Characteristic rank of a real vector bundle

1 Definition

Definition 1.1. Let $X$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}X$ be a connected, finite $CW$$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.