Stiefel-Whitney characteristic classes

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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
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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(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 2.1 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 [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 the Whitney invariant.

See also Wikipedia article.

2 Whitney-Wu formula

In this section we abbreviate PDw_i(N) to just w_i and PDw_i(f) to just \overline w_i.

Theorem 2.1 (Whitney-Wu formula).

If N is a closed smooth n-manifold, f:N\to\R^m an immersion and k>0 is an integer, then
\displaystyle \sum_{i=0}^{k}\overline w_{k-i}\cap w_i=0.

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:

\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.

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_{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

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.

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