Simplicial homology manifolds are proposed as an interesting class of geometric objects, more general than topological manifolds but still quite tractable, in which questions about the microstructure of spacetime can be naturally formulated. Their string orientations are classified by $H^3$ with coefficients in an extension of the usual group of $D$-brane charges, by cobordism classes of homology three-spheres with trivial Rokhlin invariant.
"String Orientations of Simplicial Homology Manifolds." Adv. Theor. Math. Phys. 14 (3) 991 - 1000, June 2010.