Axioms for Stiefel-Whitney homology classes of some singular spaces

Veljan, Darko

doi:10.1090/S0002-9947-1983-0690053-2

Axioms for Stiefel-Whitney homology classes of some singular spaces
HTML articles powered by AMS MathViewer

by Darko Veljan PDF

Trans. Amer. Math. Soc. 277 (1983), 285-305 Request permission

Abstract:

A system of axioms for the Stiefel-Whitney classes of certain type of singular spaces is established. The main examples of these singular spaces are Euler manifolds $\text {mod} 2$ and homology manifolds $\text {mod} 2$. As a consequence, it is shown that on homology manifolds $\text {mod} 2$ the generalized Stiefel conjecture holds.

References

J. F. Adams, On formulae of Thom and Wu, Proc. London Math. Soc. (3) 11 (1961), 741–752. MR 139177, DOI 10.1112/plms/s3-11.1.741
T. Banchoff and C. McCrory, A combinatorial formula for normal Stiefel-Whitney classes, Proc. Amer. Math. Soc. 76 (1979), no. 1, 171–177. MR 534413, DOI 10.1090/S0002-9939-1979-0534413-3
John D. Blanton and Clint McCrory, An axiomatic proof of Stiefel’s conjecture, Proc. Amer. Math. Soc. 77 (1979), no. 3, 409–414. MR 545605, DOI 10.1090/S0002-9939-1979-0545605-1
John D. Blanton and Paul A. Schweitzer, Axioms for characteristic classes of manifolds, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 1, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R.I., 1975, pp. 349–356. MR 0375339
Gregory W. Brumfiel and John W. Morgan, Homotopy theoretic consequences of N. Levitt’s obstruction theory to transversality for spherical fibrations, Pacific J. Math. 67 (1976), no. 1, 1–100. MR 431185
S. Buoncristiano, C. P. Rourke, and B. J. Sanderson, A geometric approach to homology theory, London Mathematical Society Lecture Note Series, No. 18, Cambridge University Press, Cambridge-New York-Melbourne, 1976. MR 0413113
Pierre E. Conner, Differentiable periodic maps, 2nd ed., Lecture Notes in Mathematics, vol. 738, Springer, Berlin, 1979. MR 548463
William Fulton and Robert MacPherson, Categorical framework for the study of singular spaces, Mem. Amer. Math. Soc. 31 (1981), no. 243, vi+165. MR 609831, DOI 10.1090/memo/0243
Richard Goldstein and Edward C. Turner, Stiefel-Whitney homology classes of quasi-regular cell complexes, Proc. Amer. Math. Soc. 64 (1977), no. 1, 157–162. MR 467765, DOI 10.1090/S0002-9939-1977-0467765-1
Stephen Halperin and Domingo Toledo, Stiefel-Whitney homology classes, Ann. of Math. (2) 96 (1972), 511–525. MR 312515, DOI 10.2307/1970823
Stephen Halperin and Domingo Toledo, The product formula for Stiefel-Whitney homology classes, Proc. Amer. Math. Soc. 48 (1975), 239–244. MR 365584, DOI 10.1090/S0002-9939-1975-0365584-6

Lectures on topology of real algebraic varieties

François Latour, Variétés géométriques et résolutions I. Classes caractéristiques, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 1, 1–72 (French). MR 478172
Clint McCrory, Cone complexes and PL transversality, Trans. Amer. Math. Soc. 207 (1975), 269–291. MR 400243, DOI 10.1090/S0002-9947-1975-0400243-7
Clint McCrory, A characterization of homology manifolds, J. London Math. Soc. (2) 16 (1977), no. 1, 149–159. MR 445506, DOI 10.1112/jlms/s2-16.1.149
John W. Milnor and James D. Stasheff, Characteristic classes, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR 0440554
D. Sullivan, Combinatorial invariants of analytic spaces, Proceedings of Liverpool Singularities—Symposium, I (1969/70), Lecture Notes in Mathematics, Vol. 192, Springer, Berlin, 1971, pp. 165–168. MR 0278333
D. Sullivan, Singularities in spaces, Proceedings of Liverpool Singularities Symposium, II (1969/1970), Lecture Notes in Math., Vol. 209, Springer, Berlin, 1971, pp. 196–206. MR 0339241
Laurence R. Taylor, Stiefel-Whitney homology classes, Quart. J. Math. Oxford Ser. (2) 28 (1977), no. 112, 381–387. MR 515729, DOI 10.1093/qmath/28.4.381

Euler manifolds and Stiefel-Whitney homology classes

Hassler Whitney, On the theory of sphere-bundles, Proc. Nat. Acad. Sci. U.S.A. 26 (1940), 148–153. MR 1338, DOI 10.1073/pnas.26.2.148