# Stiefel-Whitney characteristic classes

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## 1 Definition

Let $N$${{Stub}} == Definition == ; Let N be a compact smooth n-manifold (possibly with boundary). Denote by PD=PD_G:H^k(N;G)\to H_{n-k}(N,\partial N;G) the Poincaré duality isomorphism. Here for non-orientable N the coefficients in cohomology are twisted (by the orientation double covering) and the coefficients in homology are non-twisted. Let G_k=G_{k,N} be \Zz if either k=n or k>1 is odd, and \Zz_2 if either k=1 or k == References == {{#RefList:}} [[Category:Definitions]] [[Category:Forgotten in Textbooks]]N$ be a compact smooth $n$$n$-manifold (possibly with boundary). Denote by $PD=PD_G:H^k(N;G)\to H_{n-k}(N,\partial N;G)$$PD=PD_G:H^k(N;G)\to H_{n-k}(N,\partial N;G)$ the Poincaré duality isomorphism. Here for non-orientable $N$$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}$$G_k=G_{k,N}$ be $\Zz$$\Zz$ if either $k=n$$k=n$ or $k>1$$k>1$ is odd, and $\Zz_2$$\Zz_2$ if either $k=1$$k=1$ or $k$k is even.

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

See details e.g. in [Milnor&Stasheff1974, $\S$$\S$12], [Fomenko&Fuchs2016, $\S$$\S$19.C], [Skopenkov2015b, $\S\S$$\S\S$ 9,11,12].

There is an alternative definition of $PD\overline{W}_k(N)$$PD\overline{W}_k(N)$ [Skopenkov2006, $\S$$\S$2.3 `the Whitney obstruction'] analogous to definition of the Whitney invariant.

## 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.