Stiefel-Whitney characteristic classes

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1 Definition

Let $N$${{Stub}} == Definition == <wikitex>; Let $N$ be a compact smooth $n$-manifold (possibly with boundary). Denote by $PD=PD_G:H^k(N;G)\to H_{n-k}(N,\partial N;G)$ the Poincaré duality isomorphism. Here for non-orientable $N$ the coefficients in cohomology are twisted (by the orientation double covering) and the coefficients in homology are non-twisted. Let $G_k=G_{k,N}$ be $\Zz$ if either $k=n$ or $k>1$ is odd, and $\Zz_2$ if either $k=1$ or $k<n$ is even. Stiefel defined ''the homology Stiefel-Whitney class'' $PDW_k(N)\in H_{n-k}(N,\partial N;G_k)$ of $N$ to be the homology class of a degeneracy subset of a general position collection of $n+1-k$ ''tangent'' vector fields on $N$. Let $w_k(N)$ be the reduction of $W_k(N)$ modulo 2. Whitney defined ''the homology normal Stiefel-Whitney class'' $PD\overline{W}_k(f)\in H_{n-k}(N,\partial N;G_k)$ of $N$ to be the homology class of a degeneracy subset of a general position collection of $n+1-k$ ''normal'' vector fields on an immersion $f:N\subset\Rr^m$. By the Whitney-Wu formula \ref{t:wu} the reduction modulo 2 of this class (but not this class itself!) is independent of $f$ and depends only on $N$. So this reduction is denoted by $\overline{w}_k(N)$. Let $w_0(N)=\overline w_0(N)=[N]$. See details e.g. in \cite[$\S]{Milnor&Stasheff1974}, \cite[$\S.C]{Fomenko&Fuchs2016}, \cite[$\S\S$ 9,11,12]{Skopenkov2015b}. There is an alternative definition of $PD\overline{W}_k(N)$ \cite[$\S.3 `the Whitney obstruction']{Skopenkov2006} analogous to definition of [[Embeddings_just_below_the_stable_range:_classification#The_Whitney_invariant|the Whitney invariant]]. See also [[wikipedia:Stiefel–Whitney_class|Wikipedia article]]. </wikitex> == Whitney-Wu formula == <wikitex>; In this section we abbreviate $PDw_i(N)$ to just $w_i$ and $PDw_i(f)$ to just $\overline w_i$. {{beginthm|Theorem|(Whitney-Wu formula)}}\label{t:wu} If $N$ is a closed smooth $n$-manifold, $f:N\to\R^m$ an immersion and $k>0$ is an integer, then $$\sum_{i=0}^{k}\overline w_{k-i}\cap w_i=0.$$ {{endthm}} ''Proof.'' (This proof should be known but is absent from textbooks. This text is written by M. Fedorov and A. Skopenkov in frame of the course `Algebraic topology of manifolds in interesting results'.) Denote by $x_k$ the obstruction to existence of $m-k+1$ linearly independent fields on $N$. Clearly $x_k=0$. So it suffices to show that $x_k=\sum_{i=0}^k\overline w_{k-i}\cap w_i$. Take a general position collection of normal fields $u_1,\ldots,u_{m-n}$ on $N$ such that for each $i=1,\ldots,k$ the collection $u_1,\ldots,u_{m-n-i+1}$ is linearly dependent on some $(n-i)$-subcomplex $\omega^*_i$ representing $\overline w_i$. Take a general position collection of tangent fields $v_1,\ldots,v_n$ on $N$ such that for each $i=1,\ldots,k$ the collection $v_i,\ldots,v_n$ is linearly dependent on some $(n-i)$-subcomplex $\omega_i$ representing $w_i$. Define $\alpha_i:=\mathrm{vol}(u_1,\ldots,u_i)$ and $\beta_i := \mathrm{vol}(v_i, \ldots v_n)$. Denote by $C$ the following collection of $m-k+1$ vector fields on $N$: $$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.$$ This is a general position collection, so $x_k$ is represented by the set set on which $C$ is linearly dependent. Clearly, all non-zero vectors among $\alpha_1u_1,\ldots,\alpha_{m-n}u_{m-n},\beta_1v_1,\ldots,\beta_nv_n$ are linearly independent. Hence $C(x)$ is linearly dependent if and only if either: * $u_1,\ldots,u_{m-n-k+1}$ are linearly dependent at $x$ (which happens on $\omega^*_k$) or * $v_k,\ldots,v_n$ are linearly dependent at $x$ (which happens on $\omega_k$) or * $C(x)$ contains a zero vector (which happens if and only if $\alpha_{m-n-k+1+i}=\beta_i=0$ at $x$ for some $i = 1,\ldots, k-1$). Thus $x_k$ is represented by $\bigcup_{i=0}^{k}(\omega^*_{k-i}\cap\omega_i)$, where $\omega^*_{k}\cap\omega_0 = \omega^*_{k}$ and $\omega^*_0\cap\omega_k = \omega_k$ </wikitex> == References == {{#RefList:}} [[Category:Theory]] [[Category:Definitions]] [[Category:Forgotten in Textbooks]]N$ be a compact smooth $n$$n$-manifold (possibly with boundary). Denote by $PD=PD_G:H^k(N;G)\to H_{n-k}(N,\partial N;G)$$PD=PD_G:H^k(N;G)\to H_{n-k}(N,\partial N;G)$ the Poincaré duality isomorphism. Here for non-orientable $N$$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}$$G_k=G_{k,N}$ be $\Zz$$\Zz$ if either $k=n$$k=n$ or $k>1$$k>1$ is odd, and

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$\Zz_2$ if either $k=1$$k=1$ or $k<n$$k<n$ is even.

Stiefel defined the homology Stiefel-Whitney class $PDW_k(N)\in H_{n-k}(N,\partial N;G_k)$$PDW_k(N)\in H_{n-k}(N,\partial N;G_k)$ of $N$$N$ to be the homology class of a degeneracy subset of a general position collection of $n+1-k$$n+1-k$ tangent vector fields on $N$$N$. Let $w_k(N)$$w_k(N)$ be the reduction of $W_k(N)$$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)$$PD\overline{W}_k(f)\in H_{n-k}(N,\partial N;G_k)$ of $N$$N$ to be the homology class of a degeneracy subset of a general position collection of $n+1-k$$n+1-k$ normal vector fields on an immersion $f:N\subset\Rr^m$$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$$f$ and depends only on $N$$N$. So this reduction is denoted by $\overline{w}_k(N)$$\overline{w}_k(N)$.

Let $w_0(N)=\overline w_0(N)=[N]$$w_0(N)=\overline w_0(N)=[N]$.

See details e.g. in [Milnor&Stasheff1974, $\S$$\S$12], [Fomenko&Fuchs2016, $\S$$\S$19.C], [Skopenkov2015b, $\S\S$$\S\S$ 9,11,12].

There is an alternative definition of $PD\overline{W}_k(N)$$PD\overline{W}_k(N)$ [Skopenkov2006, $\S$$\S$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 $PDw_i(N)$$PDw_i(N)$ to just $w_i$$w_i$ and $PDw_i(f)$$PDw_i(f)$ to just $\overline w_i$$\overline w_i$.

Theorem 2.1 (Whitney-Wu formula).

If $N$$N$ is a closed smooth $n$$n$-manifold, $f:N\to\R^m$$f:N\to\R^m$ an immersion and $k>0$$k>0$ is an integer, then

$\displaystyle \sum_{i=0}^{k}\overline w_{k-i}\cap w_i=0.$ $$\displaystyle \sum_{i=0}^{k}\overline w_{k-i}\cap w_i=0.$$

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$$x_k$ the obstruction to existence of $m-k+1$$m-k+1$ linearly independent fields on $N$$N$. Clearly $x_k=0$$x_k=0$. So it suffices to show that $x_k=\sum_{i=0}^k\overline w_{k-i}\cap w_i$$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}$$u_1,\ldots,u_{m-n}$ on $N$$N$ such that for each $i=1,\ldots,k$$i=1,\ldots,k$ the collection $u_1,\ldots,u_{m-n-i+1}$$u_1,\ldots,u_{m-n-i+1}$ is linearly dependent on some $(n-i)$$(n-i)$-subcomplex $\omega^*_i$$\omega^*_i$ representing $\overline w_i$$\overline w_i$.

Take a general position collection of tangent fields $v_1,\ldots,v_n$$v_1,\ldots,v_n$ on $N$$N$ such that for each $i=1,\ldots,k$$i=1,\ldots,k$ the collection $v_i,\ldots,v_n$$v_i,\ldots,v_n$ is linearly dependent on some $(n-i)$$(n-i)$-subcomplex $\omega_i$$\omega_i$ representing $w_i$$w_i$.

Define $\alpha_i:=\mathrm{vol}(u_1,\ldots,u_i)$$\alpha_i:=\mathrm{vol}(u_1,\ldots,u_i)$ and $\beta_i := \mathrm{vol}(v_i, \ldots v_n)$$\beta_i := \mathrm{vol}(v_i, \ldots v_n)$. Denote by $C$$C$ the following collection of $m-k+1$$m-k+1$ vector fields on $N$$N$:

$\displaystyle u_1,\ldots,u_{m-n-k+1},\ \ \alpha_{m-n-k+2}u_{m-n-k+2}+\beta_1v_1,\ \ldots, \ \alpha_{m-n}u_{m-n}+\beta_{k-1}v_{k-1},\ \ v_k,\ldots,v_n.$ $$\displaystyle u_1,\ldots,u_{m-n-k+1},\ \ \alpha_{m-n-k+2}u_{m-n-k+2}+\beta_1v_1,\ \ldots, \ \alpha_{m-n}u_{m-n}+\beta_{k-1}v_{k-1},\ \ v_k,\ldots,v_n.$$

This is a general position collection, so $x_k$$x_k$ is represented by the set set on which $C$$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$$\alpha_1u_1,\ldots,\alpha_{m-n}u_{m-n},\beta_1v_1,\ldots,\beta_nv_n$ are linearly independent. Hence $C(x)$$C(x)$ is linearly dependent if and only if either:

• $u_1,\ldots,u_{m-n-k+1}$$u_1,\ldots,u_{m-n-k+1}$ are linearly dependent at $x$$x$ (which happens on $\omega^*_k$$\omega^*_k$) or
• $v_k,\ldots,v_n$$v_k,\ldots,v_n$ are linearly dependent at $x$$x$ (which happens on $\omega_k$$\omega_k$) or
• $C(x)$$C(x)$ contains a zero vector (which happens if and only if $\alpha_{m-n-k+1+i}=\beta_i=0$$\alpha_{m-n-k+1+i}=\beta_i=0$ at $x$$x$ for some $i = 1,\ldots, k-1$$i = 1,\ldots, k-1$).

Thus $x_k$$x_k$ is represented by $\bigcup_{i=0}^{k}(\omega^*_{k-i}\cap\omega_i)$$\bigcup_{i=0}^{k}(\omega^*_{k-i}\cap\omega_i)$.

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. Preprint of a part in English