Enhanced bounds for rho-invariants for both general and spherical 3-manifolds (Q6584688)

\textit{J. Cheeger} and \textit{M. Gromov} [J. Differ. Geom. 21, 1--34 (1985; Zbl 0614.53034)] developed the \(L^2\) rho-invariant \(\rho\) for closed, oriented Riemannian manifolds of dimension \(4k-1\) with an arbitrary representation of the fundamental group. \textit{S. Chang} and \textit{S. Weinberger} [Geom. Topol. 7, 311--319 (2003; Zbl 1037.57028)] extended the definition to topological manifolds. \textit{J. C. Cha} [Commun. Pure Appl. Math. 69, No. 6, 1154--1209 (2016; Zbl 1358.57020)] constructed bounds for the rho-invariant for topological \((4k-1)\)-dimensional manifolds. In particular, for 3-manifolds \(M\) he found upper bounds with respect to the simplicial complexity of \(M\), i.e.~the minimal number of 3-simplicies in a triangulation of \(M\).\N\NLim's principal result for all closed, orientable 3-manifolds with simplicial complexity \(n\) is that the rho-invariant satisfies \(|\rho^{(2)}(M, \phi)|\le 189540n\) for any homomorphism \(\phi\colon \pi_1(M)\to G\) to any group \(G\). This result improves Cha's calculations by an order of 2. In fact, when \(M\) has a finite fundamental group, then Lim produces an even smaller upper bound given by \(2340n\).\N\NThe methods for these bounds rely on delicate calculations when constructing chain null-homotopies. These chain homotopies are constructed by edgewise subdivision and acyclic complexes of Baumslag-Dyer-Heller [\textit{G. Baumslag} et al., J. Pure Appl. Algebra 16, 1--47 (1980; Zbl 0419.20026)]. Lim includes detailed computations in the latter part of the article using sophisticated and intricate combinatorial arguments. At the end of the paper, he applies his results to give estimates regarding 3-dimensional Lens spaces.