A topological approach to Cheeger-Gromov universal bounds for von Neumann \(\rho\)-invariants (Q2811173)

The paper under review presents a topological approach to studying certain \(\rho\)-invariants of \((4k-1)\)-dimensional manifolds.NEWLINENEWLINE\textit{J. Cheeger} and \textit{M. Gromov} [J. Differ. Geom. 21, 1--34 (1985; Zbl 0614.53034)] studied the \(L^{2}\) \(\rho\)-invariant \(\rho^{(2)} (M,\phi) \in \mathbb{R}\) which was defined for a closed smooth \((4k-1)\)-dimensional manifold \(M\) and a homomorphism \(\phi : \pi_{1} (M) \rightarrow G\) for some group \(G\). Using deep analytic methods they showed that for any such \(M\) there exists a constant \(C_{M}\) such that \(|\rho^{(2)} (M,\phi)| \leq C_{M}\) for any \(\phi\) and any \(G\).NEWLINENEWLINEThe main result of the present paper is Theorem 1.3 which says the same as the theorem of Cheeger and Gromov, but the manifold is allowed to be just topological. Naturally the methods of the proof are then also topological.NEWLINENEWLINEThe topological definition of the \(L^2\) \(\rho\)-invariant used here is as the \(L^{2}\)-signature defect of a manifold \(W\) bounding possibly several copies of \(M\) with respect to an enlargement \(\Gamma\) of \(G\), where the \(L^2\)-signature is defined using the dimension function for modules over the von Neumann algebra \(\mathbb{N} \Gamma\) due to \textit{W. Lück} [\(L^2\)-invariants: Theory and applications to geometry and \(K\)-theory. Berlin: Springer (2002; Zbl 1009.55001)]. A key result used in the proof of Theorem 1.3 is the functorial embedding of groups into acyclic groups due to Baumslag-Dyer-Heller [\textit{G. Baumslag} et al., J. Pure Appl. Algebra 16, 1--47 (1980; Zbl 0419.20026)], which enables the author to find \(W\) independent of the choice of \(\phi\).NEWLINENEWLINEThese topological methods also allow the author to obtain relationships of this invariant with various notions of topological complexity of \(3\)-manifolds. In particular \(\rho^{(2)} (M,\phi)\) is estimated from above in terms of the simplical complexity in Theorem 1.5, in terms of the Heegaard-Lickorish complexity in Theorem 1.8 and in terms of the invariants obtained from surgery presentations of \(3\)-manifolds in Theorem 1.9. The asymptotical behavior of these estimates is also studied. Furthermore in Section 1.3 a relation of \(\rho^{(2)} (M,\phi)\) to the pseudosimplicial complexity of \(3\)-manifolds is given and Theorem 1.14 contains concrete estimates for lens spaces. Yet another application is presented in section 6 where among other things Theorem 6.4 is proved, which gives for a \(3\)-manifold \(M\) obtained by surgery on a link \(L\) along a blackboard framing a relation between \(\rho^{(2)} (M,\phi)\) and the number of crossings of a planar diagram of \(L\). Remark 6.6 states that Theorem 1.9 together with Theorem 6.4 imply that the proofs of a number of results about knots and links from various other sources can give explicit examples of several interesting phenomena about knots and links.NEWLINENEWLINEThe two key technical methods used in the proofs of the estimates are a method for constructing efficient \(4\)-dimensional bordisms over a group and the use of controlled chain homotopy.NEWLINENEWLINEThe introduction which is about 10 pages long gives an informative overview of the main results of the paper and of the methods of the proofs.