Controlled algebra and the Novikov conjectures for \(K\)- and \(L\)-theory
This paper deals with the Novikov conjecture in the form of the following question: For which groups \(\Gamma\) (and rings \(R)\) does the assembly map define a split injection of spectra (up to homotopy) \(B \Gamma_\wedge C(R) \to C (R \Gamma)\)? Here \(C\) is the (non-connective) Gersten-Wagoner \(K\)-theory spectrum functor, \(K^{- \infty}\), or the periodic (Wall surgery) \(L\)-theory spectrum functor, \(\mathbb{L}^{- \infty}\). It is shown that the answer to this question is positive for \(\Gamma\) provided the classifying space \(B\Gamma\) is finite and the universal free \(\Gamma\)-space \(E\Gamma\) admits a compactification \(X\) such that (i) The \(\Gamma\)-action extends over \(X\); (ii) \(X\) is metrizable; (iii) \(X\) is contractible; (iv) Compact subsets of \(E \Gamma\) become small near the boundary \(Y = X - E \Gamma\), i.e., for every \(y \in Y\), every compact set \(K \subseteq E \Gamma\), and every neighbourhood \(U\) of \(u\) in \(X\), there is a neighborhood \(V\) of \(y\) in \(X\) such that if \(g \in \Gamma\) and \(gK \cap V \neq \emptyset\), then \(gK \subseteq U\). For the \(L\)-theory case, it is further assumed that \(R\) is equipped with an involution, and that \(K_{-i} (R)\) vanishes for sufficiently large \(i\). The proofs utilize the continuously controlled \(K\)-theory of Anderson, Connolly, Ferry and Pedersen, and Ranicki's \(L\)-theory of additive categories with involution.
scientific article; zbMATH DE number 794448