Stiefel-Whitney characteristic classes
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− | + | ''Proof.'' | |
− | (This proof should be known but is | + | (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 $x_k(N) = \sum_{i=0}^{k}\overline w_{k-i}(f)\cap w_i(N)$. | 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 $x_k(N) = \sum_{i=0}^{k}\overline w_{k-i}(f)\cap w_i(N)$. | ||
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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 $w_{i} = [\omega_i]$. | 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 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> | </wikitex> | ||
Revision as of 15:36, 14 February 2021
This page has not been refereed. The information given here might be incomplete or provisional. |
1 Definition
Let be a compact smooth -manifold (possibly with boundary). Denote by the Poincaré duality isomorphism. Here for non-orientable the coefficients in cohomology are twisted (by the orientation double covering) and the coefficients in homology are non-twisted. Let be if either or is odd, and if either or is even.
Stiefel defined the homology Stiefel-Whitney class of to be the homology class of a degeneracy subset of a general position collection of tangent vector fields on .
Whitney defined the homology normal Stiefel-Whitney class of to be the homology class of a degeneracy subset of a general position collection of normal vector fields on .
Let and be the reductions of and modulo 2.
Let .
See details e.g. in [Milnor&Stasheff1974, 12], [Fomenko&Fuchs2016, 19.C], [Skopenkov2015b, 9,11,12].
There is an alternative definition of [Skopenkov2006, 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 is a closed smooth -submanifold of and is an integer, then
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 the obstruction to existence of linearly independent fields on . Clearly . So it suffices to show that .
Take general position collection of normal fields on such that for each the collection is linearly dependent on subcomplex representing .
Take general position collection of tangent fields on such that for each the collection is linearly dependent on subcomplex representing .
Define and . Take the following collection of vector fields on
Thus all non-zero vectors among are linearly independent. Hence collection is linearly dependent if and only if contains zero vector or are linearly dependent or are linearly dependent. But contains a zero vector if and only if there exist such that . Thus is linearly dependent on .
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