## 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].

### MSC:

 57R67 Surgery obstructions, Wall groups 18F25 Algebraic $$K$$-theory and $$L$$-theory (category-theoretic aspects)

Zbl 0835.57013
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