Stiefel-Whitney characteristic classes
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== Definition == | == Definition == | ||
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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. | 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( | + | Whitney defined ''the homology normal Stiefel-Whitney class'' $PD\overline{W}_k(f)\in H_{n-k}(N,\partial N;G_k)$ of $N$ to be the homology class of a degeneracy subset of a general position collection of $n+1-k$ ''normal'' vector fields on an immersion $f:N\subset\Rr^m$. |
+ | By the Whitney-Wu formula \ref{t:wu} the reduction modulo 2 of this class (but not this class itself!) is independent of $f$ and depends only on $N$. | ||
+ | So this reduction is denoted by $\overline{w}_k(N)$. | ||
+ | |||
+ | Let $w_0(N)=\overline w_0(N)=[N]$. | ||
See details e.g. in \cite[$\S$12]{Milnor&Stasheff1974}, \cite[$\S$19.C]{Fomenko&Fuchs2016}, \cite[$\S\S$ 9,11,12]{Skopenkov2015b}. | See details e.g. in \cite[$\S$12]{Milnor&Stasheff1974}, \cite[$\S$19.C]{Fomenko&Fuchs2016}, \cite[$\S\S$ 9,11,12]{Skopenkov2015b}. | ||
− | There is an alternative definition of $PD\overline{W}_k(N)$ \cite[$\S$2.3 `the Whitney obstruction']{Skopenkov2006} analogous to definition of [[Embeddings_just_below_the_stable_range:_classification#The_Whitney_invariant|the Whitney invariant]]. | + | There is an alternative definition of $PD\overline{W}_k(N)$ \cite[$\S$2.3 `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> | ||
+ | |||
+ | == Whitney-Wu formula == | ||
+ | <wikitex>; | ||
+ | In this section we abbreviate $PDw_i(N)$ to just $w_i$ and $PDw_i(f)$ to just $\overline w_i$. | ||
+ | |||
+ | {{beginthm|Theorem|(Whitney-Wu formula)}}\label{t:wu} | ||
+ | If $N$ is a closed smooth $n$-manifold, $f:N\to\R^m$ an immersion and $k>0$ is an integer, then $$\sum_{i=0}^{k}\overline w_{k-i}\cap w_i=0.$$ | ||
+ | {{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 $x_k$ the obstruction to existence of $m-k+1$ linearly independent fields on $N$. | ||
+ | Clearly $x_k=0$. | ||
+ | So it suffices to show that $x_k=\sum_{i=0}^k\overline w_{k-i}\cap w_i$. | ||
+ | |||
+ | Take a general position collection of normal fields $u_1,\ldots,u_{m-n}$ on $N$ such that for each $i=1,\ldots,k$ the collection $u_1,\ldots,u_{m-n-i+1}$ is linearly dependent on some $(n-i)$-subcomplex $\omega^*_i$ representing $\overline w_i$. | ||
+ | |||
+ | Take a general position collection of tangent fields $v_1,\ldots,v_n$ on $N$ such that for each $i=1,\ldots,k$ the collection $v_i,\ldots,v_n$ is linearly dependent on some $(n-i)$-subcomplex $\omega_i$ representing $w_i$. | ||
+ | |||
+ | 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 of $m-k+1$ vector fields on $N$: | ||
+ | $$u_1,\ldots,u_{m-n-k+1},\ \ \alpha_{m-n-k+2}u_{m-n-k+2}+\beta_1v_1,\ \ldots, | ||
+ | \ \alpha_{m-n}u_{m-n}+\beta_{k-1}v_{k-1},\ \ v_k,\ldots,v_n.$$ | ||
+ | This is a general position collection, so $x_k$ is represented by the set set on which $C$ 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 $C(x)$ is linearly dependent if and only if either: | ||
+ | * $u_1,\ldots,u_{m-n-k+1}$ are linearly dependent at $x$ (which happens on $\omega^*_k$) or | ||
+ | * $v_k,\ldots,v_n$ are linearly dependent at $x$ (which happens on $\omega_k$) or | ||
+ | * $C(x)$ contains a zero vector (which happens if and only if $\alpha_{m-n-k+1+i}=\beta_i=0$ at $x$ for some $i = 1,\ldots, k-1$). | ||
+ | |||
+ | Thus $x_k$ is represented by $\bigcup_{i=0}^{k}(\omega^*_{k-i}\cap\omega_i)$. | ||
</wikitex> | </wikitex> | ||
== References == | == References == | ||
{{#RefList:}} | {{#RefList:}} | ||
− | + | [[Category:Theory]] | |
[[Category:Definitions]] | [[Category:Definitions]] | ||
[[Category:Forgotten in Textbooks]] | [[Category:Forgotten in Textbooks]] |
Latest revision as of 18:53, 19 February 2021
This page has not been refereed. The information given here might be incomplete or provisional. |
[edit] 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 . Let be the reduction of modulo 2.
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 an immersion . By the Whitney-Wu formula 2.1 the reduction modulo 2 of this class (but not this class itself!) is independent of and depends only on . So this reduction is denoted by .
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.
[edit] 2 Whitney-Wu formula
In this section we abbreviate to just and to just .
Theorem 2.1 (Whitney-Wu formula).
If is a closed smooth -manifold, an immersion and is an integer, thenProof. (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 a general position collection of normal fields on such that for each the collection is linearly dependent on some -subcomplex representing .
Take a general position collection of tangent fields on such that for each the collection is linearly dependent on some -subcomplex representing .
Define and . Denote by the following collection of vector fields on :
This is a general position collection, so is represented by the set set on which is linearly dependent. Clearly, all non-zero vectors among are linearly independent. Hence is linearly dependent if and only if either:
- are linearly dependent at (which happens on ) or
- are linearly dependent at (which happens on ) or
- contains a zero vector (which happens if and only if at for some ).
Thus is represented by .
[edit] 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