  This paper determines which Stiefel-Whitney numbers can be defined for
singular varieties compatibly with small resolutions. First an upper bound is
found by identifying the F_2-vector space of Stiefel-Whitney numbers invariant
under classical flops, equivalently by computing the quotient of the unoriented
bordism ring by the total spaces of RP^3 bundles. These Stiefel-Whitney numbers
are then defined for any real projective normal Gorenstein variety and shown to
be compatible with small resolutions whenever they exist. In light of Totaro's
result [Tot00] equating the complex elliptic genus with complex bordism modulo
flops, equivalently complex bordism modulo the total spaces of twisted(CP^3)
bundles, these findings can be seen as hinting at a new elliptic genus, one for
unoriented manifolds.
