Stiefel-Whitney characteristic classes

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This page has not been refereed. The information given here might be incomplete or provisional.

[edit] 1 Definition

Let N be a compact smooth n-manifold (possibly with boundary). Denote by PD=PD_G:H^k(N;G)\to H_{n-k}(N,\partial N;G) the Poincaré duality isomorphism. Here for non-orientable N the coefficients in cohomology are twisted (by the orientation double covering) and the coefficients in homology are non-twisted. Let G_k=G_{k,N} be \Zz if either k=n or k>1 is odd, and \Zz_2/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_uWff66 if either k=1 or k<n is even.

Stiefel defined the homology Stiefel-Whitney class PDW_k(N)\in H_{n-k}(N,\partial N;G_k) of N to be the homology class of a degeneracy subset of a general position collection of n+1-k tangent vector fields on N. Let w_k(N) be the reduction of W_k(N) modulo 2.

Whitney defined the homology normal Stiefel-Whitney class PD\overline{W}_k(N)\in H_{n-k}(N,\partial N;G_k) of N to be the homology class of a degeneracy subset of a general position collection of n+1-k normal vector fields on N. Let \overline{w}_k(N) be the reduction of \overline{W}_k(N) modulo 2.

See details e.g. in [Milnor&Stasheff1974, \S12], [Fomenko&Fuchs2016, \S19.C], [Skopenkov2015b, \S\S 9,11,12].

There is an alternative definition of PD\overline{W}_k(N) [Skopenkov2006, \S2.3 `the Whitney obstruction'] analogous to definition of the Whitney invariant.

See also Wikipedia article.

[edit] 2 References

  • [Fomenko&Fuchs2016] A. T. Fomenko and D. B. Fuks, Homotopical Topology. Translated from the Russian. Graduate Texts in Mathematics, 273. Springer-Verlag, Berlin, 2016. DOI 10.1007/978-3-319-23488-5.

  • [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
  • [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248--342. Available at the arXiv:0604045.
  • [Skopenkov2015b] A. Skopenkov, Algebraic Topology From Geometric Viewpoint (in Russian), MCCME, Moscow, 2015.

http://www.mccme.ru/circles/oim/home/combtop13.htm#photo

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