  We prove a topological version of the section conjecture for the profinite
completion of the fundamental group of finite CW-complexes equipped with the
action of a group of prime order $p$ whose $p$-torsion cohomology can be killed
by finite covers. As an application we derive the section conjecture for the
real points of a large class of 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.
