THE REAL SECTION CONJECTURE AND SMITH ’ S FIXED POINT THEOREM 3 Let X G denote the set of fixed points of the action of G on X

We prove a topological version of the section conjecture for the profinite completion of the fundamental group of finite CW-complexes with the homotopy type of an Eilenberg-MacLane space K(π, 1) equipped with a cellular self-map of prime order. As an application we derive the section conjecture for the real points of aspherical varieties defined over the field of real numbers and the natural analogue of the section conjecture for fixed points of finite group actions on projective curves of positive genus defined over the field of complex numbers.