Characteristic rank of a real vector bundle

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1 Definition

Definition 1.1. Let $X$${{Authors|Ludovit Balko }}{{Stub}} <!-- == Introduction == <wikitex>; ... </wikitex> --> == Definition == <wikitex>; {{beginthm|Definition}}\label{defcharrank} Let $X$ be a connected, finite $CW$-complex and $\xi$ a real vector bundle over $X$. The characteristic rank of the vector bundle $\xi$ over $X$, denoted by $\mathrm{charrank}_X(\xi)$, is the largest integer $k$, <script type="math/tex">X$ be a connected, finite $CW$$CW$-complex and $\xi$$\xi$ a real vector bundle over $X$$X$. The characteristic rank of the vector bundle $\xi$$\xi$ over $X$$X$, denoted by $\mathrm{charrank}_X(\xi)$$\mathrm{charrank}_X(\xi)$, is the largest integer $k$$k$, $0 \leq k \leq \dim(X)$$0 \leq k \leq \dim(X)$, such that every cohomology class $x \in H^j(X;\mathbb{Z}_2)$$x \in H^j(X;\mathbb{Z}_2)$, $0 \leq j \leq k$$0 \leq j \leq k$, is a polynomial in the Stiefel-Whitney classes $w_i(\xi)$$w_i(\xi)$ of $\xi$$\xi$. If $X$$X$ is closed, smooth, connected manifold, characteristic rank of manifold $X$$X$, denoted by $\mathrm{charrank}(M)$$\mathrm{charrank}(M)$ is defined as characteristic rank of $\xi=TM$$\xi=TM$, the tangent bundle of $X$$X$.

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

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

2 Motivation

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

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

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

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

3 Examples

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

4 References

• [Korbaš2010] J. Korbaš, The cup-length of the oriented Grassmannians vs a new bound for zero-cobordant manifolds, Bull. Belg. Math. Soc. Simon Stevin 17 (2010), 69–81.
• [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
• [Naolekar&Thakur2014] A. C. Naolekar and A. S. Thakur, Note on the characteristic rank of vector bundles, Math. Slovaca (2014) 64: 1525. https://doi.org/10.2478/s12175-014-0289-4

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