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Stiefel–Whitney numbers for singular varieties

Published online by Cambridge University Press:  18 January 2011

CARL McTAGUE
CARL McTAGUE*
Affiliation:
DPMMS, Wilberforce Road, Cambridge CB3 0WB. e-mail: [email protected]
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Abstract

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 F2-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 RP3 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 3 bundles, these findings can be seen as hinting at a new elliptic genus, one for unoriented manifolds.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 150 , Issue 2 , March 2011 , pp. 273 - 289
Copyright
Copyright © Cambridge Philosophical Society 2011

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References

REFERENCES

[Ati58]Atiyah, M. F.On analytic surfaces with double points. Proc. Roy. Soc. London. Ser. A 247 (1958), 237–244.Google Scholar
[Che70]Cheeger, J A combinatatorial formula for Stiefel–Whitney classes. In Topology of manifolds, vol. 1969 of Proceedings of the University of Georgia Topology of Manifolds Institute, pages xiv+514 (Markham Publishing Co., 1970).Google Scholar
[FM81]Fulton, W. and MacPherson, R.Categorical framework for the study of singular spaces. Mem. Amer. Math. Soc. 31 (243) (1981), vi+165.Google Scholar
[FM97]Fu, J. H. G. and McCrory, C.Stiefel–Whitney classes and the conormal cycle of a singular variety. Trans. Amer. Math. Soc. 349 (2) (1997), 809–835.CrossRefGoogle Scholar
[Ful98]Fulton, W.Intersection theory, volume 2 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] (Springer–Verlag, second edition, 1998).Google Scholar
[GM83]Goresky, M. and MacPherson, R.Intersection homology. II. Invent. Math. 72 (1) (1983), 77–129.CrossRefGoogle Scholar
[GM88]Goresky, M. and MacPherson, R.Stratified Morse theory, volume 14 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. (Springer-Verlag, Berlin, 1988).Google Scholar
[Gor84]Mark Goresky, R.Intersection homology operations. Comment. Math. Helv. 59 (3) (1984), 485–505.CrossRefGoogle Scholar
[GP89]Goresky, M. and Pardon, W.Wu numbers of singular spaces. Topology 28 (3) (1989), 325–367.CrossRefGoogle Scholar
[Har77]Hartshorne, R.Algebraic Geometry. Graduate Texts in Mathematics, No. 52 (Springer-Verlag, 1977).CrossRefGoogle Scholar
[KM98]Kollár, J. and Mori, S.Birational geometry of algebraic varieties Cambridge Tracts in Mathematics vol. 134 (Cambridge University Press, 1998). With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original.CrossRefGoogle Scholar
[Mac74]MacPherson, R. D.Chern classes for singular algebraic varieties. Ann. of Math. (2), 100 (1974), 423–432.CrossRefGoogle Scholar
[MP03]McCrory, C. and Parusiński, A.Virtual Betti numbers of real algebraic varieties. C. R. Math. Acad. Sci. Paris 336 (9) (2003), 763–768.Google Scholar
[MS74]Milnor, J. W. and Stasheff, J. D.Characteristic classes. Ann. Math. Stud. No. 76. (Princeton University Press, 1974).CrossRefGoogle Scholar
[Sti35]Stiefel, E.Richtungsfelder und Fernparallelismus in n-dimensionalen Mannigfaltigkeiten. Comment. Math. Helv. 8 (1) (1935), 305–353.CrossRefGoogle Scholar
[Sul71]Sullivan, D. Combinatorial invariants of analytic spaces. In Proceedings of Liverpool Singularities—Symposium, I (1969/70), pages 165–168 (Springer, 1971).Google Scholar
[Tho54]Thom, R.Quelques propriétés globales des variétés différentiables. Comment. Math. Helv. 28 (1954), 17–86.CrossRefGoogle Scholar
[Tot00]Totaro, B.Chern numbers for singular varieties and elliptic homology. Ann. of Math. (2) 151 (2) (2000) 757–791.CrossRefGoogle Scholar
[Tot02]Totaro, B. Topology of singular algebraic varieties. In Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) pages 533–541 (Higher Ed. Press, 2002).Google Scholar
[vH03]van Hamel, J.Towards an intersection homology theory for real algebraic varieties. Int. Math. Res. Not. (25) (2003), 1395–1411.CrossRefGoogle Scholar
[Whi40]Whitney, H.On the theory of sphere-bundles. Proc. Nat. Acad. Sci. U.S.A., 26 (1940), 148–153.CrossRefGoogle ScholarPubMed