# The classical topological invariants of homogeneous spaces

John Jones, Dmitriy Rumynin, Adam Thomas

###### Abstract.

We study the homogeneous spaces of a simply connected, compact, simple Lie
group $G$ through the lens of $\mathrm{K}$-theory. Our methods apply equally
well to the case where $G$ is in one of the four infinite families of
classical groups, or one of the five exceptional groups. The main examples we
study in detail are the four symmetric spaces FII, EIII, EVI, EVIII in
Cartan’s list of symmetric spaces. These are, respectively, homogeneous spaces
for $F_{4}$, $E_{6}$, $E_{7}$, $E_{8}$ with dimensions $16$, $32$, $64$,
$128$. They are the four Rosenfeld projective planes.

## Introduction

Let $G$ be a compact simply connected simple Lie group. A symmetric space for
$G$ is a homogeneous space of the form $G/K$ where $K$ is the fixed point
subgroup of an involution of $G$. Cartan classified the symmetric spaces into
seven infinite families, the symmetric spaces for the classical groups,
together with twelve symmetric spaces for the exceptional groups. A more
conceptual, modern, classification is given in [HL11] in terms of Freudenthal
magic squares.

The classical symmetric spaces are very well understood. For example, the
papers of Borel and Hirzebruch [BH58, BH59, BH60], Bott and Samelson [Bot56,
BS53] give a wealth of useful information concerning the cohomology of the
classical symmetric spaces, their homotopy groups, and the characteristic
classes of the natural bundles over them. In comparison the exceptional
symmetric spaces are not at all well understood. The paper by Piccinni
[Pic17], in particular Table A, gives an account of what is known and the
relevant references.

Here is an example which shows quite how complicated this can be. The simply
connected compact Lie group $E_{8}$ has a $128$ dimensional symmetric space
but there does not seem to be an explicit presentation of its rational
cohomology ring in the literature. To adapt a slogan from [Gar16], we
understand compact simply connected simple Lie groups only as far as we
understand $E_{8}$.

Before going into more detail we fix our conventions used throughout the
paper.

  1. (i)

Representations will always be finite-dimensional complex representations,
unless explicitly stated otherwise, and $\mathrm{K}$-theory means complex
$\mathrm{K}$-theory.

  2. (ii)

A Lie group is automatically compact and connected. All subgroups of Lie
groups are assumed to be Lie groups in their own right.

  3. (iii)

We use the term exceptional Lie group for a simply connected simple Lie group
of exceptional type.

  4. (iv)

Let $G$ be a Lie group. We make a choice of a maximal torus $T$ in $G$. Then
we use the term fundamental representation of $G$ for an irreducible
representation of $G$ whose highest weight is a fundamental weight for $T$. If
$G$ is simply connected then there are $l$ fundamental representations
$\rho_{1},\dots\rho_{l}$ of $G$, where $l$ is the rank of $G$, and the
representation ring of $G$ is the polynomial ring

$R(G)=\mathbb{Z}[\rho_{1},\dots,\rho_{l}].$

  5. (v)

Let $G$ be a simply connected Lie group. Then we use the term minimal
representation for a representation of $G$ that is faithful and irreducible of
the minimal dimension.

If $K\subset G$ is a closed subgroup of $G$ then the restriction homomorphism
is a ring homomorphism $R(G)\to R(K)$ of representation rings. Our primary
strategy is to extract topological information about $G/K$ from this ring
homomorphism. We use $\mathrm{K}$-theory to study topological invariants and
to express these invariants in terms of representation theory. In doing so, we
take advantage of the usual natural operations, $\lambda$-operations
$\lambda^{k}$ and Adams operations $\psi^{k}$, acting on these rings. Then we
use the powerful methods of representation theory, both conceptual and
computational, to get our results. This leads to systematic methods that apply
equally well to any simply connected Lie group. Our main objective is to apply
these methods to understand more about the exceptional symmetric spaces.

There is significant computational aspect to this approach, with its emphasis
on the exceptional symmetric spaces, since some big numbers occur in the
representation theory of the exceptional Lie groups. For example, the
representation ring of $E_{8}$ is generated by the eight fundamental
representations, with dimension

$3875,669600,6899079264,146325270,2450240,30380,248,147250.$

The examples we study in detail in this paper are the four Rosenfeld
projective planes, FII, EIII, EVI, EVIII in Cartan’s list, with dimensions 16,
32, 64, and 128. We plan to study the other exceptional symmetric spaces, in
detail, in a follow up to this paper.

The following three results give us a good starting point for understanding
topological invariants of a Lie group $G$, its homogeneous spaces $G/K$, and
its classifying space $BG$, in terms of $\mathrm{K}$-theory and representation
theory.

  1. (i)

Let $G$ be a Lie group with $\pi_{1}(G)$ a free abelian group. Then a theorem
due to Luke Hodgkin [Hod75] gives a simple and conceptual description of the
$\mathrm{K}$-theory of $G$ as a functor of the representation ring $R(G)$:
precisely

$\mathrm{K}^{*}(G)=\mathrm{Tor}^{*}_{R(G)}(\mathbb{Z},\mathbb{Z}).$

  2. (ii)

Suppose that $K$ is a closed subgroup of $G$ with the same rank as $G$. Then
again we get a simple and conceptual description of the $K$-theory of $G/K$
using a theorem of Harsh Pittie [Pit72]: precisely

$\mathrm{K}^{*}(G/K)=R(K)\otimes_{R(G)}\mathbb{Z}.$

  3. (iii)

If $G$ is any Lie group, the well-known Atiyah–Segal completion theorem [AS69]
gives the $\mathrm{K}$-theory of $BG$ as a functor of the representation ring
of $G$:

$\mathrm{K}^{*}(BG)=\hat{R}(G)$

where $\hat{R}(G)$ is the completion of $R(G)$ at the augmentation ideal.

### Contents

We now give a summary of this paper and try to make the conceptual relations
between the various, and quite diverse, sections clear.

### Section 1

We start with precise formulations of the theorems of Hodgkin and Pittie and
outline the proofs.

### Section 2

Next, let $G$ be a Lie group with free abelian fundamental group. We turn to
the study of the Adams operations in $\mathrm{K}^{*}(G)$. Here is an example
which shows why this is important. Representations of $K$ give vector bundles
over $G/K$ and it is natural to use characteristic classes of these bundles to
study the cohomology of $G/K$. For example, given a topological space $X$, the
Chern character gives an isomorphism of $\mathbb{Z}/2$-graded rings

$\mathrm{ch}:\mathrm{K}^{*}(X)\otimes\mathbb{Q}\to H^{*}(X;\mathbb{Q}).$

Here $H^{*}$ means the direct product of the $H^{k}$. However this does not
determine the individual groups $H^{k}(X;\mathbb{Q})$. To do this we use Adams
operations.

According to Hodgkin’s theorem the $\mathbb{Z}/2$-graded ring
$\mathrm{K}^{*}(G)$ is generated by elements in $\mathrm{K}^{1}(G)$ and we
must explain what we mean by Adams operations in the $\mathbb{Z}/2$-graded
ring $\mathrm{K}^{*}(G)$. We use the (standard) terminology $\Psi$-ring for a
ring $R$ equipped with a sequence of ring endomorphisms $\psi^{k}$, $k\geq 1$
such that $\psi^{1}=1$ and $\psi^{k}\psi^{l}=\psi^{kl}$.

The representation ring of a Lie group is an example of a $\Psi$-ring.
Following Bousfield [Bousfield] we describe the notion of a
$\mathbb{Z}/2$-graded $\Psi$-ring. We then show how the $\Psi$-ring structure
of $R(G)$ determines the $\mathbb{Z}/2$-graded $\Psi$-ring structure of
$\mathrm{K}^{*}(G)$.

### Section 3

In [Ser53] Serre proves that a Lie group of rank $l$ has the rational homotopy
type of a product of $l$ odd-dimensional spheres. The type of $G$ is this list
of odd integers. The main theme in this section is to use a functorial version
of the type of $G$.

Let $V(G)$ be the indecomposable quotient of the ring $R(G)$. In other words,

$V(G)=R(G)/(\mathbb{Z}\oplus I^{2}(G))$

where $I(G)$ is the augmentation ideal of $R(G)$. It is a free abelian group
with rank equal to the rank of $G$. It also has Adams operations, More
precisely, it is a $\Psi$-module in the usual sense. We show how to calculate
the type of $G$ from the $\Psi$-module $V(G)$.

Next let $\pi(G)\otimes\mathbb{Q}$ be the direct sum of the nonzero rational
homotopy groups of $G$. We show that there is a natural isomorphism

$\pi(G)\otimes\mathbb{Q}\to V^{*}(G)\otimes\mathbb{Q}.$

where $V^{*}(G)$ is the dual of $V(G)$. Using Adams operations we can refine
this to give a natural description of the individual rational homotopy groups.

The importance of this result is that it is functorial. It gives a way of
using representation theory to compute the homomorphism of rational homotopy
groups defined by a homomorphism of Lie groups $h:H\to G$. This turns out to
be a very efficient way of doing such calculations.

### Section 4

The previous section shows that the $\Psi$-module $V(G)$ is a very useful
invariant of $G$ and in this section we study it in much more detail. Let $G$
be a simply connected simple Lie group excluding the groups
$\mathrm{Spin}(n)$. Then in Theorem 4.3.1 we show that the $\Psi$-module
$V(G)\otimes\mathbb{Q}$ is generated by the element of $V(G)\otimes\mathbb{Q}$
defined by a minimal representation of $G$. It follows from Section 3 that the
$\mathbb{Z}/2$-graded $\Psi$-ring $\mathrm{K}^{*}(G)\otimes\mathbb{Q}$ is
generated by a single element. This in turn leads to the following result. Let
$H$ be a Lie group and $f,g:H\to G$ be two homomorphisms. Let $u$ be a minimal
representation of $G$. Then the two homomorphisms

$f_{*},g_{*}:\pi_{*}(H)\otimes\mathbb{Q}\to\pi_{*}(G)\otimes\mathbb{Q}$

are equal if and only if

$f^{*}(u)=g^{*}(u)\in V(G).$

In our examples we do not need to know the $\Psi$-module structure of
$V(\mathrm{Spin}(n))$. The details are quite intricate so we avoid following
this unnecessary, but interesting, diversion.

### Section 5

Now let $G$ be a simply connected simple Lie group, again excluding the groups
$\mathrm{Spin}(n)$. Let $U$ be the vector bundle over $BG$ defined by a
minimal representation of $G$. In this section we apply the results of Section
4 to show that the rational cohomology of $BG$ is generated by the classes
$\mathrm{ch}_{q}(U)$ with $q\geq 1$. Here $\mathrm{ch}_{q}(U)\in
H^{2q}(BG;\mathbb{Q})$ is the degree $2q$ component of the Chern character of
$U$. This is obvious for the classical groups. But it tells us something very
useful in the case of the exceptional groups. More generally, let $\rho$ be a
representation of $G$ and let $E_{\rho}$ be the vector bundle over $BG$
defined by $\rho$. We give a necessary and sufficient condition on $\rho$
which ensures that the elements $\mathrm{ch}_{q}(E_{\rho})$ with $q\geq 1$
generate the rational cohomology of $BG$.

Now let $K$ be a closed connected subgroup of $G$, with the same rank as $G$.
Then we use this result to distill a classical theorem of Borel into the
following presentation of $H^{*}(G/K;\mathbb{Q})$. Let $\pi:BK\to BG$ be the
map defined by the inclusion of $K$ in $G$. Then we can choose $BK$ in such a
way that $\pi$ is a fibre bundle with fibre $G/K$. Let $j:G/K\to BK$ be the
inclusion of a fibre. The homomorphism $j^{*}:H^{*}(BK;\mathbb{Q})\to
H^{*}(G/K;\mathbb{Q})$ is surjective since $K$ and $G$ have the same rank.
Then

$j^{*}(\mathrm{ch}_{q}(\pi^{*}(U))=0\quad\text{for $q\geq 1$}$

since $j^{*}\pi^{*}(U)$ is a trivial bundle. We show that the kernel of
$j^{*}$ is the ideal $I$ generated by $\pi^{*}(\mathrm{ch}_{q}(U))$ for $q\geq
1$. Therefore $j^{*}$ defines an isomorphism

$H^{*}(BK;\mathbb{Q})/I\to H^{*}(G/K;\mathbb{Q}).$

We can, of course, replace the $\mathrm{ch}_{q}$ by the Chern classes $c_{q}$.

### Section 6

This section contains a key idea. We can make a very clean transition from
$\mathrm{K}$-theory and rational homotopy groups to rational cohomology by
using the theory of rationally elliptic spaces due to Felix, Halperin, and
Thomas [FHT01]. Any homogeneous space is a rationally elliptic space. This
section is short but important. The theory tells us that once we know the
rational homotopy groups of a rationally elliptic space we can read off a lot
of structural information about its rational cohomology. For example, in the
case where the Euler characteristic of $\pi_{*}(X)\otimes\mathbb{Q}$ is zero,
it gives a general formula for the Poincare series and Euler characteristic of
$H^{*}(X;\mathbb{Q})$.

### Section 7

We now focus on the Rosenfeld projective planes. These are the four
homogeneous spaces $G/K$ where $G$ and $K$ are given in the following table.

$G$ | $K$ | dim. | (a) | (b) | (c)
---|---|---|---|---|---
$F_{4}$ | $\mathrm{Spin}(9)$ | 16 | FII | $P^{2}(\mathbb{O})$ | $R_{4}$
$E_{6}$ | $\mathrm{Spin}(10)\times_{C_{4}}\mathrm{U}(1)$ | 32 | EIII | $P^{2}(\mathbb{O}\otimes\mathbb{C})$ | $R_{5}$
$E_{7}$ | $\mathrm{Spin}(12)\times_{C_{2}}\mathrm{Sp}(1)$ | 64 | EVI | $P^{2}(\mathbb{O}\otimes\mathbb{H})$ | $R_{6}$
$E_{8}$ | $\mathrm{Spin}(16)/C_{2}$ | 128 | EVIII | $P^{2}(\mathbb{O}\otimes\mathbb{O})$ | $R_{7}$

The last three columns are: (a) the label in Cartan’s list of symmetric
spaces, (b) the Rosenfeld notation, (c) the notation we use. In [HL11, Table
2] they appear as the exceptional symmetric spaces of Grassmannian type. It is
usual to complete this to a list of seven, by setting
$R_{1}=P^{2}(\mathbb{R})$, $R_{2}=P^{2}(\mathbb{C})$, and
$R_{3}=P^{2}(\mathbb{H})$.

In this section we explain how to calculate the rational homotopy groups of
these Rosenfeld projective planes and set out what the theory of rationally
elliptic spaces tells us about the rational cohomology these symmetric spaces.

### Section 8

Here we provide explicit presentations of the rational cohomology of
$R_{5},R_{6},R_{7}$. We prove the presentation is correct in the case of
$H^{*}(R_{7};\mathbb{Q})$, using the results in Section 5. This is the most
complicated and seems to be the first explicit presentation of
$H^{*}(R_{7};\mathbb{Q})$. We do not go into the details of the proofs for
$H^{*}(R_{5};\mathbb{Q})$ and $H^{*}(R_{6};\mathbb{Q})$ but note that they
agree with the results of [TW74] and [Nak01].

### Section 9

In this section we explain how we do the computations in representation theory
required to complete the proofs of Theorem 4.3.1 in Section 4 and Lemma 7.0.2
in Section 7.

### Acknowledgments

This research was funded in part by the EPSRC, EP/W000466/1 (Thomas). For the
purpose of open access, the author has applied a Creative Commons Attribution
(CC BY) licence to any Author Accepted Manuscript version arising from this
submission.

##  1\. The $\mathrm{K}$ theory of Lie groups and symmetric spaces

###  1.1. The $\mathrm{K}$-theory of Lie groups

Recall that for any connected space $X$

$\mathrm{K}^{1}(X)=[X,\mathrm{U}].$

Here $[,]$ means homotopy classes of maps and $\mathrm{U}=\mathrm{U}(\infty)$
is the direct limit of the unitary groups $\mathrm{U}(n)$ where
$\mathrm{U}(n)$ is embedded as $\mathrm{U}(n)\oplus 1$ in $\mathrm{U}(n+1)$.

Now suppose $G$ is a Lie group and $\rho:G\to\mathrm{U}(n)$ is a
representation of $G$. We can compose $\rho$ with the stabilisation
homomorphism $\mathrm{U}(n)\to\mathrm{U}$ to get the element

$\beta(\rho)\in\mathrm{K}^{1}(G).$

This gives a homomorphism of abelian groups

$\beta:R(G)\to\mathrm{K}^{1}(G).$

In [Hod75] Hodgkin proves that the kernel of this homomorphism is

$\mathbb{Z}\oplus I^{2}(G)\subseteq R(G).$

As in the introduction write

$V(G)=R(G)/(\mathbb{Z}\oplus I^{2}(G))$

for the indecomposable quotient of the representation ring $R(G)$. It is a
free abelian group of rank equal to the rank of $G$. We will often identify
$V(G)$ with a subgroup of $\mathrm{K}^{1}(G)$ using $\beta$. We can now state
Hodgkin’s theorem [Hod75, Theorem 1].

######  Theorem 1.1.1.

Let $G$ be a Lie group with $\pi_{1}(G)$ free abelian. Then

$\mathrm{K}^{*}(G)=\Lambda^{*}(V(G))$

is the $\mathbb{Z}/2$-graded exterior algebra over $\mathbb{Z}$ generated by
$V(G)\subseteq\mathrm{K}^{1}(G)$.

The first step in Hodgkin’s proof is to show that $\mathrm{K}^{*}(G)$ is
torsion free. It follows from this that $\mathrm{K}^{*}(G)$ is a
$\mathbb{Z}/2$-graded Hopf algebra over $\mathbb{Z}$, and this allows him to
use facts about Hopf algebras to complete the proof. In [Ati65], Atiyah gives
a different way of completing the proof, assuming that $\mathrm{K}^{*}(G)$ is
torsion free, by using the Chern character.

This result can be reformaulted to say that

$\mathrm{K}^{*}(G)=\mathrm{Tor}^{*}_{R(G)}(\mathbb{Z},\mathbb{Z}),$

as $\mathbb{Z}/2$-graded rings.

###  1.2. The $\mathrm{K}$ theory of symmetric spaces.

Let $G$ be a Lie group and let $K$ be closed subgroup of $G$. In [AH61] Atiyah
and Hirzebruch describe a natural homomorphism

$\alpha:R(K)\to\mathrm{K}^{0}(G/K)$

This homomorphism associates to a representation $\rho:K\to\mathrm{U}(n)$ the
homogeneous vector bundle $V_{\rho}=G\times_{\rho}\mathbb{C}^{n}$. It is clear
that if the representation $\rho$ extends to $G$ then $V_{\rho}$ is isomorphic
to the product bundle $G/K\times\mathbb{C}^{n}$. In this way we get a
homomorphism

$\alpha:R(K)\otimes_{R(G)}\mathbb{Z}\to\mathrm{K}^{0}(G/K).$

Atiyah and Hirzebruch conjectured that if $G$ and $K$ have the same rank then
this ring homomorphism is surjective. In [Pit72], (see also [Ste75]), Harsh
Pittie, proves the following result.

######  Theorem 1.2.1.

Let $G$ be a Lie group such that $\pi_{1}(G)$ is a free abelian group. Let
$K\subseteq G$ be a closed connected subgroup of $G$ with the same rank as
$G$. Then

$\alpha:R(K)\otimes_{R(G)}\mathbb{Z}\to\mathrm{K}^{0}(G/K)$

is a ring isomorphism and

$\mathrm{K}^{1}(G/K)=0.$

Pittie does not explicitly state that $\mathrm{K}^{1}(G/K)=0$ but his method
obviously proves it, as we explain in the next section.

###  1.3. On the proofs.

The above results are both consequences of the Künneth theorem, due to Luke
Hodgkin [Hod75], in Atiyah–Segal equivariant $\mathrm{K}$-theory. Let $G$ be a
Lie group, such that $\pi_{1}(G)$ is free abelian. Let $X$ and $Y$ be compact
$G$-spaces. There is a spectral sequence with $E_{2}$ page

$E_{2}^{p,q}=\mathrm{Tor}^{p,q}_{R(G)}(\mathrm{K}_{G}^{*}(X),\mathrm{K}_{G}^{*}(Y))\Longrightarrow\mathrm{K}_{G}^{*}(X\times
Y).$ $d_{r}:E_{r}^{p,q}\to E_{r}^{p-r,q+r-1}$

Here $p\in\mathbb{Z}$ and $p\geq 0$ and $q\in\mathbb{Z}/2$ since
$\mathrm{K}_{G}^{*}$ is $\mathbb{Z}/2$-graded.

For example the $E_{2}$ page has two rows

$E_{2}^{p,0}=\mathrm{Tor}^{p}_{R(G)}(\mathrm{K}_{G}^{0}(X),\mathrm{K}_{G}^{0}(Y))\oplus\mathrm{Tor}^{p}_{R(G)}(\mathrm{K}_{G}^{1}(X),\mathrm{K}_{G}^{1}(Y))$
$E_{2}^{p,1}=\mathrm{Tor}^{p}_{R(G)}(\mathrm{K}_{G}^{0}(X),\mathrm{K}_{G}^{1}(Y)\oplus\mathrm{Tor}^{p}_{R(G)}(\mathrm{K}_{G}^{1}(X),\mathrm{K}_{G}^{0}(Y)).$

Hodgkin was only able to prove the convergence of the spectral sequence in
special cases. This was also studied by Snaith [Sna72]. The proof that it
converges when $\pi_{1}(G)$ is a free abelian group was completed by McLeod in
[McL79].

For example, when $X=G/K$ and $Y=G$ we can use the isomorphisms

$\mathrm{K}_{G}^{*}(G/K)=R(K),\quad\mathrm{K}_{G}^{*}(G)=\mathbb{Z},\quad\mathrm{K}_{G}^{*}(G/K\times
G)=\mathrm{K}^{*}(G/K)$

to get a spectral sequence

$E_{2}^{p,q}=\mathrm{Tor}^{p,q}_{R(G)}(R(H),\mathbb{Z})\Longrightarrow\mathrm{K}^{*}(G/H).$

Pittie proves Theorem 1.2.1 by showing that, with the given hypotheses, this
spectral sequence collapses at the $E_{2}$ page. In [Ste75] Steinberg proves,
with the given hypotheses, that $R(H)$ is a free module over $R(G)$. This
clearly proves both statements in Theorem 1.2.1.

A proof of Theorem 1.1.1 is given by taking $X=Y=G$ to get a spectral sequence

$E_{2}^{p,q}=\mathrm{Tor}^{p,q}_{R(G)}(\mathbb{Z},\mathbb{Z})\Longrightarrow\mathrm{K}^{*}(G).$

The homomorphism $\rho:R(G)\to\mathrm{K}^{1}(G)$ shows that every element in
$E_{2}^{1,*}$ is an infinite cycle. Since $E_{2}^{*,*}$ is multiplicatively
generated $E_{2}^{1,*}$ this proves the spectral sequence collapses at the
$E_{2}$ page.

##  2\. Adams operations in $\mathrm{K}^{*}(G)$

Hodgkin’s theorem tells us that if $G$ is a Lie group with $\pi_{1}(G)$ free
abelian, the $\mathbb{Z}/2$-graded ring $\mathrm{K}^{*}(G)$ is generated by
elements in $\mathrm{K}^{1}(G)$. In this section we explain what we mean by
Adams operations in a $\mathbb{Z}/2$-graded ring, and extend Hodgkin’s theorem
by working out the Adams operations in $\mathrm{K}^{*}(G)$.

###  2.1. $\Psi$-modules

By a $\Psi$-module, we mean an abelian group $A$ equipped with a sequence of
endomorphisms $\psi^{k}$, $k\geq 1$ such that

$\psi^{1}=1,\quad\psi^{k}\psi^{l}=\psi^{kl}.$

Such operations are usually referred to as Adams operations.

We need some notation. Let $A$ be a $\Psi$-module. An element $a\in A$ has
weight $r$ if $\psi^{k}a=k^{r}a$ for all $k$. Let $\mathbb{Z}(r)$, be the
$\Psi$-module freely generated (as an abelian group) by a single element of
weight $r$. In this notation

$\tilde{\mathrm{K}}^{0}(S^{2r})=\mathbb{Z}(r)$

as $\Psi$-modules (where $\tilde{\mathrm{K}}^{0}(X)$ denotes the usual reduced
K-theory of $X$). We also use the notation

$\mathbb{Z}(r_{1},\dots,r_{l})=\mathbb{Z}(r_{1})\oplus\dots\oplus\mathbb{Z}(r_{l})$
$\mathbb{Q}(r_{1},\dots,r_{l})=\mathbb{Z}(r_{1},\dots,r_{l})\otimes\mathbb{Q}.$

Here is an example which is important for us. We take for granted the usual
theory of $\Lambda$-rings (see [AT69]). Let $R$ be a $\Lambda$-ring with
augmentation ideal $I$ and suppose $I^{2}=0$. Then as a ring

$R=\mathbb{Z}\oplus I$

with product

$(n+x)(m+y)=nm+ny+mx$

where $n,m\in\mathbb{Z}$, and $x,y\in I$. It follows that if $x,y\in I$

$\lambda^{k}(x+y)=\lambda^{k}(x)+\lambda^{k}(y),\quad\lambda^{k}(xy)=\lambda^{k}(x)\lambda^{k}(y)=0.$

Furthermore if $x\in I$

$\lambda^{k}(\lambda^{l}(x))=\lambda^{kl}(x).$

Finally for $n\in\mathbb{Z}$ and $x\in I$

$\lambda^{k}(n+x)=\sum_{i=0}^{k}\binom{n}{k-i}\lambda^{i}(x).$

So the whole theory of $\Lambda$-operations in $R$ boils down to the fact that
the $\lambda^{k}$-operations define a $\Psi$-module structure on $I$. The
inductive formula for the Adams operations shows that if $x\in I$ then

$\psi^{k}(x)=(-1)^{k+1}k\lambda^{k}(x).$

If $n\in\mathbb{Z}$ and $x\in I$ then $\psi^{k}(n+x)=n+\psi^{k}(x)$, and so,
in general, it is not equal to $(-1)^{k+1}k\lambda^{k}(n+x)$.

If we start with the $\Lambda$-ring $R(G)$ where $G$ is a Lie group we can
form the $\Lambda$-ring

$S(G)=R(G)/I^{2}(G).$

Then the $\Lambda$-ring structure is determined by the $\psi$-module structure
of the augmentation ideal $I(G)/I^{2}(G)$ of $S(G)$. We now have two quotient
homomorphisms

$R(G)\to S(G)\to V(G)$

where, as in the introduction, $V(G)=R(G)/(\mathbb{Z}\oplus I^{2}(G))$. Now
$\mathbb{Z}\oplus I^{2}(G)$ is sub-$\psi$-module of $R(G)$ so $V(G)$ inherits
a $\Psi$-module structure. The first quotient is a map of $\Lambda$-rings and
the second is a map of $\Psi$-modules. It is straightforward to check that the
composition

$I(G)/I^{2}(G)\to S(G)\to V(G)$

is an isomorphism of $\Psi$-modules.

###  2.2. $\mathbb{Z}/2$-graded $\Psi$-rings

By definition, a $\Psi$-ring is a ring equipped with ring endomorphisms
$\psi^{k}$ such that $\psi^{1}=1$ and $\psi^{k}\psi^{l}=\psi^{kl}$. Any
$\Lambda$-ring becomes a $\Psi$-ring where the $\psi^{k}$ are the Adams
operations associated to the $\lambda$-operations.

In [Bou96] Bousfield gives a thorough account of the general theory of
$\mathbb{Z}/2$-graded $\Lambda$-rings. Here we give a self-contained account
of the basic theory of $\mathbb{Z}/2$-graded $\Psi$-rings, which is adequate
for our applications. This is a pared down version of the theory, but as
Bousfield points out, if the ring is torsion free the Adams operations
determine the $\lambda$-operations.

A $\mathbb{Z}/2$-graded $\Psi$-ring is a $\mathbb{Z}/2$-graded ring $A^{*}$,
which is strictly graded commutative, and is equipped with $\Psi$-module
structures $\psi^{k}$ on $A^{0}$, and $\phi^{k}$ on $A^{1}$ with the following
properties.

  1. (i)

The operations $\psi^{k}$ make $A^{0}$ into a $\Psi$-ring.

  2. (ii)

For $x\in A^{0}$ and $y\in A^{1}$, $\phi^{k}(xy)=\psi^{k}(x)\phi^{k}(y)$.

  3. (iii)

For $x,y\in A^{1},\psi^{k}(xy)=k\phi^{k}(x)\phi^{k}(y)$.

The ring $\mathrm{K}^{0}(X)$ is a $\Lambda$-ring and the corresponding Adams
operations make $\mathrm{K}^{0}(X)$ into a $\Psi$-ring. We define operations
$\phi^{k}$ on $\mathrm{K}^{1}(X)$ which give a $\Psi$-module structure on
$\mathrm{K}^{1}(X)$.

The ring $\mathrm{K}^{0}(\Sigma X)$ is a $\Lambda$-ring with augmentation
ideal $\tilde{\mathrm{K}}^{0}(\Sigma X)$. If
$x,y\in\tilde{\mathrm{K}}^{0}(\Sigma X)$ then $xy=0$. So it follows that the
operations $\lambda^{k}$ define a $\Psi$-module structure on
$\tilde{\mathrm{K}}^{0}(\Sigma X)$. Now we use the standard natural
isomorphism

$\mathrm{K}^{1}(X)\to\tilde{\mathrm{K}}^{0}(\Sigma X)$

to transfer these operations on $\tilde{\mathrm{K}}^{0}(\Sigma X)$ to
$\mathrm{K}^{1}(X)$. Define

$\phi^{k}:\mathrm{K}^{1}(X)\to\mathrm{K}^{1}(X)$

to be the endomorphism, of abelian groups, defined by

$(-1)^{k+1}\lambda^{k}:\tilde{\mathrm{K}}^{0}(\Sigma
X)\to\tilde{\mathrm{K}}^{0}(\Sigma X).$

This defines a $\Psi$-module structure on $\mathrm{K}^{1}(X)$. The sign
convention is the one needed to get the correct signs in the following lemma.

######  Lemma 2.2.1.

The operations $\psi^{k}$ on $\mathrm{K}^{0}(X)$ and $\phi^{k}$ on
$\mathrm{K}^{1}(X)$ make $\mathrm{K}^{*}(X)$ into a $\mathbb{Z}/2$-graded
$\Psi$-ring.

###### Proof.

The proof is an exercise in the theory of products in $\mathrm{K}$-theory, see
[Ati18, Chapter 2, Section 2.6]. The basic product is the external tensor
product

$\mathrm{K}^{0}(X)\otimes\mathrm{K}^{0}(Y)\to\mathrm{K}^{0}(X\times Y).$

For this particular exercise it is best to use the reduced version

$\tilde{\mathrm{K}}^{0}(X)\otimes\tilde{\mathrm{K}}^{0}(Y)\to\tilde{\mathrm{K}}^{0}(X\wedge
Y).$

We assume that $X,Y$ are connected, and we have chosen base points $x_{0}\in
X$, $y_{0}\in Y$. By definition

$X\wedge Y=X\times Y/(X\times y_{0}\cup x_{0}\times Y).$

Next take $X=Y$ and use the reduced diagonal $\Delta:X\to X\wedge X$ to get
the internal product

$\tilde{\mathrm{K}}^{0}(X)\otimes\tilde{\mathrm{K}}^{0}(X)\to\tilde{\mathrm{K}}^{0}(X)$

Finally replace $X$ by $S^{p}\wedge X$ and $Y$ by $S^{q}\wedge X$ and use the
appropriate suspension of $\Delta$ to get the product

$\tilde{\mathrm{K}^{0}}(S^{p}\wedge
X)\otimes\tilde{\mathrm{K}}^{0}(S^{q}\wedge
X)\to\tilde{\mathrm{K}}^{0}(S^{p}\wedge X\wedge S^{q}\wedge
X)\to\tilde{\mathrm{K}}^{0}(S^{p+q}\wedge X).$

We have to interchange $X$ and $S^{q}$, this introduces the sign $(-1)^{q}$,
and use the standard identification of $S^{p}\wedge S^{q}$ with $S^{p+q}$.

Now use Bott periodicity to identify $\tilde{\mathrm{K}}^{0}(S^{p}\wedge X)$
with $\tilde{\mathrm{K}}^{0}(X)$ if $p$ is even, or with $\mathrm{K}^{1}(X)$
if $p$ is odd. We end up with products

$\mathrm{K}^{1}(X)\otimes\tilde{\mathrm{K}}^{0}(X)\to\tilde{\mathrm{K}}^{0}(X),\quad\mathrm{K}^{1}(X)\otimes\mathrm{K}^{1}(X)\to\tilde{\mathrm{K}}^{0}(X).$

The verification of the formulas in the above definition follow from the fact
that the Adams operations are ring homomorphisms of the product in
$\mathrm{K}^{0}(X)$, the definition of the operations $\phi^{k}$ in
$\mathrm{K}^{1}(X)$ and the relation between the Adams operations and Bott
periodicity. ∎

###  2.3. The $\mathbb{Z}/2$-graded $\Psi$-ring $\mathrm{K}^{*}(G)$

As usual, we assume that $G$ is a Lie group with $\pi_{1}(G)$ free abelian.

######  Theorem 2.3.1.

The $\mathbb{Z}/2$-graded $\Psi$-ring $\mathrm{K}^{*}(G)$ is isomorphic to
$\Lambda(V(G))$, the $\mathbb{Z}/2$-graded exterior algebra over $\mathbb{Z}$,
generated by $V(G)\subseteq\mathrm{K}^{1}(G)$ equipped with the unique
$\mathbb{Z}/2$-graded $\Psi$-ring structure determined by the $\Psi$-module
structure on $V(G)$.

The proof of Theorem 2.3.1 follows directly from the following lemma. Recall
that both $R(G)$ and $\mathrm{K}^{1}(G)$ are $\Psi$-modules.

######  Lemma 2.3.2.

The homomorphism $\beta:R(G)\to\mathrm{K}^{1}(G)=\tilde{\mathrm{K}}^{0}(\Sigma
G)$ is a homomorphism of $\Psi$-modules.

###### Proof.

The isomorphism $\mathrm{K}^{1}(X)\to\tilde{\mathrm{K}}^{0}(\Sigma X)$ is
defined by the the so-called clutching construction in $\mathrm{K}$-theory.
Given a space $X$ we take two copies $C_{\pm}(X)$ of the cone on $X$ and glue
them together along $X$ to get

$\Sigma X=C_{+}(X)\cup_{X}C_{-}(X).$

Suppose we have a map $f:X\to U(n)$. This give us an isomorphism
$\hat{f}:X\times\mathbb{C}^{n}\to X\times\mathbb{C}^{n}$ and we can form the
vector bundle

$V(f)=C_{+}(X)\times\mathbb{C}^{n}\cup_{\hat{f}}C_{-}(X)\times\mathbb{C}^{n}$

over $\Sigma X$. If we replace $f$ by
$\lambda^{k}(f):X\to\mathrm{U}(\lambda^{k}(\mathbb{C}^{n}))$ it is clear that

$\lambda^{k}(V(f))=V(\lambda^{k}(f)).$

It is straightforward to follow through the arguments of [Ati18, Chapter 3,
Section 3.1] in this special case to convert this observation into a proof of
the lemma. ∎

In the case of a Lie group $G$ the $\alpha,\beta$ constructions of Section 1
are related in the following way. The join $G*G$ has a standard free action of
$G$ and $(G*G)/G$ is homotopy equivalent to $\Sigma G$, the suspension of $G$.
This gives us a map
$\alpha:R(G)\to\mathrm{K}^{0}((G*G)/G)=\mathrm{K}^{0}(\Sigma G)$. This map is
clearly a map of $\Lambda$-rings. On the other hand we have the map
$\beta:R(G)\to\mathrm{K}^{1}(G)$. Now we can compare these two maps by using
the isomorphism $\sigma:\mathrm{K}^{1}(G)\to K^{0}(\Sigma G)$ defined by the
clutching construction. It is not difficult to show that the following diagram
commutes

$\begin{CD}R(G)@>{\alpha}>{}>\mathrm{K}^{0}(\Sigma G)\\\
@V{\beta}V{}V@V{}V{p}V\\\
\mathrm{K}^{1}(G)@>{}>{\sigma}>\tilde{\mathrm{K}}^{0}(\Sigma G)\end{CD}$

where $p:\mathrm{K}^{0}(\Sigma G)\to\tilde{\mathrm{K}}^{0}(\Sigma G)$ is the
projection. This gives a different proof of Lemma 2.3.2 in this special case.

##  3\. The type of a Lie group

Let $G$ be a Lie group of rank $l$. Serre proves in [Ser53] that $G$ is
rationally homotopy equivalent to a product of odd dimensional spheres,
$S^{2r_{1}-1}\times\dots\times S^{2r_{l}-1}$, where

  1. (i)

$1\leq r_{1}\leq r_{2}\leq\dots\leq r_{l}$,

  2. (ii)

$\dim G=(2r_{1}-1)+\dots+(2r_{l}-1)$.

The type of $G$ is the sequence

$(2r_{1}-1,\dots,2r_{l}-1).$

Using this result, it is straightforward to calculate the
$\mathbb{Z}/2$-graded $\Psi$-ring $\mathrm{K}^{*}(G)\otimes\mathbb{Q}$ and the
$\Psi$-module $V(G)\otimes\mathbb{Q}$.

######  Theorem 3.0.1.

Let $G$ be a Lie group of rank $l$ with $\pi_{1}(G)$ free abelian. Let
$(2r_{1}-1,\dots,2r_{l}-1)$ be the type of $G$. There are isomorphisms

  1. (i)

$\mathrm{K}^{*}(G)\otimes\mathbb{Q}\to\Lambda_{\mathbb{Q}}(a_{1},\dots,a_{l}),$

where the weight of $a_{i}$ is $r_{i}$, and

  2. (ii)

$V(G)\otimes\mathbb{Q}\to\mathbb{Q}(r_{1},\dots,r_{l}).$

###### Proof.

We know that $\mathrm{K}^{1}(S^{2r-1})=\mathbb{Z}(a)$ where $a$ has weight
$r$. The result now follows directly from the Künneth formula for
$\mathrm{K}$-theory. ∎

This tells us how to compute the type of $G$ from the representation ring of
$G$: we must compute the eigenvalues of the Adams operations on $V(G)$.

###  3.1. $V(G)$ and the rational homotopy groups of $G$

If $X$ is a space and $f:S^{n}\to X$ is a map then we have the induced
homomorphism

$f^{*}:\mathrm{K}^{*}(X)\to\mathrm{K}^{*}(S^{n}).$

This clearly defines a homomorphism

$d:\pi_{n}(X)\to\mathrm{Hom}_{\Psi}(\mathrm{K}^{*}(X),\mathrm{K}^{*}(S^{n})),$

which is the Adams $d$-invariant.

We now examine the $d$-invariant

$d:\pi_{2r-1}(G)\otimes\mathbb{Q}\to\mathrm{Hom}_{\Psi}(\mathrm{K}^{1}(G),\mathbb{Q}(r)).$

If $f:S^{2r-1}\to G$ then $f^{*}:\mathrm{K}^{1}(G)\to\mathrm{K}^{1}(S^{2r-1})$
vanishes on non-trivial products so it is uniquely determined by its
restriction to the generators $V(G)\subseteq\mathrm{K}^{1}(G)$. So we abuse
notation and regard $d(f)$ as an element

$d(f)\in\mathrm{Hom}_{\Psi}(V(G),\mathbb{Q}(r)).$

Finally, we rename $\mathrm{Hom}_{\Psi}(V(G),\mathbb{Q}(r))$ as
$V^{*}_{r}(G)\otimes\mathbb{Q}$ since it is the weight $r$ subspace of the
$\Psi$-module $V^{*}(G)\otimes\mathbb{Q}$ dual to the $\Psi$-module $V(G)$.

######  Theorem 3.1.1.

Let $G$ be a Lie group such that $\pi_{1}(G)$ is torsion free. The
homomorphism

$d:\pi_{2r-1}(G)\otimes\mathbb{Q}\to V^{*}_{r}(G)\otimes\mathbb{Q}.$

is an isomorphism.

###### Proof.

If we replace $G$ by an odd sphere $S^{2r-1}$ this result is straightforward.
The full result follows because $G$ is rationally homotopy equivalent to a
product of odd dimensional spheres ∎

Suppose $h:H\to G$ is a homomorphism of Lie groups. Then Theorem 3.1.1 gives a
general method of computing the homomorphism
$h_{*}:\pi_{2r-1}(H)\otimes\mathbb{Q}\to\pi_{2r-1}(G)\otimes\mathbb{Q}$ in
terms of representation theory. Explicitly, the following diagram commutes

$\begin{CD}\pi_{2r-1}(H)\otimes\mathbb{Q}@>{h_{*}}>{}>\pi_{2r-1}(G)\otimes\mathbb{Q}\\\
@V{d}V{}V@V{}V{d}V\\\
V^{*}_{r}(H)\otimes\mathbb{Q}@>{}>{h_{*}}>V^{*}_{r}(G)\otimes\mathbb{Q}\end{CD}$

where the horizontal arrows are the linear maps induced by $h$ and the
vertical ones are the isomorphisms in Theorem 3.1.1.

###  3.2. The rational homotopy groups of a homogeneous space

Let $G$ be a Lie group. It follows from Serre’s theorem that
$\pi_{p}(G)\otimes\mathbb{Q}=0$ if $p$ is even. Now suppose $K$ is a compact
connected subgroup of $G$. It follows that the exact sequence in rational
homotopy groups of the principal $K$-bundle $G\to G/K$ splits up into exact
sequences

$0\to\pi_{2r}(G/K)\otimes\mathbb{Q}\to\pi_{2r-1}(K)\otimes\mathbb{Q}\to\pi_{2r-1}(G)\otimes\mathbb{Q}\to\pi_{2r-1}(G/K)\otimes\mathbb{Q}\to
0$

for $r\geq 1$. We refer to these exact sequences as the four-term exact
sequences. Using Theorem 3.1.1 we can rewrite these exact sequences as

$0\to\pi_{2r}(G/K)\otimes\mathbb{Q}\to V(K)^{*}_{r}\otimes\mathbb{Q}\to
V(G)^{*}_{r}\otimes\mathbb{Q}\to\pi_{2r-1}(G/K)\otimes\mathbb{Q}\to 0.$

This proves the following result.

######  Theorem 3.2.1.

Let $G$ be a Lie group with a subgroup $K$. Let $i:K\to G$ be the inclusion of
$K$ and let

$i_{*}:V^{*}(K)\otimes\mathbb{Q}\to V^{*}(G)\otimes\mathbb{Q}$

be the homomorphism of $\Psi$-modules induced by $i$. Then there are natural
isomorphisms

$\pi_{2r}(G/K)\otimes\mathbb{Q}\to\ker(i^{*}:V^{*}(K)_{2r-1}\otimes\mathbb{Q}\to
V^{*}(G)_{2r-1}\otimes\mathbb{Q}),$
$\pi_{2r-1}(G/K)\otimes\mathbb{Q}\to\mathrm{coker}(i^{*}:V^{*}(K)_{2r-1}\otimes\mathbb{Q}\to
V^{*}(G)_{2r-1}\otimes\mathbb{Q}).$

##  4\. The $\Psi$-module structure of $V(G)$

In this section we study the $\Psi$-module $V(G)$ in detail and give two
applications to the $K$-theory and rational homotopy groups of $G$. We use the
notation and terminology for $\Psi$-modules described in Section 2. We say
that a $\Psi$-module is cyclic if it is generated, as a $\Psi$-module, by one
element.

###  4.1. A basic lemma

######  Lemma 4.1.1.

  1. (i)

The $\Psi$-module $\mathbb{Q}(r_{1},\dots,r_{l})$ is a cyclic $\Psi$-module if
and only if the $r_{i}$ are distinct.

  2. (ii)

Suppose $r_{1},\dots,r_{l}$ are distinct. Then $u$ is a generator of the
$\Psi$-module $\mathbb{Q}(r_{1},\dots,r_{l})$ if and only if
$u=u_{1}+\dots+u_{l}$ where $u_{i}$ is a non-zero element of
$\mathbb{Q}(r_{1},\dots,r_{l})$ with weight $r_{i}$.

###### Proof.

Let $r$ be one of the $r_{i}$ and let $W_{r}$ be the weight $r$ subspace of
$\mathbb{Q}(r_{1},\dots,r_{l})$. So the dimension of $W_{r}$ is the number of
$r_{i}$ equal to $r$. Let $p_{r}:\mathbb{Q}(r_{1},\dots,r_{l})\to W_{r}$ be
the standard projection onto $W_{r}$. Suppose $\mathbb{Q}(r_{1},\dots,r_{l})$
is a cyclic $\Psi$-module. Then, since $p_{r}$ is a map of $\Psi$-modules
$W_{r}$ is also a cyclic $\Psi$-module. Let $w$ be a non-zero element of
$W_{r}$. Then $\psi^{k}w=k^{r}w$ and so the $\Psi$-module generated by $w$
must be $1$-dimensional and the dimension of $W_{r}$ must be one. It follows
that the $r_{1},\dots,r_{l}$ must be distinct.

Recall that for a linear map $A:V\to V$, a vector $v\in V$ is called a cyclic
vector for $A$ if

$v,A(v),\dots,A^{l-1}(v)$

is a basis for $V$.

The eigenvalues of $\psi^{k}$ are $k^{r_{1}},\dots,k^{r_{l}}$. If
$r_{1},\dots,r_{l}$ are distinct then it is straightforward to check that
there exists a cyclic vector for $\psi^{k}$. If $u$ is such a cyclic vector,
it follows that

$u,\psi^{k}u,\psi^{k^{2}}u,\dots\psi^{k^{l}-1}(u)$

are a basis for $\mathbb{Q}(r_{1},\dots,r_{l})$ and so
$\mathbb{Q}(r_{1},\dots,r_{l})$ is generated as a $\Psi$-module by $u$.

This completes the proof of the first part of the theorem. The second part
follows directly from the above characterisation of cyclic vectors. ∎

###  4.2. Applications

For this section we assume that $G$ is a simply connected simple Lie group
that is not isomorphic to $\mathrm{Spin}(4n)$.

######  Theorem 4.2.1.

The $\Psi$-module $V(G)\otimes\mathbb{Q}$ is cyclic.

###### Proof.

Theorem 3.0.1 shows that the $\Psi$-module $V(G)\otimes\mathbb{Q}$ is
$\mathbb{Q}(r_{1},\dots,r_{l})$ where the type of $G$ is
$(2r_{1}-1,\dots,2r_{l}-1)$. The only simply connected simple Lie groups with
a repeated integer in their type are the groups $\mathrm{Spin}(4n)$. ∎

######  Corollary 4.2.2.

The $\mathbb{Z}/2$-graded $\Psi$-ring $\mathrm{K}^{*}(G)\otimes\mathbb{Q}$ is
generated by one element.

###### Proof.

This follows immediately from Theorem 4.2.1. ∎

######  Corollary 4.2.3.

Let $H$ be a Lie group and suppose $f,g:H\to G$ are homomorphisms of Lie
groups. Let $u\in R(G)$ be such that the element of $V(G)\otimes\mathbb{Q}$
defined by $u$ generates the $\Psi$-module $V(G)\otimes\mathbb{Q}$. Then the
two homomorphisms

$f_{*},g_{*}:\pi_{*}(H)\otimes\mathbb{Q}\to\pi_{*}(G)\otimes\mathbb{Q}$

are equal if and only if

$f^{*}(u)=g^{*}(u)\in V(G)\otimes\mathbb{Q}$

###### Proof.

Since $V(G)\otimes\mathbb{Q}$ is generated as a $\Psi$ module by the element
defined by $u$ it follows that if $f^{*}(u)=g^{*}(u)\in V(G)\otimes\mathbb{Q}$
if and only if the homomorphisms $f^{*},g^{*}:V(G)\otimes\mathbb{Q}\to
V(H)\otimes\mathbb{Q}$ are equal. The result now follows directly from Theorem
3.1.1. ∎

###  4.3. A ‘most convenient’ generator of the cyclic $\Psi$-module $V(G)$

We now assume that $G$ is a simply connected simple Lie group with the
exception of the groups $\mathrm{Spin}(n)$.

######  Theorem 4.3.1.

The element of $V(G)\otimes\mathbb{Q}$ defined by a minimal representation of
$G$ is a generator of the $\Psi$-module $V(G)\otimes\mathbb{Q}$.

This result for the classical groups $\mathrm{SU}(n)$ and $\mathrm{Sp}(n)$ is
straightforward. Indeed the $\Lambda$-ring $R(\mathrm{SU}(n))$ is generated by
its usual representation on $\mathbb{C}^{n}$. This is obviously a faithful
representation. It follows that $V(\mathrm{SU}(n))\otimes\mathbb{Q}$ is
generated as a $\Psi$-module by the element defined by this representation.
The same argument applies to $R(\mathrm{Sp}(n))$. It is generated as a
$\Lambda$-ring by its usual representation on $\mathbb{H}^{n}=\mathbb{C}^{2n}$
and this is obviously faithful.

The case of $G_{2}$ is also straightforward. The fundamental representations
are $v$, the $7$-dimensional representation on the imaginary quaternions, and
$a$ the $14$-dimensional adjoint representation. So $R(G_{2})$ is the
polynomial ring $\mathbb{Z}[v,a]$. The representation $v$ is obviously
faithful and it is easy to check that

$\lambda^{2}v=v+a.$

So the $\Lambda$-ring $R(G_{2})$ is generated by $v$.

In Section 9, we prove this for $F_{4},E_{6},E_{7},E_{8}$ by computation using
Magma. It is inevitable that this will need some computational input simply
because of the $8$ large numbers given by the dimensions of the fundamental
representations of $E_{8}$ but there is more to it than that and this is
explained in Section 9.

##  5\. On the rational cohomology of $BG$ and $G/K$

We now use the results of the previous section to throw some light on the
rational cohomology of $BG$ and the homogeneous spaces $G/K$ where $G$ is an
exceptional Lie group.

###  5.1. On the rational cohomology of $BG$.

Let $G$ be an exceptional Lie group of type

$(2r_{1}-1,\dots,2r_{l}-1)$

where $l$ is the rank of $G$. Then the integers $r_{1},\dots,r_{l}$ are
distinct. It is well known, see [Bor53], that

$H^{*}(BG;\mathbb{Q})=\mathbb{Q}[a_{2j}:j=r_{1},\dots,r_{l}],$

where $a_{2j}\in H^{2j}(BG;\mathbb{Q})$. Of course these generators are not
unique but we only need one generator in each of the dimensions $2j$ for
$j=r_{1},\dots,r_{l}$.

From now on, if $\rho$ is a representation of $G$ we will write
$\mathrm{ch}_{k}(\rho)\in H^{2k}(BG;\mathbb{Q})$ for the $k$-th component of
the Chern character of the vector bundle over $BG$ associated to $\rho$. We
extend this convention to any characteristic class and indeed to the
associated bundle over $X/G$, where $G$ acts freely on $X$.

We now ask what might constitute a good choice of generators? Our basic
strategy is to use representation theory to do computations. So we are led to
the following question. Is there a representation $u$ of $G$ such that the
elements $\mathrm{ch}_{k}(u)$ where $k\geq 1$ generate the ring
$H^{*}(BG;\mathbb{Q})$?

######  Theorem 5.1.1.

If $u\in R(G)$, then $H^{*}(BG;\mathbb{Q})$ is generated as a
$\mathbb{Q}$-algebra by

$\mathrm{ch}_{r_{1}}(u),\dots,\mathrm{ch}_{r_{l}}(u)$

if and only if the element of $V(G)\otimes\mathbb{Q}$ defined by $u$ generates
$V(G)\otimes\mathbb{Q}$ as a $\Psi$-module.

###### Proof.

If $X$ is a topological space and $x\in\mathrm{K}^{0}(X)$ then

$\mathrm{ch}_{q}(\psi^{k}x)=k^{q}\mathrm{ch}_{q}(x).$

So we make $H^{\mathrm{ev}}(X;\mathbb{Q})$ into a $\Psi$-ring by defining
Adams operations as follows. For $y\in H^{2q}(X;\mathbb{Q})$ define
$\psi^{k}(y)$ by

$\psi^{k}(y)=k^{q}y.$

It follows that if $X$ is homotopy equivalent to a CW complex with a finite
number of cells in each dimension then

$\mathrm{ch}:\mathrm{K}^{0}(X)\otimes\mathbb{Q}\to
H^{\mathrm{ev}}(X;\mathbb{Q})$

is an isomorphism of $\Psi$-rings.

We use the ad-hoc notation $W(G)$ for the indecomposable quotient of the
$\Psi$-ring $H^{\mathrm{ev}}(BG;\mathbb{Q})$. It is isomorphic to the
$\Psi$-module $\mathbb{Q}(r_{1},\dots,r_{l})$. We get a basis for this
$\Psi$-module by choosing homogeneous generators $a_{r_{1}},\dots,a_{r_{l}}$
for the graded ring $H^{*}(BG;\mathbb{Q})$. We retain the notation

$\mathrm{ch}:V(G)\otimes\mathbb{Q}\to W(G)$

for the homomorphism of $\Psi$-modules defined by the Chern character. This
homomorphism is an isomorphism of $\Psi$-modules, since
$\mathrm{ch}:\mathrm{K}^{0}(BG)\otimes\mathbb{Q}\to
H^{\mathrm{ev}}(BG;\mathbb{Q})$ is an isomorphism of $\Psi$-rings. Lemma 4.1.1
completes the proof. ∎

Combining this theorem with Theorem 4.3.1 leads to the following result.

######  Theorem 5.1.2.

Let $G$ be an exceptional Lie group with type $(2r_{1}-1,\dots,2r_{l}-1)$. Let
$u$ be a minimal representation of $G$. Then $H^{*}(BG;\mathbb{Q})$ is
generated as a $\mathbb{Q}$-algebra by

$\mathrm{ch}_{r_{1}}(u),\dots,\mathrm{ch}_{r_{l}}(u).$

###  5.2. On the rational cohomology of homogeneous spaces

Let $G$ be a Lie group with free abelian fundamental group and let $K\subseteq
G$ be a closed subgroup. Let $i:K\to G$ be the inclusion and $\pi=Bi:BK\to BG$
the map of classifying spaces defined by $i$. Then there is the usual
fibration

$\begin{CD}G/K@>{j}>{}>BK@>{\pi}>{}>BG\end{CD}$

The induced map of cohomology $\pi^{*}:H^{*}(BG;\mathbb{Q})\to
H^{*}(BK;\mathbb{Q})$ makes $H^{*}(BK;\mathbb{Q})$ into a module over
$H^{*}(BG;\mathbb{Q})$. Then we have the following theorem, see [Bor53,
Section 26].

######  Theorem 5.2.1.

Let $G$ be an exceptional Lie group and let $u\in R(G)$ be a minimal
representation of $G$. Let $K$ be a closed connected subgroup of $G$, $i:K\to
G$ be the inclusion of $K$, and $\pi=Bi$ be the map of classifying spaces
defined by $i$. Finally, let $j:G/K\to BK$ be the inclusion of a fibre of
$\pi=Bi:BK\to BG$ .

Suppose the rank of $K$ is equal to the rank of $G$. Then the rational
cohomology of $G/K$ is given by

$H^{*}(G/K;\mathbb{Q})=H^{*}(BK;\mathbb{Q})/I$

where $I$ is the ideal in $H^{*}(BK;\mathbb{Q})$ generated by the elements

$\mathrm{ch}_{q}(i^{*}u),\quad q\geq 1.$

###### Proof.

Consider the following commutative diagram.

$\begin{CD}R(K)\otimes\mathbb{Q}@>{}>{}>\mathrm{K}^{*}(G/K)\otimes\mathbb{Q}@>{\mathrm{ch}}>{}>H^{*}(G/K;\mathbb{Q})\\\
@A{}A{}A@A{j^{*}}A{}A@A{}A{j^{*}}A\\\
R(K)\otimes\mathbb{Q}@>{}>{}>\mathrm{K}^{*}(BK;\mathbb{Q})@>{}>{\mathrm{ch}}>H^{*}(BK;\mathbb{Q})\end{CD}$

By Theorem 1.2.1,
$R(K)\otimes\mathbb{Q}\to\mathrm{K}^{*}(G/K)\otimes\mathbb{Q}$ is surjective.
Since the Chern character is a rational isomorphism it follows that the
composite of the two homomorphisms in the top row is surjective. The
commutativity of the diagram shows the the right hand vertical arrow is
surjective. A standard application of the Leray-Hirsch theorem tell us that
$j^{*}:H^{*}(BK;\mathbb{Q})\to H^{*}(G/K;\mathbb{Q})$ is surjective with
kernel the ideal in $H^{*}(BK;\mathbb{Q})$ generated by the elements

$\pi^{*}(x),\text{ for }x\in H^{q}(BG),q\geq 1.$

Applying Theorem 5.1.2 completes the proof. ∎

###  5.3. An example

The Grassmann manifold $G_{k}(\mathbb{C}^{n})$ is the homogeneous space

$\mathrm{U}(n)/(\mathrm{U}(k)\times\mathrm{U}(n-k)).$

We assume that $k\leq n-k$.

Write $V_{m}$ for the universal vector bundle over $B\mathrm{U}(m)$. This is
the bundle defined by the usual action of $\mathrm{U}(m)$ on $\mathbb{C}^{m}$.
There are two obvious bundles $V_{k}$ and $V_{n-k}$ on
$B(\mathrm{U}(k)\times\mathrm{U}(n-k))$ and
$H^{*}(B(\mathrm{U}(k)\times\mathrm{U}(n-k));\mathbb{Q})$ is the polynomial
algebra generated by

$a_{2}=\mathrm{ch}_{1}(V_{k}),\dots,a_{2k}=\mathrm{ch}_{k}(V_{k}),$
$b_{2}=\mathrm{ch}_{1}(V_{n-k}),\dots,b_{2(n-k)}=\mathrm{ch}_{n-k}(V_{n-k}).$

The restriction of $V_{n}$ to $B(\mathrm{U}(k)\times\mathrm{U}(n-k))$ is
$V_{k}\oplus V_{n-k}$. So the relations given by Theorem 5.2.1 are simply

$\mathrm{ch}_{i}(V_{k})+\mathrm{ch}_{i}(V_{n-k})=0\quad\text{for $1\leq i\leq
n$.}$

This gives a presentation of $H^{*}(G_{k}(\mathbb{C}^{n});\mathbb{Q})$ with
$n$ generators and $n$ relations.

We should decode these relations. The first $k$ relations simply say that

$a_{2i}+b_{2i}=0\quad\text{for $1\leq i\leq k$}$

Now recall that $\mathrm{ch}_{k+1}(V_{k})$ is not necessarily zero. But it is
a polynomial in $\mathrm{ch}_{1}(V_{k}),\dots,\mathrm{ch}_{k}(V_{k})$. So
$\mathrm{ch}_{k+1}(V_{k})+\mathrm{ch}_{k+1}(V_{n-k})=0$ tells us that
$b_{2k+2}$ is a polynomial in $a_{2},\dots,a_{2k}$. This carries on to show
that $b_{2k+2},\dots,b_{2(n-k)}$ are also polynomials in $a_{2},\dots,a_{2k}$.
These polynomials are given by the Newton identities for symmetric
polynomials. For $i=n-k+1,\dots,n$ the equations

$\mathrm{ch}_{i}(V_{k})+\mathrm{ch}_{i}(V_{n-k})=0$

tell us that a particular polynomial in $a_{2},\dots,a_{2k}$ is equal to
another particular polynomial in $b_{2},\dots,b_{2(n-k)}$. Now when we
substitute the previous formulas for the $b_{2i}$ in terms of the $a_{2j}$
this shows that two polynomials in the $a_{2i}$ are equal. This gives $k$
relations in the $a_{2i}$. We end up with a presentation with $k$ generators
and $k$ relations.

This calculation does not use the Schubert cell decomposition of the Grassmann
manifold.

##  6\. Rationally elliptic spaces.

A simply connected topological space is rationally elliptic if both
$H^{*}(X;\mathbb{Q})$ and $\pi_{*}(X)\otimes\mathbb{Q}$ are finite
dimensional. Any compact simply connected homogeneous space is rationally
elliptic. The space $S^{3}\vee S^{3}$ is a simple example of a space that not
rationally elliptic: its rational cohomology is finite dimensional but its
rational homotopy is not. The space $\mathbb{C}\mathbb{P}^{\infty}$ is not
rationally elliptic: its rational homotopy is finite dimensional but its
rational cohomology is not. If $X$ is rationally elliptic we write

$\chi_{H}(X),\quad\chi_{\pi}(X)$

for the Euler characteristic of $H^{*}(X;\mathbb{Q})$ and
$\pi_{*}(X)\otimes\mathbb{Q}$ respectively.

###  6.1. The main structural theorem on rationally elliptic spaces

A very thorough account of the theory of rationally elliptic spaces can be
found in [FHT01, Section 32] together with references to the original papers.
The following theorem is a combination of Proposition 32.10 and Proposition
32.16 in [FHT01].

######  Theorem 6.1.1.

Suppose that $X$ is a rationally elliptic space. Then

$\chi_{H}(X)\geq 0,\quad\chi_{\pi}(X)\leq 0.$

Furthermore, the following statements are equivalent:

  1. (i)

$\chi_{H}(X)>0$,

  2. (ii)

$H^{*}(X;\mathbb{Q})$ is concentrated in even degrees,

  3. (iii)

$H^{*}(X;\mathbb{Q})$ is the quotient of a polynomial ring
$\mathbb{Q}[a_{1},\dots,a_{q}]$ where the $a_{i}$ have even degree, by an
ideal generated by a regular sequence of length $q$,

  4. (iv)

$\dim\pi_{\text{even}}(X)\otimes\mathbb{Q}=\dim\pi_{\text{odd}}(X)\otimes\mathbb{Q}$.

It follows that the minimum number of generators, $q$ in (iii), is equal to
$\dim\pi_{\text{even}}(X)\otimes\mathbb{Q}$. Furthermore a minimal set of
generators in a fixed degree $2s$ is in one-to-one correspondence with a basis
for $\pi_{2s}(X)\otimes\mathbb{Q}$. Furthermore, a minimal set of relations in
degree $2t$ is in one-to-one correspondence with a basis of
$\pi_{2t-1}(X)\otimes\mathbb{Q}$.

###  6.2. The Poincaré polynomial of a rationally elliptic space with positive
Euler characteristic.

Now suppose $X$ is a rationally elliptic space with positive Euler
characteristic. Let

$q=\dim\pi_{\mathrm{ev}}(X)\otimes\mathbb{Q}=\dim\pi_{\mathrm{odd}}(X)\otimes\mathbb{Q}.$

and let $(2n_{1},\dots,2n_{q})$, $(2m_{1}-1,\dots,2m_{q}-1)$ be the degrees of
a basis of the rational homotopy groups of $X$. Then the Poincaré polynomial
of $X$ is

$P(t)=\frac{(1-t^{2n_{1}})\cdots(1-t^{2n_{q}})}{(1-t^{2m_{1}})\cdots(1-t^{2m_{q}})}.$

This follows from the Theorem 6.1.1 and the following standard algebraic fact.
Let $A$ be a graded commutative algebra over $\mathbb{Q}$ of the form

$A=\mathbb{Q}[a_{1},\dots,a_{q}]/(r_{1},\dots,r_{q})$

where the $a_{i}$ have even degree and the $r_{i}$ form a regular sequence.
Then the Poincaré polynomial of $A$ is

$\frac{(1-t^{|r_{1}|})\cdots(1-t^{|r_{q}|})}{(1-t^{|a_{1}|})\cdots(1-t^{|a_{q}|})}.$

##  7\. The rational homotopy groups of the Rosenfeld projective planes

The rational homotopy groups of the Rosenfeld projective planes were first
calculated by Svjetlana Terzić in [Ter11]. We give a proof following the main
strategy of this paper, using Theorem 3.2.1 and a computation in
representation theory.

######  Theorem 7.0.1.

The non-zero rational homotopy groups of $R_{5}$, $R_{6}$ and $R_{7}$ are
given by

  1. (i)

$\pi_{n}(R_{5})\otimes\mathbb{Q}=\mathbb{Q}$, if $n=2,8,17,23$,

  2. (ii)

$\pi_{n}(R_{6})\otimes\mathbb{Q}=\mathbb{Q}$, if $n=4,8,12,23,27,35$,

  3. (iii)

$\pi_{n}(R_{7})\otimes\mathbb{Q}=\mathbb{Q}$, if $n=8,12,16,20,35,39,47,59$.

Recall, from [Ada96] that there are homomorphisms

$h_{6}:\mathrm{Spin}(10)\to E_{6},\quad h_{7}:\mathrm{Spin}(12)\to E_{7},\quad
h_{8}:\mathrm{Spin}(16)\to E_{8}.$

The first two are injective and the kernel of $h_{8}$ is a central subgroup of
$\mathrm{Spin}(16)$ of order $2$ which does not contain the kernel of the
universal covering $\mathrm{Spin}(16)\to\mathrm{SO}(16)$.

######  Lemma 7.0.2.

  1. (i)

The kernel of $(h_{6})_{*}:V^{*}(\mathrm{Spin}(10))\to V^{*}(E_{6})$ has
dimension $1$.

  2. (ii)

The kernel of $(h_{7})_{*}:V^{*}(\mathrm{Spin}(12))\to V^{*}(E_{7})$ has
dimension $2$.

  3. (iii)

The kernel of $(h_{8})_{*}:V^{*}(\mathrm{Spin}(16))\to V^{*}(E_{8})$ has
dimension $4$.

The proof is computational and we postpone it until Section 9.

###  7.1. The rational homotopy groups of $R_{5}$

Recall that $R_{5}=E_{6}/(\mathrm{Spin}(10)\times_{C_{4}}S^{1})$. Let $N_{5}$
be the homogeneous space $E_{6}/\mathrm{Spin}(10)$, so $N_{5}$ is an
$S^{1}$-bundle over $R_{5}$.

First we use the four-term exact sequences of Section 3.2 to compute the
rational homotopy groups of $N_{5}$. The type of $\mathrm{Spin}(10)$ is

$(3,7,9,11,15)$

and the type of $E_{6}$ is

$(3,9,11,15,17,23)$

The following table summarises the four term exact sequences for $N_{5}$.

$r$ | $3$ | $7$ | $9$ | $11$ | $15$ | $17$ | $23$
---|---|---|---|---|---|---|---
$\pi_{r+1}(N_{5})$ |  | $\mathbb{Q}$ |  |  |  |  |
$\pi_{r}(\mathrm{Spin}(10))$ | $\mathbb{Q}$ | $\mathbb{Q}$ | $\mathbb{Q}$ | $\mathbb{Q}$ | $\mathbb{Q}$ |  |
$\pi_{r}(E_{6})$ | $\mathbb{Q}$ |  | $\mathbb{Q}$ | $\mathbb{Q}$ | $\mathbb{Q}$ | $\mathbb{Q}$ | $\mathbb{Q}$
$\pi_{r}(N_{5})$ |  |  |  |  |  | $\mathbb{Q}$ | $\mathbb{Q}$

Our conventions are

  1. (i)

homotopy groups are rational homotopy groups,

  2. (ii)

a blank entry in the table means the relevant group is $0$, as does not
appearing in the table.

The two middle rows are the homotopy groups of $\mathrm{Spin}(10)$ and
$E_{6}$, so we must calculate the homomorphism

$\pi_{r}(\mathrm{Spin}(10))\to\pi_{r}(E_{6})$

for $r=3,7,9,11,15$. When $r=3$ this homomorphism has to be an isomorphism and
for $i=r$ it has to be zero, so it has a one dimensional kernel. Using Theorem
3.2.1 and Lemma 7.0.2 we know that the kernel of
$\pi(\mathrm{Spin}(10))\to\pi(E_{6})$ is one dimensional. It must follow that
for $r=9,11,15$ this homomorphism is injective and hence an isomorphism. This
gives the complete table.

A simple argument with the homotopy exact sequence of the circle bundle
$N_{5}\to R_{5}$ completes the calculation of the rational homotopy groups of
$R_{5}$.

###  7.2. The rational homotopy groups of $R_{6}$

Recall that $R_{6}=E_{7}/(\mathrm{Spin}(12))\times_{C_{2}}S^{3})$. Let $N_{6}$
be the homogeneous space $E_{7}/\mathrm{Spin}(12)$, so $N_{6}$ is an $S^{3}$
bundle over $R_{6}$.

As before we begin by computing the homotopy groups of $N_{6}$ using the four-
term exact sequences. The type of $\mathrm{Spin}(12)$ is

$(3,7,11,11,15,19,23)$

and the type of $E_{7}$ is

$(3,11,15,19,23,27,35).$

The following table, with the same conventions as the previous case,
summarises the four term exact sequences of $N_{6}$.

$r$ | $3$ | $7$ | $11$ | $15$ | $19$ | $23$ | $27$ | $35$
---|---|---|---|---|---|---|---|---
$\pi_{r+1}(N_{6})$ |  | $\mathbb{Q}$ | $\mathbb{Q}$ |  |  |  |  |
$\pi_{r}(\mathrm{Spin}(12))$ | $\mathbb{Q}$ | $\mathbb{Q}$ | $\mathbb{Q}\oplus\mathbb{Q}$ | $\mathbb{Q}$ | $\mathbb{Q}$ |  |  |
$\pi_{r}(E_{7})$ | $\mathbb{Q}$ |  | $\mathbb{Q}$ | $\mathbb{Q}$ | $\mathbb{Q}$ | $\mathbb{Q}$ | $\mathbb{Q}$ | $\mathbb{Q}$
$\pi_{r}(N_{5})$ |  |  |  |  |  | $\mathbb{Q}$ | $\mathbb{Q}$ | $\mathbb{Q}$

This time we must calculate the homomorphism

$\pi_{r}(\mathrm{Spin}(12))\to\pi_{r}(E_{7}).$

Elementary arguments with the four-term exact sequences show that when $r=3$
this must be an isomorphism and when $r=7,11$ it must have one-dimensional
kernel. Once more we know from Theorem 3.2.1 and Lemma 7.0.2 that the kernel
of $\pi(\mathrm{Spin}(12))\to\pi(E_{7})$ is two-dimensional. Therefore the
entries in the top row must be $0$ from column $15$ onwards. Finally the
homotopy exact sequence of the $S^{3}$ bundle $N_{6}\to R_{6}$ completes the
proof.

###  7.3. The rational homotopy groups of $R_{7}$

First we must explain what the half-spin or semi-spin group
$\mathrm{HSpin}(2n)$ is. The centre of $\mathrm{Spin}(4n)$ is $C_{2}\times
C_{2}$. So there are three non-trivial central subgroups of order two in
$\mathrm{Spin}(4n)$. If $4n\neq 8$ the outer automorphism group of
$\mathrm{Spin}(4n)$ is cyclic of order two. The non-trivial outer automorphism
interchanges two of these central subgroups, and this gives two different, but
isomorphic, $C_{2}$ quotients of $\mathrm{Spin}(4n)$. These are the half-spin
groups. It is usual to choose one to work with and call this
$\mathrm{HSpin}(4n)$. The third quotient is $\mathrm{SO}(4n)$. In the case of
$\mathrm{Spin}(8)$ the outer autmorphism group is the symmetric group
$\Sigma_{3}$ and this acts transitively on the three central subgroups of
order $2$. So all three $C_{2}$ quotients of $\mathrm{Spin}(8)$ are isomorphic
to $\mathrm{SO}(8)$.

Now $R_{7}=E_{8}/\mathrm{HSpin}(16)$, the type of $\mathrm{HSpin}(16)$ is

$(3,7,11,15,15,19,23,27),$

and the type of $E_{8}$ is

$(3,15,23,27,35,39,47,59)$

The following table, with the usual conventions, summarises the conclusions of
the calculations with the four-term exact sequences for $R_{7}$.

$r$ | $3$ | $7$ | $11$ | $15$ | $19$ | $23$ | $27$ | $35$ | $39$ | $47$ | $59$
---|---|---|---|---|---|---|---|---|---|---|---
$\pi_{r+1}(R_{7})$ |  | $\mathbb{Q}$ | $\mathbb{Q}$ | $\mathbb{Q}$ | $\mathbb{Q}$ |  |  |  |  |  |
$\pi_{r}(\mathrm{HSpin}(16))$ | $\mathbb{Q}$ | $\mathbb{Q}$ | $\mathbb{Q}$ | $\mathbb{Q}\oplus\mathbb{Q}$ | $\mathbb{Q}$ | $\mathbb{Q}$ | $\mathbb{Q}$ |  |  |  |
$\pi_{r}(E_{8})$ | $\mathbb{Q}$ |  |  | $\mathbb{Q}$ |  | $\mathbb{Q}$ | $\mathbb{Q}$ | $\mathbb{Q}$ | $\mathbb{Q}$ | $\mathbb{Q}$ | $\mathbb{Q}$
$\pi_{r}(R_{7})$ |  |  |  |  |  |  |  | $\mathbb{Q}$ | $\mathbb{Q}$ | $\mathbb{Q}$ | $\mathbb{Q}$

This time we must compute

$\pi_{r}(\mathrm{HSpin}(16))\to\pi_{r}(E_{8}).$

The usual arguments with the four-term exact sequences show that it must be an
isomorphism when $r=3$ and have one dimensional kernel when $r=7,11,15,19$. As
in the other cases the proof is completed by Theorem 3.2.1 and Lemma 7.0.2.

###  7.4. What does the theory of elliptic spaces tell us?

Applying the results in Section 6, we read off the following information about
the rational cohomology of the Rosenfeld projective planes
$R_{5},R_{6},R_{7}$.

  1. (i)

$H^{*}(R_{5},\mathbb{Q})=\frac{\mathbb{Q}[a_{2},a_{8}]}{(r_{18},r_{24})}$
$P=\left(\frac{1-t^{18}}{1-t^{2}}\right)\left(\frac{1-t^{24}}{1-t^{8}}\right),\quad\chi=27.$

  2. (ii)

$H^{*}(R_{6},\mathbb{Q})=\frac{\mathbb{Q}[a_{4},a_{8},a_{12}]}{(r_{24},r_{28},r_{36}}$
$P=\left(\frac{1-t^{24}}{1-t^{8}}\right)\left(\frac{1-t^{28}}{1-t^{4}}\right)\left(\frac{1-t^{36}}{1-t^{12}}\right),\quad\chi=63.$

  3. (iii)

$H^{*}(R_{7},\mathbb{Q})=\frac{\mathbb{Q}[a_{8},a_{12},a_{16},a_{20}]}{(r_{36},r_{40},r_{48},r_{60})}$
$P=\left(\frac{1-t^{36}}{1-t^{12}}\right)\left(\frac{1-t^{40}}{1-t^{8}}\right)\left(\frac{1-t^{48}}{1-t^{16}}\right)\left(\frac{1-t^{60}}{1-t^{20}}\right),\quad\chi=135.$

In all three cases the subscripts on the generators and the relations are the
degrees. Furthermore, in all three cases the relations form a regular
sequence.

These results agree with [Pic17, Table A] which also contains references to
other proofs. It is worth repeating here that one of our main aims is to give
a systematic account of these results. In the next section we show how to get
explicit presentations for the rings $H^{*}(R_{5};\mathbb{Q})$,
$H^{*}(R_{6};\mathbb{Q})$ and $H^{*}(R_{7};\mathbb{Q})$ by using Theorem
5.2.1.

##  8\. The rational cohomology of $R_{5},R_{6},R_{7}$

###  8.1. The general strategy

We have the following general situation: two Lie groups $K$ and $G$ of the
same rank and a homomorphism

$h:K\to G$

with finite kernel. From this we get a quotient homomorphism

$p:K\to\bar{K}=K/\ker(h)$

and an embedding

$i:\bar{K}\to G.$

Let $R=G/\bar{K}$ and let

$j:R\to B\bar{K}$

be the inclusion of $R$ as a fibre of $Bi:B\bar{K}\to BG$. This gives a
diagram of spaces

$\begin{CD}BK@>{Bh}>{}>BG\\\ @V{Bp}V{}V@V{}V{1}V\\\
R@>{}>{j}>B\bar{K}@>{}>{Bi}>BG\end{CD}$

Since the kernel of $p$ is finite $(Bp)^{*}:H^{*}(B\bar{K};\mathbb{Q})\to
H^{*}(BK;\mathbb{Q})$ is an isomorphism. We get a surjective homomorphism

$\phi=j^{*}\circ((Bp)^{*})^{-1}:H^{*}(BK;\mathbb{Q})\to H^{*}(R;\mathbb{Q})$

and an isomorphism

$H^{*}(BK;\mathbb{Q})/I\to H^{*}(R;\mathbb{Q})$

where $I$ is the kernel of $\phi$.

In our examples $G$ is one of $E_{6},E_{7},E_{8}$ and in these cases we can
choose a minimal representation $w$ of $G$. From Theorem 5.2.1 it follows that

$I=(\mathrm{ch}_{q}(h^{*}(w)):q\geq 1).$

To convert this into an explicit presentation we must compute
$\mathrm{ch}(h^{*}(w))\in H^{*}(BK;\mathbb{Q})$.

We now explain how to calculate characteristic classes using this data; this
is used in the detailed study of $R_{5}$ in [JRT23].

Let $\rho$ be a (complex) representation of $\bar{K}$ . This gives a
representation $p^{*}(\rho)$ of $K$ in which $\ker(h)$ acts trivially. We
continue to write $\rho$ and $p^{*}(\rho)$ for the vector bundles over
$B\bar{K}$ and $BK$, respectively, associated to these representations. We
write

$E_{\bar{\rho}}=j^{*}(\bar{\rho})$

for the complex vector bundle over $R$ associated to $\bar{\rho}$.

It follows from the definition of $\phi:H^{*}(K;\mathbb{Q})\to
H^{*}(R;\mathbb{Q})$ above that

$c_{k}(E_{\bar{\rho}})=\phi(c_{k}(\rho)).$

This allows us to calculate

$c_{k}(E_{\bar{\rho}})\in H^{2k}(R;\mathbb{Z})\mod\text{torsion}.$

Of course we can use this method to calculate Pontryagin classes, the Chern
character, or any other characteristic class.

###  8.2. The rational cohomology of $R_{5}$

Adams shows in [Ada96, Chapter 8] that there is a homomorphism

$h:\mathrm{Spin}(10)\times\mathrm{U}(1)\to E_{6}$

such that the kernel of $h$ is a central subgroup $C_{4}$. If we first choose
one of the two minimal representations $w$ of $E_{6}$ then we can subsequently
choose $h$ so that

$h^{*}(w)=\xi^{4}+\rho_{10}\otimes\xi^{2}+\delta_{10}^{-}\otimes\xi.$

Here $\rho_{10}$ is the $10$-dimensional vector representation of
$\mathrm{Spin}(10)$, $\delta_{10}^{\pm}$ are the two spin representations of
$\mathrm{Spin}(10)$, and $\xi$ is the standard representation of
$\mathrm{U}(1)$ on $\mathbb{C}$. This relation between representations leads
to the relations in the presentation of $H^{*}(R_{5};\mathbb{Q})$.

By definition

$R_{5}=E_{6}/(\mathrm{Spin}(10)\times_{C_{4}}\mathrm{U}(1)).$

We get a surjective homomorphism

$\phi:H^{*}(B(\mathrm{Spin}(10)\times\mathrm{U}(1));\mathbb{Q})\to
H^{*}(R_{5};\mathbb{Q})$

whose kernel is generated by the $\mathrm{ch}_{q}(h^{*}(w))$ with $q\geq 1$.

Next we compute

$\mathrm{ch}_{q}(h^{*}(w))\in
H^{2q}(B(\mathrm{Spin}(10)\times\mathrm{U}(1));\mathbb{Q})$

in terms of the Pontryagin classes $p_{i}(\rho_{10})\in
H^{4i}(B\mathrm{Spin}(10);\mathbb{Q})$, the Euler class $e(\rho_{10})\in
H^{10}(B\mathrm{Spin}(10);\mathbb{Q})$, and the first Chern class $c_{1}(\xi)$
in $H^{2}(B\mathrm{U}(1);\mathbb{Q})$. This leads to the following theorem.

######  Theorem 8.2.1.

The cohomology ring $H^{*}(R_{5};\mathbb{Q})$ is isomorphic to

$\frac{\mathbb{Q}[a_{2},a_{8}]}{(r_{18},r_{24})},$

where the generators are

$a_{2}=\phi(c_{1}(\xi)),\quad a_{8}=\phi(p_{2}(\rho_{10})),$

and the relations are

$\displaystyle r_{18}$
$\displaystyle=-39936a_{2}^{9}+1728a_{2}^{5}a_{8}+a_{2}a_{8}^{2},$
$\displaystyle r_{24}$
$\displaystyle=-50429952a_{2}^{12}+3068928a_{2}^{8}a_{8}-11808a_{2}^{4}a_{8}^{2}-a_{8}^{3}.$

The integral cohomology of $R_{5}$ has been calculated by Toda and Watanabe,
in [TW74]. Although our presentation is different to the one in [TW74,
Corollary C] the two presentations define the same ring over $\mathbb{Q}$.

###  8.3. The rational cohomology of $R_{6}$

In this case Adams shows in [Ada96, Chapter 8] that there is a homomorphism

$h:\mathrm{Spin}(12)\times\mathrm{Sp}(1)\to E_{7}$

such that the kernel of $h$ is a central subgroup of order $2$. Now let $w$ be
the minimal representation of $E_{7}$. Adams also shows that we may choose $h$
so that

$h^{*}(w)=\rho_{12}\otimes\zeta+\delta_{12}^{+}.$

Here $\rho_{12}$ is the $12$ dimensional vector representation of
$\mathrm{Spin}(12)$, $\delta^{\pm}_{12}$ are the spin representations, and
$\zeta$ is the representation of $\mathrm{Sp}(1)$ on
$\mathbb{H}=\mathbb{C}^{2}$. This relation between representations leads to
the relations in the presentation of $H^{*}(R_{6};\mathbb{Q})$.

By definition

$R_{6}=E_{6}/(\mathrm{Spin}(12)\times_{C_{2}}\mathrm{Sp}(1)).$

This gives a surjective homomorphism

$\phi:H^{*}(B(\mathrm{Spin}(12)\times\mathrm{Sp}(1));\mathbb{Q})\to
H^{*}(R_{6};\mathbb{Q})$

whose kernel is generated by the $\mathrm{ch}_{q}(h^{*}(w))$ with $q\geq 1$.
So we compute

$\mathrm{ch}_{q}(h^{*}(w))\in
H^{2q}(B(\mathrm{Spin}(12)\times\mathrm{Sp}(1);\mathbb{Q})$

in terms of the Pontryagin classes $p_{i}(\rho_{12})\in
H^{4i}(B\mathrm{Spin}(12);\mathbb{Q})$, the Euler class $e(\rho_{12})\in
H^{12}(B\mathrm{Spin}(12);\mathbb{Q})$, and the Chern class $c_{2}(\zeta)\in
H^{4}(B\mathrm{Sp}(1);\mathbb{Q})$.

######  Theorem 8.3.1.

The cohomology ring $H^{*}(R_{6};\mathbb{Q})$ is isomorphic to

$\frac{\mathbb{Q}[a_{4},a_{8},a_{12}]}{(r_{24},r_{28},r_{36})},$

where the generators are

$a_{4}=\phi(p_{1}(\rho_{12}))=2c_{2}(\zeta),\quad
a_{8}=\phi(p_{2}(\rho_{12})),\quad a_{12}=\phi(p_{3}(\rho_{12})),$

and the relations are

$\displaystyle\ r_{24}$
$\displaystyle=38367a_{4}^{6}-131436a_{4}^{4}a_{8}+88272a_{4}^{2}a_{8}^{2}-1600a_{8}^{3}-273024a_{4}^{3}a_{12}+55296a_{4}a_{8}a_{12}-10368a_{12}^{2},$
$\displaystyle r_{28}$
$\displaystyle=63a_{4}^{7}+96a_{4}^{5}a_{8}+48a_{4}^{3}a_{8}^{2}+640a_{4}a_{8}^{3}-1686a_{4}^{4}a_{12}-4656a_{4}^{2}a_{8}a_{12}+160a_{8}^{2}a_{12}-1152a_{4}a_{12}^{2},$
$\displaystyle r_{36}$
$\displaystyle=19503a_{4}^{9}+41184a_{4}^{7}a_{8}-150816a_{4}^{5}a_{8}^{2}-156672a_{4}^{3}a_{8}^{3}+19200a_{4}a_{8}^{4}-127224a_{4}^{6}a_{12}-{}$
$\displaystyle\phantom{==}908448a_{4}^{4}a_{8}a_{12}+264576a_{4}^{2}a_{8}^{2}a_{12}+12800a_{8}^{3}a_{12}-1806336a_{4}^{3}a_{8}^{2}+36864a_{4}a_{8}a_{12}^{2}+18432a_{12}^{3}.$

In [Nak01], Nakagawa gives a presentation of $H^{*}(R_{6};\mathbb{Q})$. Ours
is different but as in the case of $R_{5}$ it is not difficult to check that
the two presentations give isomorphic algebras over $\mathbb{Q}$.

###  8.4. The rational cohomology of $R_{7}$

This time, see [Ada96, Chapter 7], there is a homomorphism

$h:\mathrm{Spin}(16)\to E_{8}$

such that the kernel of $h$ is generated by a central element of order $2$ in
$\mathrm{Spin}(16)$ of which is not in the kernel of the covering map
$\mathrm{Spin}(16)\to\mathrm{SO}(16)$. This gives an embedding

$i:\mathrm{HSpin}(16)\to E_{8}.$

By definition

$R_{7}=E_{8}/\mathrm{HSpin}(16)$

is the Rosenfeld projective plane of dimension $128$.

Let $w$ be the adjoint representation of $E_{8}$. This is the minimal
representation of $E_{8}$. Then we may choose $h$ so that in the natural
notation

$h^{*}(w)=\lambda^{2}\rho_{16}+\delta_{16}^{+},$

see [Ada96, Theorem 6.1]. This relation between representations once more
leads to the relations in the presentation of $H^{*}(R_{7};\mathbb{Q})$.

As before we get a surjective ring homomorphism

$\phi:H^{*}(B\mathrm{Spin}(16);\mathbb{Q})\to H^{*}(R_{7};\mathbb{Q}).$

This time we compute $\mathrm{ch}(h^{*}w)$ in terms of the Pontryagin classes
and the Euler class of $\rho_{16}$ in $H^{*}(B\mathrm{Spin}(16);\mathbb{Q})$.

######  Theorem 8.4.1.

The cohomology ring $H^{*}(R_{7};\mathbb{Q})$ is isomorphic to

$\frac{\mathbb{Q}[a_{8},a_{12},a_{16},a_{20}]}{(r_{36},r_{40},r_{48},r_{60})},$

where the generators are

$a_{8}=\phi(p_{2}(\rho_{16})),\quad a_{12}=\phi(p_{3}(\rho_{16})),\quad
a_{16}=\phi(p_{4}(\rho_{16}))\quad a_{20}=\phi(p_{5}(\rho_{16})),$

and the relations are

$\displaystyle\ r_{36}$
$\displaystyle=275a_{8}^{3}a_{12}-6150a_{8}^{2}a_{20}+5400a_{8}a_{12}a_{16}-756a_{12}^{3}-10800a_{16}a_{20},$
$\displaystyle r_{40}$
$\displaystyle=275a_{8}^{5}+4080a_{8}^{3}a_{16}+945a_{8}^{2}a_{12}^{2}-26460a_{8}p_{12}p_{20}-25920a_{8}a_{20}^{2}+27216a_{12}^{2}a_{16}+26460a_{20}^{2},$
$\displaystyle r_{48}$
$\displaystyle=-225875a_{8}^{6}-8037000a_{8}^{4}a_{16}+4233600a_{8}^{3}a_{12}^{2}-23020200a_{8}^{2}a_{12}a_{20}-29160000a_{8}^{2}a_{16}^{2}+{}$
$\displaystyle\phantom{==}28576800a_{8}a_{12}^{2}a_{16}-166698000a_{8}a_{20}^{2}+3000564a_{12}^{4}+57153600a_{12}a_{16}a_{20}+46656000a_{16}^{3},$
$\displaystyle r_{60}$
$\displaystyle=-2868125a_{8}^{6}a_{12}+22312500a_{8}^{5}a_{20}-36945000a_{8}^{4}a_{12}a_{16}-3307500a_{8}^{3}a_{12}^{3}-{}$
$\displaystyle\phantom{==}390600000a_{8}^{3}a_{16}a_{20}+222264000a_{8}^{2}a_{12}^{2}a_{20}-243000000a_{8}^{2}a_{12}a_{16}^{2}+71442000a_{8}a_{12}^{3}a_{16}-{}$
$\displaystyle\phantom{==}972405000a_{8}a_{12}a_{20}^{2}+1360800000a_{8}a_{16}^{2}a_{20}-18003384a_{12}^{5}+1000188000a_{12}^{2}a_{16}a_{20}-{}$
$\displaystyle\phantom{==}699840000a_{12}a_{16}^{3}-463050000a_{20}^{3}.$

This seems to be the first explicit presentation of $H^{*}(R_{7};\mathbb{Q})$
in the literature.

###  8.5. The proof for $R_{7}$

The ring $H^{*}(B\mathrm{Spin}(16);\mathbb{Q})$ is given by

$H^{*}(B\mathrm{Spin}(16);\mathbb{Q})=\mathbb{Q}[p_{1},p_{2},p_{3},p_{4},e,p_{5},p_{6},p_{7}]$

where the $p_{i}=p_{i}(\rho_{16})\in H^{4i}(B\mathrm{Spin}(16);\mathbb{Q})$
are the Pontryagin classes of $\rho_{16}$, the $16$-dimensional real vector
bundle over $B\mathrm{Spin}(16)$ and $e=e(\rho_{16})\in
H^{16}(B\mathrm{Spin}(16);\mathbb{Q})$ is the Euler class of the same bundle.
Following on from the previous section we now do the calculation of

$\mathrm{ch}(h^{*}(w))=\mathrm{ch}(\lambda^{2}(\rho_{16}))+\mathrm{ch}(\delta_{16}^{+}).$

We provide some details about this computation in Section 8.6.

We know that $H^{*}(R_{7};\mathbb{Q})$ is generated by $4$ elements of degrees
$8,12,16,20$. Since the map $\phi:H^{*}(B\mathrm{Spin}(16);\mathbb{Q})\to
H^{*}(R_{7};\mathbb{Q})$ is surjective we can take

$a_{8}=\phi(p_{2}),\quad a_{12}=\phi(p_{3}),\quad a_{20}=\phi(p_{5})$

as the generators in degree 8, 12, and 20. However, the ring
$H^{*}(B\mathrm{Spin}(16);\mathbb{Q})$ has two generators in degree $16$ so we
have to make a choice of an element in $H^{16}(BE_{8};\mathbb{Q})$ which maps
to a generator in $H^{16}(BR_{7};\mathbb{Q})$.

However, $\phi(p_{1})=0$ since $H^{4}(R_{5};\mathbb{Q})$ is zero. By
definition $a_{8}=\phi(p_{2})$, and $a_{12}=\phi(p_{3})$. The computations
show that

$\mathrm{ch}_{8}(h^{*}(w))=\frac{1}{336}p_{2}^{2}+\frac{1}{28}p_{4}+\frac{1}{2}e\mod
p_{1}.$

Here mod $p_{1}$ means up to an element of the form $xp_{1}$. This allows us
to make a choice of the $16$ dimensional generator and we choose
$a_{16}=\phi(p_{4})$. It follows that

$\phi(e)=-\frac{1}{168}a_{8}^{2}-\frac{1}{14}a_{16}.$

Next we compute $\mathrm{ch}_{12}(v)$ as a polynomial in
$p_{1},p_{2},p_{3},p_{4},e,p_{5},p_{6}$; it does not involve $p_{7}$ since
$p_{7}$ has degree $28$. Apply $\phi$ to this polynomial using the above
formulas. It gives

$\phi(p_{6})=\frac{13}{1512}a_{8}^{3}+\frac{3}{14}a_{8}a_{16}-\frac{1}{20}a_{12}^{2}.$

Repeating this with $\mathrm{ch}_{14}(v)$ gives

$\phi(p_{7})=\frac{1}{168}a_{8}^{2}a_{12}-\frac{1}{12}a_{8}a_{20}+\frac{1}{14}a_{12}a_{16}.$

We now have formulas for $\phi$ applied to each of the $8$ generators of
$H^{*}(B\mathrm{Spin}(16);\mathbb{Q})$ as polynomials in
$a_{4},a_{12},a_{16},a_{20}$. This allows us to factorise the homomorphism
$\phi$ as

$\begin{CD}H^{*}(\mathrm{Spin}(16);\mathbb{Q})@>{\theta}>{}>\mathbb{Q}[a_{4},a_{12},a_{16},a_{20}]@>{q}>{}>H^{*}(R_{7};\mathbb{Q})\end{CD}$

where $q$ is the quotient homomorphism, and $\theta$ is defined by the above
formulas for $\phi(p_{i})$ for $i=1,\dots,7$ and $\phi(e)$. It follows that
$\ker(q)$ is generated by the $4$ elements

$\theta(\mathrm{ch}_{18}(h^{*}w)),\quad\theta(\mathrm{ch}_{20}(h^{*}w)),\quad\theta(\mathrm{ch}_{24}(h^{*}w)),\quad\theta(\mathrm{ch}_{30}(h^{*}w)).$

This gives us four polynomials

$r_{36},r_{40},r_{48},r_{60}\in\mathbb{Q}[a_{8},a_{12},a_{16},a_{20}],$

such that

$H^{*}(R_{7})=\mathbb{Q}[a_{8},a_{12},a_{16},a_{20}]/(r_{36},r_{40},r_{48},r_{60}).$

Calculating these four polynomials in detail gives the relations in Theorem
8.4.1.

###  8.6. Comments on the computation

The powers of $2$ in these calculation need care for two reasons. The first is
that the kernel of the homomorphism $h$ in our three examples is
$C_{4},C_{2},C_{2}$ and this introduces some extra factors of $2$ in the
detailed calculations.

The second is that to compute $\mathrm{ch}_{q}(h^{*}(w))$ we must choose
generators for $H^{*}(B\mathrm{Spin}(2n);\mathbb{Q})$. The standard choices
are the Pontryagin classes $p_{i}\in H^{4i}$ and the Euler class in $H^{2n}$.
The Pontryagin classes are integral but they do not generate the integral
cohomology. For example if $V$ is a real spin bundle then $p_{1}(V)$ is
divisible by $2$. See [BW95, Dua18, Tho62] for more details. However they do
generate $H^{*}(B\mathrm{Spin}(2n);\Lambda)$ if $2$ is invertible in
$\Lambda$.

The paper [SUZ18], in particular Section 3, is very helpful for doing these
calculations. We illustrate our methods by outlining one method of computing

$\mathrm{ch}(h^{*}(w))=\mathrm{ch}(\lambda^{2}\rho_{16})+\mathrm{ch}(\delta_{16}^{+})$

in terms of the Pontryagin classes and Euler class in
$H^{*}(B\mathrm{Spin}(16);\mathbb{Q})$. This is the computation which leads to
the presentation of $H^{*}(R_{7};\mathbb{Q})$.

We begin with the computation of $\mathrm{ch}(\delta_{16}^{+})$. The standard
method is to use the splitting principle, as described in Jay Wood’s paper
[Woo98], and then use the correspondence between symmetric functions and the
Pontryagin classes and the Euler class to express
$\mathrm{ch}(\delta_{16}^{+})$ in terms of the generators chosen above. The
explicit expansion up to degree 16 is

$\displaystyle\mathrm{ch}(\delta_{16}^{+})={}$ $\displaystyle
128+16p_{1}+\frac{1}{3}p_{1}^{2}+\frac{4}{3}p_{2}+\frac{1}{360}p_{1}^{3}+\frac{1}{30}p_{1}p_{2}+\frac{2}{15}p_{3}+\frac{1}{80640}p_{1}^{4}+\frac{1}{3360}p_{1}^{2}p_{2}+{}$
$\displaystyle\frac{1}{315}p_{1}p_{3}+\frac{1}{5040}p_{2}^{2}+\frac{17}{1260}p_{4}+\frac{1}{2}e+\cdots$

Again, the standard method for calculating $\mathrm{ch}(\lambda^{2}\rho_{16})$
is to use the splitting principle to express
$\mathrm{ch}_{k}(\lambda^{2}\rho_{16})$ as a polynomial in the Pontryagin
classes and the Euler class. The explicit expansion up to degree 16 is

$\displaystyle\mathrm{ch}(\lambda^{2}\rho_{16})={}$ $\displaystyle
120+14p_{1}+\frac{7}{6}p_{1}^{2}-\frac{4}{3}p_{2}+\frac{7}{180}p_{1}^{3}-\frac{1}{30}p_{1}p_{2}-\frac{2}{15}p_{3}+\frac{1}{1440}p_{1}^{4}-\frac{1}{72}p_{1}p_{3}+{}$
$\displaystyle\frac{1}{360}p_{2}^{2}+\frac{1}{45}p_{4}+\cdots$

##  9\. Computations in representation theory

The proofs of Theorem 4.3.1 and Lemma 7.0.2 both require computations in
representation theory which follow a standard operating procedure.

  1. (i)

First we need detailed information about the $\Lambda$-ring $R(G)$ where $G$
is an appropriate Lie group. Sometimes this is in the literature, but usually
in the case of $E_{6},E_{7},E_{8}$ we use Magma to get this information. The
main issue is that if we start with a representation $\rho$ of $G$ and ask
Magma to compute $\lambda^{k}(\rho)$ it will give the answer as a
decomposition into a sum of irreducible representations. However we want to
express it as a polynomial in a chosen set of generators. All computational
programmes currently use the fundamental representations as the choice of
generators, so that is our default choice. We now have to use Magma to convert
the decomposition of $\lambda^{k}(w)$, where $w$ is fundamental
representation, into a polynomial in the fundamental representations.

  2. (ii)

Next we have to soften this information to allow us to extract what we need.
The first step in this process is to pass from the $\Lambda$-ring $R(G)$ to
the quotient $\Lambda$-ring

$S(G)=R(G)/I^{2}(G).$

Our standard notation is $[x]\in S(G)$ for the image of the element $x$ in
$R(G)$ under the quotient homomorphism $R(G)\to S(G)$. If $x\in R(G)$ we write
$\epsilon_{x}\in\mathbb{Z}$ for the augmentation of $x$. If $x,y\in R(G)$ then

$(x-\epsilon_{x})(y-\epsilon_{y})\in I^{2}(G)$

and it follows that in $S(G)$

$[xy]=\epsilon_{y}[x]+\epsilon_{x}[y]-\epsilon_{x}\epsilon_{y}.$

This will give the $\lambda$-operations in $S(G)$. The only practical way to
handle the arithmetic in this step of the process to use Magma.

  3. (iii)

Both of the results we prove using this procedure involve the $\Psi$-module

$V(G)=S(G)/\mathbb{Z}.$

Note that $S(G)$ is a $\Lambda$-ring but $V(G)$ is not. However, while the
quotient map $S(G)\to V(G)$ is not a map of $\Lambda$-rings, it is a map of
$\Psi$-rings.

###  9.1. The proof of Theorem 4.3.1

We illustrate this procedure in the proof of Theorem 4.3.1 in the case
$G=F_{4}$. The fundamental representations of $F_{4}$ are, in Bourbaki order
([Bou68, Ch. VI, Planches I-IX]),

$w_{1},w_{2},w_{3},w_{4},\quad\text{ of dimension}\quad 52,1274,273,26,\text{
respectively.}$

In $R(F_{4})$ the $\lambda$-operations are given by

$\lambda^{2}w_{4}=w_{1}+w_{3},$ $\lambda^{3}w_{4}=w_{1}w_{4}+w_{2}-w_{4},$
$\lambda^{4}w_{4}=w_{1}^{2}+w_{1}w_{3}-w_{2}-w_{4}^{2}.$

This is the input for step (i) of the standard operating procedure.

Next we reduce these formulas to the $\Lambda$-ring $S(F_{4})$. This gives the
following formulas.

$\lambda^{1}[w_{4}]=[w_{4}],$ $\lambda^{2}[w_{4}]=[w_{1}]+[w_{3}],$
$\lambda^{3}[w_{4}]=26[w_{1}]+[w_{2}]+51[w_{4}]-1352,$
$\lambda^{4}[w_{4}]=377[w_{1}]-[w_{2}]+52[w_{3}]-52[w_{4}]-16224.$

Now we reduce to $V(F_{4})=S(F_{4})/\mathbb{Z}$. This leads to the $4\times 4$
matrix.

$\begin{pmatrix}0&1&26&377\\\ 0&0&1&-1\\\ 0&1&0&52\\\ 1&0&51&-52\\\
\end{pmatrix}$

The columns of this matrix are the coordinates, in the basis defined by the
fundamental representations $w_{1},w_{2},w_{3},w_{4}$, of the elements in
$V(F_{4})$ defined by
$[w_{4}],\lambda^{2}[w_{4}],\lambda^{3}[w_{4}],\lambda^{4}[w_{4}]$. The
determinant of this matrix is $351$.

Now $V(F_{4})\otimes\mathbb{Q}$ is the indecomposable quotient of the ring
$R(F_{4})\otimes\mathbb{Q}$. This shows that

$w_{4},\lambda^{2}w_{4},\lambda^{3}w_{4},\lambda^{4}w_{4}$

generate the ring $R(F_{4})\otimes\mathbb{Q}$ and so
$R(F_{4})\otimes\mathbb{Q}$ is generated as a $\Lambda$-ring by $w_{4}$. In
turn this shows that $S(F_{4}\otimes\mathbb{Q})$ is generated as a
$\Lambda$-ring by $[w_{4}]$, and $V(F_{4})\otimes\mathbb{Q}$ is generated as a
$\Psi$-module by the element defined by $[w_{4}]$.

Implementing this procedure for $E_{6}$, $E_{7}$, and $E_{8}$ leads to the
following three matrices.

  1. (i)

For $E_{6}$

$\begin{pmatrix}1&0&0&1&-702&-3483\\\ 0&0&0&351&3834&17496\\\
0&1&0&-27&-77&-324\\\ 0&0&1&0&-54&-236\\\ 0&0&0&79&756&2511\\\
0&0&0&-405&-2759&2754\\\ \end{pmatrix}$ $\det=-27208467.$

  2. (ii)

For $E_{7}$

$\begin{pmatrix}0&0&0&0&-42504&-834648&-4655288\\\
0&0&0&0&8645&207424&2129083\\\ 0&0&0&0&968&20902&166704\\\
0&0&0&1&0&-267&-2848\\\ 0&0&1&0&-132&-856&-5831\\\
0&1&0&1&56&-14742&-179760\\\ 1&0&1&0&-7371&-79648&-560651\\\ \end{pmatrix}$
$\det=-1997102661696.$

  3. (iii)

For $E_{8}$

$\begin{pmatrix}0&0&0&0&0&-401581533&-48147450080&-2346940420190\\\
0&0&0&0&0&6661497&860320742&49370806120\\\
0&0&0&0&0&185628&22371987&1105454390\\\ 0&0&0&0&1&-1&-7999&-514878\\\
0&0&0&1&0&-3627&-112746&-2002508\\\ 0&0&1&0&248&34255&-9767140&-735062326\\\
0&1&0&247&-496&-5059262&-205995340&-5451187498\\\
1&1&495&30380&2573495&304455619&12658360729&209067977980\\\ \end{pmatrix}$
$\det=22804152835143344418390$

###  9.2. The proof of Lemma 7.0.2

This time we start with the case of $E_{6}$. For ease of notation, let
$j=h_{6}$ be the inclusion map of $\mathrm{Spin}(10)$ into $E_{6}$ as in
Section 7. The fundamental representations of $E_{6}$ are

$a,\quad\rho_{1},\quad\rho_{2},\quad\lambda^{2}\rho_{1},\quad\lambda^{2}\rho_{2},\quad\lambda^{3}\rho_{1}=\lambda^{3}\rho_{2},$

where $a$ is the $78$-dimensional adjoint representation and $\rho_{1}$,
$\rho_{2}$ are the two inequivalent $27$-dimensional minimal representations.

The fundamental representations of $\mathrm{Spin}(10)$ are

$v_{1},\quad v_{2}=\lambda^{2}v_{1},\quad
v_{3}=\lambda^{3}v_{1},\quad\delta^{-}_{10},\quad\delta_{10}^{+},$

where $v_{1}$ is the $10$-dimensional representation, and $\delta_{10}^{\pm}$
are the two $16$-dimensional spin representations.

Restricting the six fundamental $E_{6}$-representations to $\mathrm{Spin}(10)$
yields the following formulas, as in [Min77, Lemma 2].

  1. (i)

$j^{*}(a)=v_{2}+\delta^{+}+\delta^{-}+1$,

  2. (ii)

$j^{*}(\rho_{1})=v_{1}+\delta^{-}+1$,

  3. (iii)

$j^{*}(\rho_{2})=v_{1}+\delta^{+}+1$,

  4. (iv)

$j^{*}(\lambda^{2}\rho_{1})=v_{1}+v_{2}+v_{3}+\delta^{-}+v_{1}\delta^{-}$,

  5. (v)

$j^{*}(\lambda^{2}\rho_{2})=v_{1}+v_{2}+v_{3}+\delta^{+}+v_{1}\delta^{+}$,

  6. (vi)

$j^{*}(\lambda^{3}\rho_{1})=v_{2}+2v_{3}+v_{2}\delta^{+}+v_{2}\delta^{-}+v_{1}v_{3}$.

Next we reduce these formulas to the $\Lambda$-ring $S(G)$. This produces the
following six equations in $S(\mathrm{Spin}(10))$.

  1. (i)

$j^{*}[a]=[v_{2}]+[\delta^{+}]+[\delta^{-}]$ \+ 1,

  2. (ii)

$j^{*}[\rho_{1}]=[v_{1}]+[\delta^{-}]$ \+ 1,

  3. (iii)

$j^{*}[\rho_{2}]=[v_{1}]+[\delta^{+}]+1$,

  4. (iv)

$j^{*}[\lambda^{2}\rho_{1}]=17[v_{1}]+[v_{2}]+[v_{3}]+11[\delta^{-}]-160$,

  5. (v)

$j^{*}[\lambda^{2}\rho_{2}]=17[v_{1}]+[v_{2}]+[v_{3}]+11[\delta^{+}]-160$,

  6. (vi)

$j^{*}[\lambda^{3}\rho_{1}]=120[v_{1}]+33[v_{2}]+12[v_{3}]+45[\delta^{-}]+45[\delta^{+}]-2640$.

This gives the matrix with 5 rows and 6 columns which is the matrix of
$j^{*}:V(E_{6})\to V(\mathrm{Spin}(10))$ in the chosen bases for $V(E_{6})$
and $V(\mathrm{Spin}(10))$.

$\begin{pmatrix}0&1&1&17&17&120\\\ 1&0&0&1&1&33\\\ 0&0&0&1&1&12\\\
1&1&0&11&0&45\\\ 1&0&1&0&11&45\\\ \end{pmatrix}$

The proof of part 1 of Lemma 7.0.2 is given by showing that the rank of this
matrix is $2$ and the nullity is $4$. It follows that the transpose has rank
$4$ and nullity $1$. This shows that the kernel of
$j_{*}:V^{*}(\mathrm{Spin}(10))\otimes\mathbb{Q}\to
V^{*}(E_{6})\otimes\mathbb{Q}$ is one dimensional.

The same procedure gives the corresponding matrices for the homomorphisms
$h_{7}:\mathrm{Spin}(12)\to E_{7}$ and $h_{8}:\mathrm{Spin}(16)\to E_{8}$.
However in this case we have to use Magma to calculate Adams operations
information and to soften it to give the linear maps
$h_{7}^{*}:V(E_{7})\otimes\mathbb{Q}\to V(\mathrm{Spin}(12)\otimes\mathbb{Q}$
and $h_{8}^{*}:V(E_{8})\otimes\mathbb{Q}\to
V(\mathrm{Spin}(16))\otimes\mathbb{Q}$.

Using the standard bases defined by fundamental representations, we find that
the matrices of $h_{7}^{*}$ and $h_{8}^{*}$ are, respectively,

$\begin{pmatrix}0&34&220&25840&1890&88&2\\\ 1&0&66&890&56&2&0\\\
0&2&12&1320&34&0&0\\\ 0&0&1&145&24&1&0\\\ 2&12&200&9120&220&0&0\\\
0&2&0&120&210&24&1\\\ \end{pmatrix}$

and

$\begin{pmatrix}160&9056&597840&1005303376&13865488&125552&560&0\\\
-1&-259&-15473&-49227299&-344942&4241&128&1\\\
0&144&4480&17099280&296400&3168&16&0\\\ 1&-3&1808&-1025376&-23023&-240&-1&0\\\
0&16&560&2093696&22048&112&0&0\\\ 0&-2&-105&-370711&3367&121&1&0\\\
16&816&58608&94909584&920192&4368&0&0\\\
-1&15&-5568&-12391764&-22477&4944&119&1\\\ \end{pmatrix}.$

In both cases the rank is $4$. Taking account of the fact that the first is a
$6\times 7$ matrix and the second is an $8\times 8$ matrix this shows that the
nullity of the transpose of the first is $2$ and for the second it is $4$.
This completes the proof.

## References

  * [Ada96] J. F. Adams.  Lectures on exceptional Lie groups.  Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1996.
  * [AH61] M. F. Atiyah and F.E.P. Hirzebruch.  Vector bundles and homogeneous spaces, volume Vol.III of Proc. Symp. Pure Math. Amer. Math. Soc., 1961.
  * [AS69] M. F. Atiyah and G. B. Segal.  Equivariant $K$-theory and completion.  J. Differential Geometry, 3:1–18, 1969.
  * [AT69] M. F. Atiyah and D. O. Tall.  Group representations, $\lambda$-rings and the $J$-homomorphism.  Topology, 8:253–297, 1969.
  * [Ati65] M. F. Atiyah.  On the $K$-theory of compact Lie groups.  Topology, 4:95–99, 1965.
  * [Ati18] M. Atiyah.  K-theory.  CRC Press, 2018.
  * [BH58] A. Borel and F. Hirzebruch.  Characteristic classes and homogeneous spaces. I.  Amer. J. Math., 80:458–538, 1958.
  * [BH59] A. Borel and F. Hirzebruch.  Characteristic classes and homogeneous spaces. II.  Amer. J. Math., 81:315–382, 1959.
  * [BH60] A. Borel and F. Hirzebruch.  Characteristic classes and homogeneous spaces. III.  Amer. J. Math., 82:491–504, 1960.
  * [Bor53] Armand Borel.  Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts.  Ann. of Math. (2), 57:115–207, 1953.
  * [Bot56] Raoul Bott.  An application of the Morse theory to the topology of Lie-groups.  Bull. Soc. Math. France, 84:251–281, 1956.
  * [Bou68] N. Bourbaki.  Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines.  Actualités Scientifiques et Industrielles, No. 1337. Hermann, Paris, 1968.
  * [Bou96] A. K. Bousfield.  On $\lambda$-rings and the $K$-theory of infinite loop spaces.  $K$-Theory, 10(1):1–30, 1996.
  * [BS53] R. Bott and H. Samelson.  On the Pontryagin product in spaces of paths.  Comment. Math. Helv., 27:320–337, 1953.
  * [BW95] D. J. Benson and Jay A. Wood.  Integral invariants and cohomology of $B{\rm Spin}(n)$.  Topology, 34(1):13–28, 1995.
  * [Dua18] Haibao Duan.  The characteristic classes and weyl invariants of spinor groups, 2018\.
  * [FHT01] Y. Felix, S. Halperin, and J.C. Thomas.  Rational Homotopy Theory.  Graduate Texts in Mathematics. Springer New York, 2001.
  * [Gar16] Skip Garibaldi.  $E_{8}$, the most exceptional group.  Bull. Amer. Math. Soc. (N.S.), 53(4):643–671, 2016.
  * [HL11] Yongdong Huang and Naichung Conan Leung.  A uniform description of compact symmetric spaces as Grassmannians using the magic square.  Math. Ann., 350(1):79–106, 2011.
  * [Hod75] Luke Hodgkin.  The equivariant Künneth theorem in $K$-theory.  In Topics in $K$-theory. Two independent contributions, volume Vol. 496 of Lecture Notes in Math., pages 1–101. Springer, Berlin-New York, 1975.
  * [JRT23] John Jones, Dmitriy Rumynin, and Adam Thomas.  On the 32-dimensional rosenfeld projective plane, 2023.
  * [McL79] John McLeod.  The Künneth formula in equivariant $K$-theory.  In Algebraic topology, Waterloo, 1978 (Proc. Conf., Univ. Waterloo, Waterloo, Ont., 1978), volume 741 of Lecture Notes in Math., pages 316–333. Springer, Berlin, 1979.
  * [Min77] Haruo Minami.  $K$-groups of $EIII$ and $FII$.  Osaka Math. J., 14(1):173–177, 1977.
  * [Nak01] Masaki Nakagawa.  The mod 2 cohomology ring of the symmetric space $E{\rm VI}$.  J. Math. Kyoto Univ., 41(3):535–556, 2001.
  * [Pic17] Paolo Piccinni.  On the cohomology of some exceptional symmetric spaces.  In Special metrics and group actions in geometry, volume 23 of Springer INdAM Ser., pages 291–305. Springer, Cham, 2017.
  * [Pit72] Harsh V. Pittie.  Homogeneous vector bundles on homogeneous spaces.  Topology, 11:199–203, 1972.
  * [Ser53] Jean-Pierre Serre.  Groupes d’homotopie et classes de groupes abéliens.  Ann. of Math. (2), 58:258–294, 1953.
  * [Sna72] V. P. Snaith.  On the Künneth formula spectral sequence in equivariant $K$-theory.  Proc. Cambridge Philos. Soc., 72:167–177, 1972.
  * [Ste75] Robert Steinberg.  On a theorem of Pittie.  Topology, 14:173–177, 1975.
  * [SUZ18] Hisham Sati, Silviu-Marian Udrescu, and Erika Zogla.  Computations of characteristic classes and genera: a practical toolkit for beginners and practitioners.  Grad. J. Math., 3(2):60–93, 2018.
  * [Ter11] Svjetlana Terzić.  On real cohomology generators of compact homogeneous spaces.  Sarajevo J. Math., 7(20)(2):277–287, 2011.
  * [Tho62] Emery Thomas.  On the cohomology groups of the classifying space for the stable spinor groups.  Bol. Soc. Mat. Mexicana (2), 7:57–69, 1962.
  * [TW74] Hirosi Toda and Takashi Watanabe.  The integral cohomology ring of ${\bf F}_{4}/{\bf T}$ and ${\bf E}_{6}/{\bf T}$.  J. Math. Kyoto Univ., 14:257–286, 1974.
  * [Woo98] Jay A. Wood.  The Chern character for classical matrix groups.  Proc. Amer. Math. Soc., 126(4):1237–1244, 1998.