Spaces with high topological complexity (Q2877948)

Topological complexity was introduced by Farber as a measure of how difficult it is to find an algorithm which controls a robot motion planning. The topological complexity \(TC(X)\) of a topological space \(X\) is the least integer (or infinite) \(n\) for which there exists a covering of \(n\) open sets \(\{U_1,\dots,U_n\}\) of \(X\times X\) and continuous sections \(U_i\to X^I\) of the path fibration \(X^I\to X\times X\). Later on, Iwase and Sakai introduced the monoidal topological complexity \(TC^M(X)\) imposing in the previous definition that each section preserves the diagonal, that is \(s_i(x,x)=c_x\) where \(c_x\) denotes the constant path on \(x\). Up to know the closest relation between these two invariants is \(TC(X)\leq TC^M(X)\leq TC(X)+1\). In this paper the authors find an interesting upper bound for the monoidal topological complexity. Namely, if \(X\) is a \((p-1)\)-connected complex of dimension \(np+r\), \(n\in\mathbb Z\), \(0\leq r<p\), then \(TC^M(X)\leq 2n+1\). As a consequence, if, under the same hypothesis, \(TC^M(X)\geq 2n\), then the Lusternik-Schnirelmann cateogry of \(X\) equals \(n+1\).NEWLINENEWLINETo this end the authors use the interesting characterizations of monoidal topological complexity based on the Whitehead and Ganea constructions which are modified in the fibrewise sense. For that, the fibrewise fat wedge and the fibrewise Ganea space are introduced.NEWLINENEWLINEThe corresponding weak and stable versions, \(wTC^M(X)\) and \(\sigma TC^M(X)\), of the monoidal topological complexity are also studied and compared with the classical topological complexity. Indeed, under the same connectivity and dimensional hypothesis of the result above, it is shown that any of the following conditions implies that \(TC(X)=wTC^M(X)\) (resp. \(TC(X)=\sigma TC^M(X)\)):NEWLINENEWLINE(a) \(wTC^M(X)=2n+1\) (resp. \(\sigma TC^M(X)=2n+1\)).NEWLINENEWLINE(b) \(wTC^M(X)=2n\) and \(2r+1<p\) (resp. \(\sigma TC^M(X)=2n\) and \(2r+1<p\)).NEWLINENEWLINE(c) \(wTC^M(X)=2n-1\), \(wcat(X)=n\) and \(r+1<p\) (resp. \(\sigma TC^M(X)=2n-1\), \(\sigma cat(X)=n\) and \(r+1<p\)).NEWLINENEWLINEHere, \(wcat\) and \(\sigma cat\) denote the weak and stable Lusternik-Schnirelmann category.