An upper bound for topological complexity

In [11], a new approximating invariant TCD for topological complexity was introduced called D-topological complexity. In this paper, we explore more fully the properties of TCD and the connections between TCD and invariants of Lusternik–Schnirelmann type. We also introduce a new TC-type invariant TC˜ that can be used to give an upper bound for TC,(1)TC(X)≤TCD(X)+TC˜(X). This then entails a connectivity-dimension estimateTC(X)≤TCD(X)+⌈2dimX−kk+1⌉, where X is a finite dimensional simplicial complex with k-connected universal cover X˜. The above inequality is a refinement of an estimate given by Dranishnikov [5].