Stiefel-Whitney characteristic classes
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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$19.C]{Fomenko&Fuchs2016}. | + | See e.g. \cite[$\S$19.C]{Fomenko&Fuchs2016}, \cite[$\S\S$ 9,11,12]{Skopenkov2015}. |
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== References == | == References == | ||
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[[Category:Definitions]] | [[Category:Definitions]] |
Revision as of 13:23, 28 March 2019
This page has not been refereed. The information given here might be incomplete or provisional. |
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 e.g. [Fomenko&Fuchs2016, 19.C], [Skopenkov2015, 9,11,12].
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.
- [Skopenkov2015] M. Skopenkov, When is the set of embeddings finite up to isotopy? Intern. J. Math. 26:7 (2015), http://arxiv.org/abs/1106.1878