An upper bound for topological complexity

In arXiv:1711.10132 a new approximating invariant ${mathsf{TC}}^{mathcal{D}}$ for topological complexity was introduced called $mathcal{D}$-topological complexity. In this paper, we explore more fully the properties of ${mathsf{TC}}^{mathcal{D}}$ and the connections between ${mathsf{TC}}^{mathcal{D}}$ and invariants of Lusternik-Schnirelmann type. We also introduce a new $mathsf{TC}$-type invariant $widetilde{mathsf{TC}}$ that can be used to give an upper bound for $mathsf{TC}$, $$mathsf{TC}(X)le {mathsf{TC}}^{mathcal{D}}(X) + leftlceil frac{2dim X -k}{k+1}rightrceil,$$ where $X$ is a finite dimensional simplicial complex with $k$-connected universal cover $tilde X$. The above inequality is a refinement of an estimate given by Dranishnikov.