Stiefel-Whitney characteristic classes

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(Definition)
(Definition)
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Whitney defined ''the homology normal Stiefel-Whitney class'' $PD\overline{W}_k(N)\in H_{n-k}(N,\partial N;G_k)$ of $N$ to be the homology class of a degeneracy subset of a general position collection of $n+1-k$ ''normal'' vector fields on $N$. Let $\overline{w}_k(N)$ be the reduction of $\overline{W}_k(N)$ modulo 2.
Whitney defined ''the homology normal Stiefel-Whitney class'' $PD\overline{W}_k(N)\in H_{n-k}(N,\partial N;G_k)$ of $N$ to be the homology class of a degeneracy subset of a general position collection of $n+1-k$ ''normal'' vector fields on $N$. Let $\overline{w}_k(N)$ be the reduction of $\overline{W}_k(N)$ modulo 2.
See details e.g. in \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 [[http://www.map.mpim-bonn.mpg.de/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 [[http://www.map.mpim-bonn.mpg.de/Embeddings_just_below_the_stable_range:_classification#The_Whitney_invariant|the Whitney invariant]].

Revision as of 13:40, 28 March 2019


This page has not been refereed. The information given here might be incomplete or provisional.

1 Definition

Let N be a compact smooth n-manifold (possibly with boundary). Denote by PD=PD_G:H^k(N;G)\to H_{n-k}(N,\partial N;G) the Poincaré duality isomorphism. Here for non-orientable N the coefficients in cohomology are twisted (by the orientation double covering) and the coefficients in homology are non-twisted. Let G_k=G_{k,N} be \Zz if either k=n or k>1 is odd, and \Zz_2 if either k=1 or k<n is even.

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(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 normal vector fields on N. Let \overline{w}_k(N) be the reduction of \overline{W}_k(N) modulo 2.

See details e.g. in [Milnor&Stasheff1974, \S12], [Fomenko&Fuchs2016, \S19.C], [Skopenkov2015b, \S\S 9,11,12].

There is an alternative definition of PD\overline{W}_k(N) [Skopenkov2006, \S2.3 `the Whitney obstruction'] analogous to definition of [Whitney invariant].

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.

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