  Using Lusztig's total positivity in split real Lie groups V. Fock and A.
Goncharov have introduced spaces of positive (framed) representations. For
general semisimple Lie groups a generalization of Lusztig's total positivity
was recently introduced by O. Guichard and A. Wienhard. They also introduced
the associated space of positive representations. Here we consider the
corresponding spaces of positive framed representations of the fundamental
group of a punctured surface. We give several parametrizations of the spaces of
framed positive representations. Using these parametrizations, we describe
their topology and their homotopy type. We show that the number of connected
components of the space of framed positive representations agrees with the
number of connected components of the space of positive representations, and
determine this number for simple Lie groups. Along the way, we also
parametrize, for an arbitrary semisimple Lie group, the space of
representations of the fundamental group of a punctured surface which are
transverse with respect to a fixed ideal triangulation of the surface.
