Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Stiefel-Whitney classes and
the conormal cycle of a singular variety


Authors: Joseph H. G. Fu and Clint McCrory
Journal: Trans. Amer. Math. Soc. 349 (1997), 809-835
MSC (1991): Primary 14P25, 57R20; Secondary 14P15, 49Q15
DOI: https://doi.org/10.1090/S0002-9947-97-01815-1
MathSciNet review: 1401519
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A geometric construction of Sullivan's Stiefel-Whitney homology classes of a real analytic variety $X$ is given by means of the conormal cycle of an embedding of $X$ in a smooth variety. We prove that the Stiefel-Whitney classes define additive natural transformations from certain constructible functions to homology. We also show that, for a complex analytic variety, these classes are the mod 2 reductions of the Chern-MacPherson classes.


References [Enhancements On Off] (What's this?)

  • [A] E. Akin, Stiefel-Whitney homology classes and bordism, Trans. A.M.S. 205 (1975), 341-359. MR 50:11288
  • [B] T. Banchoff, Stiefel-Whitney homology classes and singularities of projections for polyhedral manifolds, Proc. Symp. Pure Math., vol. 27, part 1, A.M.S., 1975, pp. 333-347. MR 51:14090
  • [Be] K. Bekka, Regular stratifications of subanalytic sets, Bull. London Math. Soc. 25 (1993), 7-16. MR 93f:32013
  • [BH] A. Borel and A. Haefliger, La classe d'homologie fondamentale d'un espace analytique, Bull. Soc. Math. France 89 (1961), 461-513. MR 26:6990
  • [BV] D. Burghelea and A. Verona, Local homological properties of analytic sets, Manus. Math. 7 (1972), 55-62. MR 46:9386
  • [BDK] J. L. Brylinski, A. Dubson and M. Kashiwara, Formule de l'indice pour les modules holonomes et obstruction d'Euler locale, C. R. Acad. Sci. Paris, ser. A, 293 (1981), 573-576. MR 83a:32010
  • [C] S. Cairns, A simple triangulation method for smooth manifolds, Bull. A.M.S. 67 (1961), 389-390. MR 26:6978
  • [Ch] J. Cheeger, A combinatorial formula for Stiefel-Whitney classes, Topology of Manifolds (J. Cantrell and C. H. Edwards, eds.), Markham, New York, 1970, pp. 470-471. MR 42:3793
  • [CK] M. Coste and K. Kurdyka, On the link of a stratum in a real algebraic set, Topology 31 (1992), 323-336. MR 93d:14088
  • [E] M. ElHaouari, Sur les classes de Stiefel-Whitney en theorie bivariante, Bull. Soc. Math. Belgique (1987), 151-186. MR 88a:57033
  • [F1] J. Fu, Monge-Ampère functions I, Indiana U. Math. J. 38 (1989), 745-771. MR 91d:49048
  • [F2] -, On Verdier's specialization formula for Chern classes, Math. Ann. 291 (1991), 247-251. MR 92k:32073
  • [F3] -, Curvature of singular spaces via the normal cycle, Proc. Symp. Pure Math., vol. 54, part 2, A.M.S., 1993, pp. 211-221. MR 94f:53126
  • [F4] -, Curvature measures of subanalytic sets, Amer. J. Math. 116 (1994), 819-880. MR 95g:32016
  • [F5] -, Curvature measures and Chern classes of singular varieties, J. Diff. Geometry 39 (1994), 251-280. MR 95e:32013
  • [FM] W. Fulton and R. MacPherson, Categorical framework for the study of singular spaces, Memoirs A.M.S. 31 (1981). MR 83a:55015
  • [GM] M. Goresky and R. MacPherson, Stratified Morse Theory, Springer-Verlag, New York, 1988. MR 90d:57039
  • [HT] S. Halperin and D. Toledo, Stiefel-Whitney homology classes, Annals of Math. 96 (1972), 511-525. MR 47:1072
  • [Ha1] R. Hardt, Slicing and intersection theory for chains modulo $\nu $ associated with real analytic varieties, Trans. A.M.S. 183 (1973), 327-340. MR 49:3195
  • [Ha2] -, Sullivan's local Euler characteristic theorem, Manus. Math. 12 (1974), 87-92. MR 49:3196
  • [Ha3] -, Topological properties of subanalytic sets, Trans. A.M.S. 211 (1975), 57-70. MR 52:751
  • [Ha4] -, Stratification of real analytic mappings and images, Invent. Math. 28 (1975), 193-208. MR 51:8453
  • [Ha5] -, Triangulation of subanalytic sets and proper light subanalytic maps, Invent. Math. 38 (1977), 207-217. MR 56:12302
  • [Ha6] -, Semi-algebraic local triviality in semi-algebraic mappings, Amer. J. Math. 102 (1980), 291-302. MR 81d:32012
  • [HM] R. Hardt and C. McCrory, Steenrod operations in subanalytic homology, Compositio Math. 39 (1979), 333-371. MR 82b:55019
  • [HZ] R. Harvey and J. Zweck, Stiefel-Whitney currents (to appear).
  • [Hi1] H. Hironaka, Subanalytic sets, Number Theory, Algebraic Geometry and Commutative Algebra, Kinokuniya, Tokyo, 1973, pp. 453-493. MR 51:13275
  • [Hi2] -, Triangulations of algebraic sets, Proc. Symp. Pure Math., vol. 29, A.M.S., 1975, pp. 165-185. MR 51:10331
  • [H] M. Hirsch, Differential Topology, Springer-Verlag, New York, 1976. MR 56:6669
  • [Hu] D. Husemoller, Fibre Bundles, second edition, Springer-Verlag, New York, 1975. MR 51:6805
  • [KS] M. Kashiwara and P. Schapira, Sheaves on Manifolds, Springer-Verlag, Berlin, 1990. MR 92a:58132
  • [M] R. MacPherson, Chern classes for singular algebraic varieties, Annals of Math. 100 (1974), 423-432. MR 50:13587
  • [Mc] C. McCrory, Euler singularities and homology operations, Proc. Symp. Pure Math., vol. 27, part 1, A.M.S., 1975, pp. 371-380. MR 51:14089
  • [Sa] C. Sabbah, Quelques remarques sur la géométrie des espaces conormaux, Astérisque 130 (1985), 161-192. MR 87f:32031
  • [Sc] P. Schapira, Operations on constructible functions, J. Pure Appl. Algebra (1991), 83-93. MR 92h:32012
  • [SY] M. Shiota and M. Yokoi, Triangulations of subanalytic sets and locally subanalytic manifolds, Trans. A.M.S. 286 (1984), 727-750. MR 86m:32014
  • [St] E. Stiefel, Richtungsfelder und Fernparallelismus in Mannigfaltigkeiten, Comm. Math. Helvetici (1936), 3-51.
  • [S] D. Sullivan, Combinatorial invariants of analytic spaces, Proc. Liverpool Singularities Symposium, Lecture Notes in Math., vol. 192, Springer-Verlag, 1971, pp. 165-169. MR 43:4063
  • [T] R. Thom, Quelques propriétés globales des variétés différentiables, Comm. Math. Helvetici 28 (1954), 17-86. MR 15:890a
  • [V] J.-L. Verdier, Spécialisation des classes de Chern, Astérisque 82-83 (1981), 149-159. MR 83m:14015
  • [W] H. Whitney, On the theory of sphere bundles, Proc. Nat. Acad. Sci. U.S.A. 26 (1940), 148-153. MR 1:220b

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 14P25, 57R20, 14P15, 49Q15

Retrieve articles in all journals with MSC (1991): 14P25, 57R20, 14P15, 49Q15


Additional Information

Joseph H. G. Fu
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
Email: fu@math.uga.edu

Clint McCrory
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
Email: clint@math.uga.edu

DOI: https://doi.org/10.1090/S0002-9947-97-01815-1
Keywords: Stiefel-Whitney class, real analytic set, conormal cycle, characteristic cycle, polar cycle, integral current
Received by editor(s): October 2, 1995
Additional Notes: Research supported in part by NSF grant DMS-9403887. First author also partially supported by NSF grant DMS-9404366. We thank Adam Parusiński for his encouragement and help.
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society