Stiefel-Whitney Numbers for Singular Varieties

Details Stiefel-Whitney Numbers for Singular Varieties Stiefel-Whitney Numbers for Singular Varieties

2010-04-25

arxiv.org/abs/1004.4348v2

Math. Proc. Camb. Phil. Soc. 150 (2011), 273-289

Mathematics

Algebraic Topology

17 pages, final revised version

Scientific paper

10.1017/S0305004110000630

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.

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