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We apply some methods of homology and K-theory to special classes of braneswrapping homologically nontrivial cycles. We treat the classification offour-geometries in terms of compact stabilizers (by analogy with Thurston'sclassification of three-geometries) and derive the K-amenability of Lie groupsassociated with locally symmetric spaces listed in this case. More complicatedexamples of T-duality and topology change from fluxes are also considered. Weanalyse 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 detailsthe conjectured T-duality to ${\mathbb R}P^7\times X^3$ with no flux. In thesimple case of $X^3 = {\mathbb T}^3$, T-dualizing the circles reduces toduality 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.

Homology and K--Theory Methods for Classes of Branes Wrapping Nontrivial Cycles