Stiefel-Whitney characteristic classes
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== Definition == | == Definition == | ||
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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. | 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 e.g. \cite[$\S$19.C]{Fomenko&Fuchs2016}, \cite[$\S\S$ 9,11,12]{ | + | See details e.g. in \cite[$\S$12]{Milnor&Stasheff1974}, \cite[$\S$19.C]{Fomenko&Fuchs2016}, \cite[$\S\S$ 9,11,12]{Skopenkov2015b}. |
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+ | There is an alternative definition of $PD\overline{W}_k(N)$ \cite[$\S$2.3 `the Whitney obstruction']{Skopenkov2006} analogous to definition of [[Embeddings_just_below_the_stable_range:_classification#The_Whitney_invariant|the Whitney invariant]]. | ||
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+ | See also [[wikipedia:Stiefel–Whitney_class|Wikipedia article]]. | ||
</wikitex> | </wikitex> | ||
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[[Category:Definitions]] | [[Category:Definitions]] | ||
+ | [[Category:Forgotten in Textbooks]] |
Revision as of 15:58, 30 April 2019
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1 Definition
Let be a compact smooth -manifold (possibly with boundary). Denote by the Poincaré duality isomorphism. Here for non-orientable the coefficients in cohomology are twisted (by the orientation double covering) and the coefficients in homology are non-twisted. Let be if either or is odd, and if either or is even.
Stiefel defined the homology Stiefel-Whitney class of to be the homology class of a degeneracy subset of a general position collection of tangent vector fields on . Let be the reduction of modulo 2.
Whitney defined the homology normal Stiefel-Whitney class of to be the homology class of a degeneracy subset of a general position collection of normal vector fields on . Let be the reduction of modulo 2.
See details e.g. in [Milnor&Stasheff1974, 12], [Fomenko&Fuchs2016, 19.C], [Skopenkov2015b, 9,11,12].
There is an alternative definition of [Skopenkov2006, 2.3 `the Whitney obstruction'] analogous to definition of the Whitney invariant.
See also Wikipedia article.
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, 2020. Accepted for English translation by `Moscow Lecture Notes' series of Springer. Preprint of a part