Stiefel-Whitney characteristic classes
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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.
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 zero vector or
are linearly dependent at
or
are linearly dependent at
.
The collection
contains a zero vector if and only if
at
for some
.
Thus ???
is represented by
.
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