Characteristic rank of a real vector bundle

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1 Definition

Definition 1.1. Let X be a connected, finite CW-complex and \xi a real vector bundle over X. The characteristic rank of the vector bundle \xi over X, denoted by \mathrm{charrank}_X(\xi), is the largest integer k, 0 \leq k \leq \dim(X), such that every cohomology class x \in H^j(X;\mathbb{Z}_2), 0 \leq j \leq k, is a polynomial in the Stiefel-Whitney classes w_i(\xi) of \xi. If X is closed, smooth, connected manifold, characteristic rank of manifold X, denoted by \mathrm{charrank}(M) is defined as characteristic rank of \xi=TM, the tangent bundle of X.

Bundle dependency then admits the following definition of the upper characteristic rank of CW-complex X.

Definition 1.2. The upper characteristic rank of CW-complex X, \mathrm{ucharrank}(X) is maximum of \mathrm{charrank}_X(\xi) as \xi varies over all vector bundles over X.

2 Motivation

For closed, smooth, connected d-dimensional manifold M unorientedly cobordant to zero, there exist an element in \mathbb{Z}_2-cohomology algebra H^\ast(M;\mathbb{Z}_2) of M, which cannot be expressed as a polynomial in Stiefel-Whitney classes w_i(M) of (tangent bundle of) manifold M. For this type of manifold there is the following estimate of \mathbb{Z}_2 cuplength of manifold M.

Theorem 2.1.[Korbaš2010] Let M be a closed smooth connected d-dimensional manifold unorientedly cobordant to zero. Let H^r(M;\mathbb{Z}_2), r < d, be the first nonzero reduced cohomology group of M. Let z (z < d - 1) be an integer such that for j \leq z each element of H^j(M;\mathbb{Z}_2) can be expressed as a polynomial in the Stiefel-Whitney classes of the manifold M. Then we have that

(1)\mathrm{cup}(M)\leq 1+\frac{d-z-1}{r}.

This leads the definition of characteristic rank of manifold M, which is the largest possible z which fulfills the conditions of the theorem above. Making the definition of characteristic rank bundle dependent, i.e. replacing manifold M with connected, finite CW-complex and tangent bundle TM of M with a real vector bundle \xi, gives definition 1.1 [Naolekar&Thakur2014].

3 Examples

The \mathbb{Z}_2 cohomology ring of n-dimensional real projective space is H^\ast(\mathbb{R}P^n;\mathbb{Z}_2)=\mathbb{Z}_2[w_1(\gamma_n^1)]/(w_1(\gamma_n^1)^{n+1}), where w_1(\gamma_n^1) is the first Stiefel-Whitney characteristic class of canonical line bundle \gamma_n^1 over \mathbb{R}P^n [Milnor&Stasheff1974]. It is then clear, that \mathrm{charrank}_{\mathbb{R}P^n}(\gamma_n^1)=n. On the other hand, we have w_1(\mathbb{R}P^{k})=0 for k odd and w_1(\mathbb{R}P^{k})=w_1(\gamma_k^1) for k even. In this case \mathrm{charrank}(\mathbb{R}P^n) equals 0 if k is odd, and equals k if k is even.

4 References

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