On the $32$-dimensional Rosenfeld projective plane

In [JRT23] we made a systematic study of the classical topological invariants of homogeneous spaces with a particular emphasis on the twelve compact symmetric spaces for the exceptional Lie groups. Our main examples in that paper are the three Rosenfeld projective planes of dimensions $32,64$ and $128$. In this paper we make a much more detailed study of the $32$-dimensional Rosenfeld projective plane.

Throughout we will use the notation $R$ for the $32$-dimensional Rosenfeld projective plane. Explicitly,

which is a symmetric space for the exceptional Lie group $E_{6}$. It is called $R_{5}$ in [JRT23], EIII in Cartan’s list of compact symmetric spaces, and also known as $P^{2}(\mathbb{O}\otimes\mathbb{C})$. The subgroup $C_{4}$ is the cyclic subgroup of $\mathrm{Spin}(10)\times S^{1}$ generated by $(\epsilon,i)\in\mathrm{Spin}(10)\times S^{1}$. Here $\epsilon$ is the element of the centre of $\mathrm{Spin}(10)$ which acts as multiplication by $i$ on $\delta_{10}^{+}$ and, therefore, multiplication by $-i$ on $\delta_{10}^{-}$. As usual, $\delta_{10}^{\pm}$ are the two $16$-dimensional complex spin representations of $\mathrm{Spin}(10)$.

We also use the following notation

These three homogeneous spaces for $F_{4}$ have dimensions $16,31,30$ respectively, and all three are submanifolds of $R$.

We study $R$ by using the action of $F_{4}\subset E_{6}$ on $R$. The idea for this approach comes from Atiyah and Berndt [AB03]. The action of $F_{4}$ on $R$ has three orbit types. The principal (generic) orbit of codimension $1$ is $S$ and there are $2$ special orbits $P$ and $Q$. This orbit structure is described in the Appendix to [AB03] and the paper [LM01]. It can also be derived from [Ada96, Chapter 14]. In standard terminology, the action of $F_{4}$ on $R$ has cohomogeneity $1$, and the orbit space $R/F_{4}$ is a closed interval.

Using Mostert’s cohomogeneity one theorem [Mos57, Mos57a, GGZ18, Bre72] we get the following result.

An equivalent way to describe $R$ is as the double mapping cylinder of the maps $P\leftarrow S\to Q$ given by the projections in the sphere bundles $S_{P}$ and $S_{Q}$.

The $16$-dimensional real vector bundle $N_{P}$ is the bundle over $P=F_{4}/\mathrm{Spin}(9)$ associated to the $16$-dimensional real spin representation of $\mathrm{Spin}(9)$. This is also isomorphic to the tangent bundle of $P$. The $2$-dimensional real bundle $N_{Q}$ over $Q=F_{4}/(\mathrm{Spin}(7)\times_{C_{2}}\mathrm{Spin}(2))$ is associated to the obvious representation $\mathrm{Spin}(7)\times_{C_{2}}\mathrm{Spin}(2)\to\mathrm{SO}(2)$.

This geometric decomposition of $R$ allows us to give a new proof of a theorem originally proved by Toda and Watanabe [TW74].

There is an equivalent way to present the cohomology of $R$ using Poincaré duality. Choosing a generator (fundamental class) $[R]$ in $H_{32}(R)\cong\mathbb{Z}$ yields a non-degenerate bilinear pairing

This tells us that if we know all products that end up in the top degree, then we know all products. The following table, proved in Theorem 3.4, gives all products to the top degree.

Recall that $R$ is a $16$-dimensional smooth complex subvariety of $\mathbb{C}\mathbb{P}^{26}$, often called the fourth Severi variety [Zak85]. It is also a generalised flag variety, see [LV12, §1b]. We write $T_{c}(R)$ for the 16-dimensional complex tangent bundle of $R$. As an application, we explain how to calculate the Chern classes of $T_{c}(R)$.

In Section 1 we explain the role that the triality automorphism of $\mathrm{Spin}(8)$ plays in this study of $R$. Section 2 is devoted to the calculation of $H^{*}(Q)$. In Section 3 we give a proof of Theorem 2. Finally, in Section 4 we turn to the calculation of the characteristic classes of vector bundles over $R$, proving Theorem 3.

One of the special features of $\mathrm{Spin}(8)$ is triality. Recall that $\mathrm{Spin}(8)$ has three $8$-dimensional real representations, the vector representation $u_{8}$ and the two spin representations $\delta^{\pm}_{8}$. No two of these representations are isomorphic but given any two there is an outer automorphism of $\mathrm{Spin}(8)$ which transforms one to the other. Indeed the group $\mathrm{Out}(\mathrm{Spin}(8))$ of outer automorphisms of $\mathrm{Spin}(8)$ can be identified with $\Sigma_{3}$, the group of permutations of the set $\{u_{8},\delta_{8}^{+},\delta_{8}^{-}\}$.

The representation $u_{8}$ gives a transitive action of $\mathrm{Spin}(8)$ on $S^{7}$ with stabiliser $\mathrm{Spin}(7)$. By triality, the same is true for $\delta_{8}^{+}$ and $\delta^{-}_{8}$. So we get three (conjugacy classes) of embeddings

and each of the homogeneous spaces

are $\mathrm{Spin}(8)$-equivariariantly diffeomorphic to $S^{7}$.

Now we have the usual embeddings

The embeddings $j_{+}$, $j_{-}$ are conjugate in $\mathrm{Spin}(9)$ and we get two conjugacy classes of embeddings

The embeddings $i,j$ are conjugate in $F_{4}$, see [Ada81].

First we identify the homogeneous spaces

Next we identify the homogeneous spaces

The first is the total space of the fibre bundle

The fibre of this bundle is the Stiefel manifold $V_{2}(\mathbb{R}^{9})$. Let $U_{9}$ be the real $9$-dimensional real vector bundle over $F_{4}/\mathrm{Spin}(9)$ associated to the $9$-dimensional vector representation of $\mathrm{Spin}(9)$. It follows from the previous lemma that $F_{4}/i(\mathrm{Spin}(7))$ is $F_{4}$-equivariantly diffeomorphic to the fibrewise Stiefel manifold $V_{2}(U_{9})$, that is the fibre bundle over $P$ with fibre over $x\in P$ equal to $V_{2}(U_{9,x})$, where $U_{9,x}$ is the fibre of $U_{9}$ over $x$.

The second is the total space of the fibre bundle

Let $\Delta_{9}$ be the $16$-dimensional real vector bundle over $P=F_{4}/\mathrm{Spin}(9)$ associated to the (real) spin representation of $\mathrm{Spin}(9)$. This time the previous lemma tells us that $F_{4}/j(\mathrm{Spin}(7))$ is $F_{4}$-equivariantly diffeomorphic to $S(\Delta_{9})$, the sphere bundle of $\Delta_{9}$.

As mentioned above, the embeddings $i,j:\mathrm{Spin}(7)\to F_{4}$ are conjugate. This proves the following theorem.

This theorem is quite surprising. We get two fibre bundles

with diffeomorphic total spaces and bases. Clearly, since the fibres of these bundles are not diffeomorphic, there is no fibre preserving diffeomorphism. The $E_{2}$ pages of the Serre spectral sequences of the two fibre bundles look very different but they converge to the same answer.

We know that the cohomology of both $R$ and $Q$ is torsion free and zero in odd degrees. From [JRT23, Section 7.4], we see that the Poincaré polynomial of $R$ is

and recall from Section 2.2 that the Poincaré polynomial of $Q$ is

The following table, which highlights the very small difference between $R$ and $Q$, lists the non-zero Betti numbers of $R$ and $Q$.

Let $i:P\to R$ be the embedding of $P$ in $R$. Then we have the usual induced homomorphisms $i_{*},i^{*}$ in homology and cohomology, respectively. However both $P$ and $R$ are orientable manifolds and $i$ is a codimension $16$ embedding, so we also have the umkehr homomorphisms

Recall that $i_{!}$ is defined by the following commutative diagram

where the vertical arrows are the Poincaré duality isomorphisms. Now let $j:Q\to R$ be the codimension $2$ embedding of $Q$. In this case we have $j_{*},j^{*}$ and

We know that the cohomology groups of $P,Q,R$ are zero in odd degrees.

First, we define $t\in H^{2}(R)$ and $w\in H^{8}(R)$ by