  In this paper we show that the moduli space of framed torsion-free sheaves on
a certain class of toric surfaces admits a filtrable Bialynicki-Birula
decomposition determined by the torus action. The irreducibility of this moduli
space follows immediately as a corollary. Moreover, using its filtrable
decomposition we show that the moduli space shares the same homotopy type of an
invariant proper compact subvariety having the same fixed points set. We start
our study from the moduli space of framed torsion-free sheaves on the
projective plane. Afterwards, we generalize the results to toric surfaces under
a few assumptions.
