Lusternik-Schnirelmann category and based topological complexities of motion planning

Abstract

Farber and Rudyak introduced topological complexity $\mathbf{TC}(X)$ of motion planning and its higher analogs ${\mathbf{TC}}_n(X)$ to measure the complexity of assigning paths to point tuples. Motivated by motion planning where a robotic system starts at the home configuration and possibly comes back after passing through a list of locations, we define three other classes of topological complexities ${\mathbf{LTC}}_n(X)$, ${\mathbf{ltc}}_n(X)$ and ${\mathbf{tc}}_n(X)$. We will compare these notions and compute the latter for some familiar classes of spaces.

arXiv comment7 pages; comments added: LTC_2(X) was introduced earlier by My Ismail Mamouni and Derfoufi Younes and they concluded that LTC_2(X)=TC(X)