Stiefel-Whitney characteristic classes
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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 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 $ | + | 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 $\beta_i := \mathrm{vol}(v_i, \ldots v_n)$. | Define $\alpha_i:=\mathrm{vol}(u_1,\ldots,u_i)$ and $\beta_i := \mathrm{vol}(v_i, \ldots v_n)$. | ||
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\ \alpha_{m-n}u_{m-n}+\beta_{k-1}v_{k-1},\ \ v_k,\ldots,v_n.$$ | \ \alpha_{m-n}u_{m-n}+\beta_{k-1}v_{k-1},\ \ v_k,\ldots,v_n.$$ | ||
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 | 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 $C(x)$ contains | + | $\alpha_1u_1,\ldots,\alpha_{m-n}u_{m-n},\beta_1v_1,\ldots,\beta_nv_n$ are linearly independent. Hence $C(x)$ is linearly dependent if and only if either $C(x)$ contains $\alpha_{m-n-k-1+i}u_{m-n-k-1+i} + \beta_{i}v_i = 0$ for some $i$, or $u_1,\ldots,u_{m-n-k+1}$ are linearly dependent at $x$, or $v_k,\ldots,v_n$ are linearly dependent at $x$. |
− | + | Thus $x_k$ is represented by $\bigcup_{i=0}^{k}(\omega^*_{k-i}\cap\omega_i)$, where $\omega^*_{k}\cap\omega_0 = \omega^*_{k}$ and $\omega^*_0\cap\omega_k = \omega_k$ | |
− | Thus | + | |
</wikitex> | </wikitex> | ||
Revision as of 18:45, 19 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.
![G_k=G_{k,N}](/images/math/3/a/5/3a528d552c2c5c273c0762e39ceb8991.png)
![\Zz](/images/math/4/a/c/4accad044940b02ba4c5fdc82933e033.png)
![k=n](/images/math/c/4/d/c4dc5bfa91f4b50702a6b365fcc76fb2.png)
![k>1](/images/math/5/d/c/5dcdb97e423849cfe92b330cca86d8a7.png)
Tex syntax errorif either
![k=1](/images/math/a/6/f/a6f0672a50348fdc06bc34fdc560cae9.png)
![k<n](/images/math/a/9/e/a9e25883c601aab53aa0d46441c24ab1.png)
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.
2 Whitney-Wu formula
In this section we abbreviate to just
and
to just
.
Theorem 2.1 (Whitney-Wu formula).
If![N](/images/math/e/2/5/e25ec8b0af895735d0fe10be2ae08fc9.png)
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![f:N\to\R^m](/images/math/0/b/e/0be72a4c0fffe25548eff4d0f07f92af.png)
![k>0](/images/math/b/4/e/b4eca4d5200d283fb85f12a9a3ac5084.png)
![\displaystyle \sum_{i=0}^{k}\overline w_{k-i}\cap w_i=0.](/images/math/0/8/7/0879c7139db7cda26c7f28ef5fcb05a7.png)
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 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
:
![\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.](/images/math/9/8/7/98749debc88be2b867e7af9cccbaa68f.png)
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
contains
for some
, or
are linearly dependent at
, or
are linearly dependent at
.
Thus
is represented by
, where
and
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