  In this paper, we study the existence of multiple positive solutions for a
class of fractional Schr\"{o}dinger-Poisson systems involving sign-changing
potential and critical nonlinearities on an unbounded domain. With the help of
Nehari manifold and Ljusternik-Schnirelmann category, we investigate how the
coefficient $g(x)$ of the critical nonlinearity affects the number of positive
solutions. Moreover, we present a novel relationship between the number of
positive solutions and the category of the global maximum set of $g(x)$.
