Stiefel-Whitney characteristic classes

Revision as of 15:43, 14 February 2021

1 Definition

Let ${{Stub}} == Definition == <wikitex>; 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 $k<n$ is even. Stiefel defined ''the homology Stiefel-Whitney class'' $PDW_k(N)\in H_{n-k}(N,\partial N;G_k)$ of $N$ to be the homology class of a degeneracy subset of a general position collection of $n+1-k$ ''tangent'' vector fields on $N$. 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 $N\subset\Rr^m$. Let $w_k(N)$ and $\overline{w}_k(N)$ be the reductions of $W_k(N)$ and $\overline{W}_k(N)$ modulo 2. Let $w_0(N)=\overline w_0(N)=[N]$. See details e.g. in \cite[$\S]{Milnor&Stasheff1974}, \cite[$\S.C]{Fomenko&Fuchs2016}, \cite[$\S\S$ 9,11,12]{Skopenkov2015b}. There is an alternative definition of $PD\overline{W}_k(N)$ \cite[$\S.3 `the Whitney obstruction']{Skopenkov2006} analogous to definition of [[Embeddings_just_below_the_stable_range:_classification#The_Whitney_invariant|the Whitney invariant]]. See also [[wikipedia:Stiefel–Whitney_class|Wikipedia article]]. </wikitex> == The Wu formula == <wikitex>; {{beginthm|Theorem|(Wu formula)}}\label{t:wu} If $N$ is a closed smooth $n$-submanifold of $\R^m$ and $k>0$ is an integer, then $$\sum_{i=0}^{k}\overline w_{k-i}(f)\cap w_i(N)=0.$$ {{endthm}} ''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 $x_k(N)$ the obstruction to existence of $m-k+1$ linearly independent fields on $N$. Clearly $x_k(N)=0$. So it suffices to show that $x_k(N) = \sum_{i=0}^{k}\overline w_{k-i}(f)\cap w_i(N)$. Take general position collection of normal fields $u_1,\ldots,u_{m-n}$ on $N$ such that for each $i=1,\ldots,k$ the collection $u_1,\ldots,u_{m-n-i+1}$ is linearly dependent on $n-i$ subcomplex $\omega^*_i$ representing $\overline w_{i} = [\omega^*_i]$. Take general position collection of tangent fields $v_1,\ldots,v_n$ on $N$ such that for each $i=1,\ldots,k$ the collection $u_i,\ldots,u_n$ is linearly dependent on $n-i$ subcomplex $\omega_i$ representing $w_{i} = [\omega_i]$. Define $\alpha_i:=\mathrm{vol}(u_1,\ldots,u_i)$ and $\beta_i := \mathrm{vol}(v_i, \ldots v_n)$. Take the following collection $C$ of $m-k$ vector fields on $N$ $$u_1,\ \ldots,\ u_{m-n-k+1},\ \alpha_{m-n-k+2}u_{m-n-k+2}+\beta_1v_1,\ \ldots, \ \alpha_{m-n}u_{m-n}+\beta_{k-1}v_{k-1},\ v_k,\ \ldots,\ v_n.$$ Thus all non-zero vectors among $\alpha_1u_1,\ldots,\alpha_{m-n}u_{m-n},\beta_1v_1,\ldots,\beta_nv_n$ are linearly independent. Hence collection $C$ is linearly dependent if and only if $C$ contains zero vector or $u_1,\ldots,u_{m-n-k+1}$ are linearly dependent or $v_k,\ldots,v_n$ are linearly dependent. But $C$ contains a zero vector if and only if there exist $j$ such that $\alpha_{m-n-k+1+j} = \beta_j=0$. Thus $C$ is linearly dependent on $\bigcup_{i=0}^{k}(\omega^*_{k-i}\cap\omega_i)$. </wikitex> == References == {{#RefList:}} [[Category:Definitions]] [[Category:Forgotten in Textbooks]]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 $k<n$ is even.

Stiefel defined the homology Stiefel-Whitney class $PDW_k(N)\in H_{n-k}(N,\partial N;G_k)$ of $N$ to be the homology class of a degeneracy subset of a general position collection of $n+1-k$ tangent vector fields on $N$.

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 $N\subset\Rr^m$.

Let $w_k(N)$ and $\overline{w}_k(N)$ be the reductions of $W_k(N)$ and $\overline{W}_k(N)$ modulo 2.

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

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

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

See also Wikipedia article.

2 The Wu formula

Theorem 2.1 (Wu formula). If $N$ is a closed smooth $n$-submanifold of $\R^m$ and $k>0$ is an integer, then

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

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

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

Take general position collection of tangent fields $v_1,\ldots,v_n$ on $N$ such that for each $i=1,\ldots,k$ the collection $u_i,\ldots,u_n$ is linearly dependent on $n-i$ subcomplex $\omega_i$ representing $w_{i} = [\omega_i]$.

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

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

This is a general position collection, so the $x_k(N)$ is represented by set where $C$ is linearly dependent. Clearly, all non-zero vectors among $\alpha_1u_1,\ldots,\alpha_{m-n}u_{m-n},\beta_1v_1,\ldots,\beta_nv_n$ are linearly independent. Hence collection $C$ is linearly dependent if and only if either $C$ contains zero vector or $u_1,\ldots,u_{m-n-k+1}$ are linearly dependent or $v_k,\ldots,v_n$ are linearly dependent. But $C$ contains a zero vector if and only if there exist $j$ such that $\alpha_{m-n-k+1+j} = \beta_j=0$. Thus $C$ is linearly dependent on $\bigcup_{i=0}^{k}(\omega^*_{k-i}\cap\omega_i)$.

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. Accepted for English translation by `Moscow Lecture Notes' series of Springer. Preprint of a part