Structure-set rank conjecture for algebraic and geometric structures

Let $M$ be a closed oriented topological manifold of dimension $n=4k-1$ with $k>1$, and let $\pi_1(M)=\Gamma$. Let $\widetilde{\mathcal S}_{\mathrm{alg}}(M)$ and $\widetilde{\mathcal S}_{\mathrm{geom}}(M)$ denote the reduced algebraic and geometric structure sets considered in the paper, with free rank taken after passing to the relevant coinvariants.

Structure-set rank conjecture. The free ranks satisfy

$$\operatorname{frank}\widetilde{\mathcal S}_{\mathrm{alg}}(M)\geq N_{\mathrm{fin}}(\Gamma),\qquad \operatorname{frank}\widetilde{\mathcal S}_{\mathrm{geom}}(M)\geq N_{\mathrm{fin}}(\Gamma).$$

The conjecture is motivated by evidence from residually finite groups and by results giving positive free rank when the fundamental group has torsion. The general lower bound remains open.