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
. Let
be the reduction of
modulo 2.
See details e.g. in [Fomenko&Fuchs2016, 19.C], [Skopenkov2015b,
9,11,12].
2 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.
- [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