Stiefel-Whitney characteristic classes

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

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)$$PD=PD_G:H^k(N;G)\to H_{n-k}(N,\partial N;G)$ the Poincaré duality isomorphism. Here for non-orientable $N$$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<n$$k<n$ 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(f)\in H_{n-k}(N,\partial N;G_k)$$PD\overline{W}_k(f)\in H_{n-k}(N,\partial N;G_k)$ of $N$$N$ to be the homology class of a degeneracy subset of a general position collection of $n+1-k$$n+1-k$ normal vector fields on an immersion $f:N\subset\Rr^m$$f:N\subset\Rr^m$. By the Whitney-Wu formula 2.1 the reduction modulo 2 of this class (but not this class itself!) is independent of $f$$f$ and depends only on $N$$N$. So this reduction is denoted by $\overline{w}_k(N)$$\overline{w}_k(N)$.

Let $w_0(N)=\overline w_0(N)=[N]$$w_0(N)=\overline w_0(N)=[N]$.

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.

See also Wikipedia article.

2 Whitney-Wu formula

In this section we abbreviate $PDw_i(N)$$PDw_i(N)$ to just $w_i$$w_i$ and $PDw_i(f)$$PDw_i(f)$ to just $\overline w_i$$\overline w_i$.

Theorem 2.1 (Whitney-Wu formula).

If $N$$N$ is a closed smooth $n$$n$-manifold, $f:N\to\R^m$$f:N\to\R^m$ an immersion and $k>0$$k>0$ is an integer, then

$\displaystyle \sum_{i=0}^{k}\overline w_{k-i}\cap w_i=0.$ $$\displaystyle \sum_{i=0}^{k}\overline w_{k-i}\cap w_i=0.$$

Proof. (This proof should be known but is absent from textbooks. This text is written by M. Fedorov and A. Skopenkov in frame of the course `Algebraic topology of manifolds in interesting results'.)

Denote by $x_k$$x_k$ the obstruction to existence of $m-k+1$$m-k+1$ linearly independent fields on $N$$N$. Clearly $x_k=0$$x_k=0$. So it suffices to show that $x_k=\sum_{i=0}^k\overline w_{k-i}\cap w_i$$x_k=\sum_{i=0}^k\overline w_{k-i}\cap w_i$.

Take a general position collection of normal fields $u_1,\ldots,u_{m-n}$$u_1,\ldots,u_{m-n}$ on $N$$N$ such that for each $i=1,\ldots,k$$i=1,\ldots,k$ the collection $u_1,\ldots,u_{m-n-i+1}$$u_1,\ldots,u_{m-n-i+1}$ is linearly dependent on some $(n-i)$$(n-i)$-subcomplex $\omega^*_i$$\omega^*_i$ representing $\overline w_i$$\overline w_i$.

Take a general position collection of tangent fields $v_1,\ldots,v_n$$v_1,\ldots,v_n$ on $N$$N$ such that for each $i=1,\ldots,k$$i=1,\ldots,k$ the collection $v_i,\ldots,v_n$$v_i,\ldots,v_n$ is linearly dependent on some $(n-i)$$(n-i)$-subcomplex $\omega_i$$\omega_i$ representing $w_i$$w_i$.

Define $\alpha_i:=\mathrm{vol}(u_1,\ldots,u_i)$$\alpha_i:=\mathrm{vol}(u_1,\ldots,u_i)$ and $\beta_i := \mathrm{vol}(v_i, \ldots v_n)$$\beta_i := \mathrm{vol}(v_i, \ldots v_n)$. Denote by $C$$C$ the following collection of $m-k+1$$m-k+1$ vector fields on $N$$N$:

$\displaystyle u_1,\ldots,u_{m-n-k+1},\ \ \alpha_{m-n-k+2}u_{m-n-k+2}+\beta_1v_1,\ \ldots, \ \alpha_{m-n}u_{m-n}+\beta_{k-1}v_{k-1},\ \ v_k,\ldots,v_n.$ $$\displaystyle u_1,\ldots,u_{m-n-k+1},\ \ \alpha_{m-n-k+2}u_{m-n-k+2}+\beta_1v_1,\ \ldots, \ \alpha_{m-n}u_{m-n}+\beta_{k-1}v_{k-1},\ \ v_k,\ldots,v_n.$$

This is a general position collection, so $x_k$$x_k$ is represented by the set set on which $C$$C$ is linearly dependent. Clearly, all non-zero vectors among $\alpha_1u_1,\ldots,\alpha_{m-n}u_{m-n},\beta_1v_1,\ldots,\beta_nv_n$$\alpha_1u_1,\ldots,\alpha_{m-n}u_{m-n},\beta_1v_1,\ldots,\beta_nv_n$ are linearly independent. Hence $C(x)$$C(x)$ is linearly dependent if and only if either:

• $u_1,\ldots,u_{m-n-k+1}$$u_1,\ldots,u_{m-n-k+1}$ are linearly dependent at $x$$x$ (which happens on $\omega^*_k$$\omega^*_k$) or
• $v_k,\ldots,v_n$$v_k,\ldots,v_n$ are linearly dependent at $x$$x$ (which happens on $\omega_k$$\omega_k$) or
• $C(x)$$C(x)$ contains a zero vector (which happens if and only if $\alpha_{m-n-k+1+i}=\beta_i=0$$\alpha_{m-n-k+1+i}=\beta_i=0$ at $x$$x$ for some $i = 1,\ldots, k-1$$i = 1,\ldots, k-1$).

Thus $x_k$$x_k$ is represented by $\bigcup_{i=0}^{k}(\omega^*_{k-i}\cap\omega_i)$$\bigcup_{i=0}^{k}(\omega^*_{k-i}\cap\omega_i)$.

3 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. Preprint of a part in English