Complexifiable characteristic classes

In the theory of characteristic classes (in the sense of Milnor and Stasheff [MilnorStasheff], whom we follow in terminology and notation in this article), it is well-known how the Chern classes are mapped to even Stiefel-Whitney classes when converting complex vector space bundles to real vector space bundles by forgetting the complex structure. In the other direction, we have the fibre-wise complexification: Given a real vector bundle $F\rightarrow B$ with fibre ${\mathbb{R}}^{n}$, its complexification is the complex vector bundle $F^{\mathbb{C}}:=F\otimes_{\mathbb{R}}{\mathbb{C}}\rightarrow B$ obtained by declaring complex multiplication on $F\oplus F$ in each fibre ${\mathbb{R}}^{n}\oplus{\mathbb{R}}^{n}$ by $i(x,y):=(-y,x)$ for the imaginary unit $i$. The Pontrjagin classes of a real vector bundle are (up to a sign) constructed as Chern classes of its complexification. Conversely, which classes of a real vector bundle can be attributed to its complexification? These are the complexifiable characteristic classes which we determine in this article, under the request that they are characteristic classes in the sense of [MilnorStasheff].

Consider a real vector bundle $F\rightarrow B$ and a complex vector bundle $E\rightarrow B$ over the same paracompact Hausdorff base space $B$ (we keep the latter assumption on $B$ throughout this article).

Not every complex vector bundle is real-generated; as the odd degree Chern classes have the property $c_{2k+1}(\overline{E})=-c_{2k+1}(E)$ on the complex conjugate bundle $\overline{E}$, it is an easy exercise to show that no complex vector bundle with some nonzero and non-$2$-torsion odd Chern class can admit a real generator bundle. This makes it seem possible that the subcategory of real-generated vector bundles could admit information additional to its Chern classes, in terms of complexifiable classes of the real generator bundles. However, we will see that the Chern classes already contain all of the relevant information.

We will now give a complete classification of the complexifiable characteristic classes. Denote by ${\mathbb{Z}}_{2}:={\mathbb{Z}}/2{\mathbb{Z}}$ the group with two elements.

The implication (i)$\Rightarrow$(ii) follows easily from the fact that the square of the $n$-th Stiefel-Whitney class of a real vector bundle is the mod-$2$-reduction of the $n$-th Chern class of the complexified vector bundle. The proof of the implication (ii)$\Rightarrow$(i) is prepared with several intermediary steps leading to it. One ingredient, Lemma 1, follows essentially from work of Cartan on fibrations of H-spaces (at Cartan’s time called Hopf spaces). But this only allows us to show that complexifiable characteristic classes in cohomology with ${\mathbb{Z}}_{2}$–coefficients are contained in the ideal generated by the squares of the Stiefel-Whitney classes. To show that they constitute exactly the subring generated by the squares of the Stiefel-Whitney classes, which is much smaller, we need the technical decomposition of Lemma 2 that we prove by induction.

By their naturality, characteristic classes are uniquely determined on the universal bundle over the classifying space ($B{\mathcal{O}}$ for real vector bundles). As the cohomology ring $\operatorname{H}^{*}(B{\mathcal{O}},{\mathbb{Z}}_{2})$ is generated by the Stiefel-Whitney classes of the universal bundle, all modulo $2$ characteristic classes are polynomials in the Stiefel-Whitney classes, and Theorem 1 tells us which of them are complexifiable.

We build on this result to investigate which integral cohomology classes are complexifiable. To express our result, we use Feshbach’s description [Feshbach] of the cohomology ring of the classifying space $B{\mathcal{O}}$ with ${\mathbb{Z}}$–coefficients. Generators for this ring are known since Thomas [polynomialAlgebras], [realGrassmann], and all the relations between its generators are known since Brown [Brown] and Feshbach [Feshbach]. Consider the Steenrod squaring operation $Sq^{1}$ and the mod–2–reduction homomorphism

As generators for $\operatorname{H}^{*}(B{\mathcal{O}},{\mathbb{Z}})$, Feshbach uses Pontrjagin classes and classes $V_{I}$ with index sets $I$ that are finite nonempty subsets of $\left\{\frac{1}{2}\right\}\cup{\mathbb{N}},$ admitting mod–2–reductions

where $\omega_{i}$ is the $i$-th Stiefel Whitney class of the universal bundle over $B{\mathcal{O}}$. In particular, we have a generator $V_{\{\frac{1}{2}\}}$. We give the details of Feshbach’s description in the appendix. Our final result now takes the following shape.

And conversely, we can say the following.

In this section, we shall prove Theorem 1, after developing all the tools we need to do so. For this entire section, we only consider cohomology with ${\mathbb{Z}}_{2}$–coefficients. We write ${\mathbb{N}}$ for the natural numbers without $0$.
Let $F\rightarrow B$ be a real vector bundle over a paracompact Hausdorff base space. Let $c$ be a complexifiable polynomial in the Stiefel-Whitney classes $w_{i}$. Let ${\mathcal{O}}$ be the direct limit of the orthogonal groups, $U$ the direct limit of the unitary groups and $EU$ the universal total space to the classifying space $BU$ for stable complex vector bundles. Let $B{\mathcal{O}}:=EU/{\mathcal{O}}$, via the inclusion ${\mathcal{O}}\subset U$ induced by the canonical inclusion ${\mathbb{R}}\subset{\mathbb{C}}$. Let $\gamma({\mathbb{R}}^{\infty})$ be the universal bundle over $B{\mathcal{O}}$, and denote its Stiefel-Whitney classes by $\omega_{i}:=w_{i}(\gamma({\mathbb{R}}^{\infty}))$. Let $\varepsilon$ be the trivial vector bundle.

The above lemma allows us to write the characteristic class $c$ under investigation as a sum over products with squares of Stiefel-Whitney classes,

with $r_{j}$ some polynomials in the Stiefel-Whitney classes. We must inductively identify squares of Stiefel-Whitney classes as factors of the remainders $r_{j}$, until we achieve the decomposition claimed in the following lemma.

Before giving the proof of Lemma 2, we shall introduce two notations just to make that proof more readable.

Note that the terms $\left(r_{\vec{j}}(\gamma({\mathbb{R}}^{\infty}))\cup\bigcup\limits_{p\hskip 2.84526pt\in\hskip 2.84526ptI(\vec{j})}\omega_{p}^{2}\right)$ with   $\left(2\sum\limits_{p\hskip 2.84526pt\in\hskip 2.84526ptI(\vec{j})}p>\deg c\right)$ must vanish in any decomposition of $c(\gamma({\mathbb{R}}^{\infty}))-c(\varepsilon)$. That is why we do not let them contribute in the last definition.

As seen in Lemma 1, $c(\gamma({\mathbb{R}}^{\infty}))-c(\varepsilon)$ lies in $\ker f^{*}=\langle\omega_{i}^{2}\rangle_{i\in{\mathbb{N}}}$, so there is a decomposition

for some $m$ $\in$ ${\mathbb{N}}\cup\{0\}$, some $i_{\vec{j}_{1}}$ $\in$ ${\mathbb{N}}$, and some $r_{\vec{j}_{1}}(\gamma({\mathbb{R}}^{\infty}))$ $\in$ $\operatorname{H}^{*}(B{\mathcal{O}},{\mathbb{Z}}_{2})$. We will show that there is a lower degree remainder $r_{\vec{j}_{1}}(\gamma({\mathbb{R}}^{\infty}))-r_{\vec{j}_{1}}(\varepsilon)$ in this decomposition that lies in $\ker f^{*}$. Then, that lower degree remainder admits a decomposition as a linear combination of squares $\omega_{i_{\vec{j}_{2}}}^{2}$ with coefficients $r_{\vec{j}_{2}}(\gamma({\mathbb{R}}^{\infty}))$ in $\operatorname{H}^{*}(B{\mathcal{O}},{\mathbb{Z}}_{2})$, leading to a new decomposition of $c(\gamma({\mathbb{R}}^{\infty}))-c(\varepsilon)$. So, inductively, we will replace a lower degree remainder in any given decomposition of $c(\gamma({\mathbb{R}}^{\infty}))-c(\varepsilon)$ by a linear combination the coefficients of which are remainders with longer index vectors. That is why after a finite number of these steps, the index vectors $\vec{j}$ will no longer appear, because the sums $\left(2\sum\limits_{p\hskip 2.84526pt\in\hskip 2.84526ptI(\vec{j})}p\right)$ will exceed the degree of $c$. This is the moment when all lower degree remainders are eliminated and the decomposition described in Lemma 2 is achieved.

To carry out this strategy, we first need to introduce the following truncation procedure.

We will build on our results obtained for ${\mathbb{Z}}_{2}$–coefficients and use the mod–2–reduction homomorphism

to prove the theorems with ${\mathbb{Z}}$–coefficients stated in the introduction. Define the element $V_{I}\in\operatorname{H}^{*}(B{\mathcal{O}},{\mathbb{Z}})$ as in the appendix, and let $v_{I}$ be the characteristic class that is $V_{I}$ on the universal bundle.