Stiefel-Whitney characteristic classes

Revision as of 15:23, 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 $N$. Let $w_k(N)$ be the reduction of $W_k(N)$ 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 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[$\S.3 `the Whitney obstruction']{Skopenkov2006} analogous to definition of [[Embeddings_just_below_the_stable_range:_classification#The_Whitney_invariant|the Whitney invariant]]. See also [[wikipedia:Stiefel–Whitney_class|Wikipedia article]]. </wikitex> == References == {{#RefList:}} [[Category:Definitions]] [[Category:Forgotten in Textbooks]]N$ 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 $N$.

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

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

.

\begin{proof} (This proof should be known but is, in this short and explicit form, absent from textbooks. This text is written by M. Fedorov and A. Skopenkov in frame of the course `Algorithms for recognition of realizability of hypergraphs'.)

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

Take general position collection of normal fields $u_1,\ldots,u_{m-n}$ on $f(N)$ 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 $f(N)$ 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 $w_{i} = [\omega_i]$.

Now take the following collection of $m-k$ vector fields on $f(N)$

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

where $\alpha_i=\mathrm{vol}(u_1,\ldots,u_i)$ and $\beta_i = \mathrm{vol}(v_i, \ldots v_n)$.

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 $\alpha_{m-n-k+1+j} = \beta_j=0$.

This means $C$ 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