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We show that the parametrised topological complexity of Cohen, Farber andWeinberger gives an invariant of group epimorphisms. We extend various boundsfor the topological complexity of groups to obtain bounds for the parametrisedtopological complexity of epimorphisms. Several applications are given,including an alternative computation of the parametrised topological complexityof the planar Fadell--Neuwirth fibrations which avoids calculations involvingcup products. We also prove a homotopy invariance result for parametrisedtopological complexity of fibrations over different bases.

Parametrised topological complexity of group epimorphisms