  We apply some methods of homology and K-theory to special classes of branes
wrapping homologically nontrivial cycles. We treat the classification of
four-geometries in terms of compact stabilizers (by analogy with Thurston's
classification of three-geometries) and derive the K-amenability of Lie groups
associated with locally symmetric spaces listed in this case. More complicated
examples of T-duality and topology change from fluxes are also considered. We
analyse D-branes and fluxes in type II string theory on ${\mathbb C}P^3\times
\Sigma_g \times {\mathbb T}^2$ with torsion $H-$flux and demonstrate in details
the conjectured T-duality to ${\mathbb R}P^7\times X^3$ with no flux. In the
simple case of $X^3 = {\mathbb T}^3$, T-dualizing the circles reduces to
duality between ${\mathbb C}P^3\times {\mathbb T}^2 \times {\mathbb T}^2$ with
$H-$flux and ${\mathbb R}P^7\times {\mathbb T}^3$ with no flux.
