# Stiefel-Whitney characteristic classes

## 1 Definition

Let $N$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}N$ be a compact smooth $n$$n$-manifold (possibly with boundary). Denote by $PD=PD_G:H^k(N;G)\to H_{n-k}(N,\partial N;G)$$PD=PD_G:H^k(N;G)\to H_{n-k}(N,\partial N;G)$ the PoincarÃ© duality isomorphism. Here for non-orientable $N$$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.