\delta, \varepsilon}(X,Y; p_X, R)\) are defined to be the equivalence classes of finitely generated \(n\)-dimensional \(\varepsilon\)-connected \(\varepsilon\)-quadratic complexes on \(p_X\) that are \(\varepsilon\)-Poincaré over \(X-Y\), where the equivalence relation is generated by finitely generated \(\delta\)-connected \(\delta\)-cobordisms that are \(\delta\)-Poincaré over \(X-Y\). Finally, for a fixed subset \(Y\) of \(X\), let \(\mathcal F\) be a family of subsets of \(X\) such that \(Z \supset Y\) for each \(Z \in \mathcal F\). The authors introduce also \(\varepsilon\)-controlled projective \(L\)-groups \(L_n^{\mathcal F, \delta,\varepsilon}(Y; p_X, R)\) as equivalence classes of finitely generated \(n\)-dimensional \(\varepsilon\)-Poincaré \(\varepsilon\)-projective quadratic complexes \(((C,p), \psi)\) on \(p_Y\) such that \([C,p]=0\) in \(\widetilde K_0^{n, \varepsilon}(Z; p_Z, R)\) for each \(Z \in \mathcal F\), where \(\widetilde K_0^{n, \varepsilon}(Z; p_Z, R)\) is an abelian group defined as the set of equivalence classes \([C,p]\) of finitely generated \(\varepsilon\)-projective chain complexes on \(p_Z\), see [loc. cit.]. Relations between the various controlled \(L\)-groups are discussed; in particular, the authors study the homology exact sequence of a pair, excision properties and the Mayer-Vietoris sequence, showing that controlled \(L\)-theory is very close to being a generalized homology theory. In particular, when \(X\) is a finite polyhedron and the control map \(p_X:M \to X\) is a fibration, they show that controlled \(L\)-groups are generalized homology groups. Finally, the authors discuss the controlled surgery obstructions and apply this to prove a classical theorem of Novikov on the topological invariance of the rational Pontrjagin classes.

For the entire collection see [Zbl 1104.57001].

For the entire collection see [Zbl 1104.57001].

Reviewer: Fulvia Spaggiari (Modena)

### MSC:

57R67 | Surgery obstructions, Wall groups |

18F25 | Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) |