Group actions and higher topological complexity of lens spaces

In this paper, we obtain an upper bound on the higher topological complexity of the total spaces of fibrations. As an application, we improve the usual dimensional upper bound on higher topological complexity of total spaces of some sphere bundles. We show that this upper bound on the higher topological complexity of the total spaces of fibrations can be improved using the notion of higher subspace topological complexity. We also show that the usual dimensional upper bound on the higher topological complexity of any path-connected space can be improved in the presence of positive dimensional compact Lie group action. We use these results to compute the exact value of higher topological complexity of higher dimensional lens spaces in many cases.