The Transition Function of G₂ over S⁶
We obtain explicit formulas for the trivialization functions of the SU(3) principal bundle G₂→S⁶ over two affine charts. We also calculate the explicit transition function of this fibration over the equator of the six-sphere. In this way, we obtain a new proof of the known fact that this fibration c...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2019 |
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| Формат: | Стаття |
| Мова: | Англійська |
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Інститут математики НАН України
2019
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/210310 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | The Transition Function of G₂ over S⁶ / Á. Gyenge // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 9 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860244350013800448 |
|---|---|
| author | Gyenge, Á. |
| author_facet | Gyenge, Á. |
| citation_txt | The Transition Function of G₂ over S⁶ / Á. Gyenge // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 9 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We obtain explicit formulas for the trivialization functions of the SU(3) principal bundle G₂→S⁶ over two affine charts. We also calculate the explicit transition function of this fibration over the equator of the six-sphere. In this way, we obtain a new proof of the known fact that this fibration corresponds to a generator of π₅(SU(3)).
|
| first_indexed | 2025-12-07T21:25:06Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 15 (2019), 078, 16 pages
The Transition Function of G2 over S6
Ádám GYENGE
Mathematical Institute, University of Oxford, UK
E-mail: Adam.Gyenge@maths.ox.ac.uk
Received May 23, 2019, in final form September 26, 2019; Published online October 09, 2019
https://doi.org/10.3842/SIGMA.2019.078
Abstract. We obtain explicit formulas for the trivialization functions of the SU(3) principal
bundle G2 → S6 over two affine charts. We also calculate the explicit transition function of
this fibration over the equator of the six-sphere. In this way we obtain a new proof of the
known fact that this fibration corresponds to a generator of π5(SU(3)).
Key words: G2; six-sphere; octonions; fibration; transition function
2010 Mathematics Subject Classification: 57S15; 55R10; 55R25
1 Introduction
The well-known classification of simple Lie groups shows that G2 is the smallest among the
exceptional types. Further interesting properties and applications of it are numerous. In this
paper we revisit the compact real form of G2 from the viewpoint of differential geometry.
We identify G2 with AutO ⊂ SO(7), the automorphism group of the Cayley octonions. It
is a classical fact that there is a fibration p : G2 → S6, which makes G2 a locally trivial SU(3)-
bundle over S6. It is also known that the principal SU(3)-bundles over S6 are classified by
π5(SU(3)) = Z.
A natural question is that to which element in π5(SU(3)) = Z does the fibration G2 → S6
correspond? In other words, what is the homotopy class of the transition function S5 → SU(3)
of the above fibration, where S5 ⊂ S6 is the equator of the six-sphere?
Theorem 1.1 ([3, Proposition 2]). The class of the transition function of the fibration G2 → S6
is a generator of π5(SU(3)).
Let U1 = S6\{S} and U2 = S6\{N} be two affine charts on S6, where S and N are the south
and north poles. Here we consider S6 as the unit six-sphere
S6 =
{
(0, x2, . . . , x8) ∈ O :
∑
i
|xi|2 = 1
}
in the 7-dimensional vector space of purely imaginary octonions, such that S = (0,−1, 0, . . . , 0)
and N = (0, 1, 0, . . . , 0). Our first result is an explicit formula for the trivialization functions
ψ1 : p−1(U1)→ U1 × SU(3)
and
ψ2 : p−1(U2)→ U2 × SU(3)
deduced in Propositions 3.3 and 3.4 below.
mailto:Adam.Gyenge@maths.ox.ac.uk
https://doi.org/10.3842/SIGMA.2019.078
2 Á. Gyenge
Using the octonion multiplication the linear subspace {x1 = x2 = 0} of O can naturally be
identified with C3. For the precise description of this identification see Section 3.2 below. Let
S5 ⊂ S6 be the equator. This is a (real) submanifold of C3 given as
S5 =
{
(u, v, w) ∈ C3 : |u|2 + |v|2 + |w|2 = 1
}
.
The transition function of the SU(3)-bundle G2 → S6 between the two affine charts of S6 is the
“gluing map” t12 : U1 ∩ U2 → SU(3) for which
ψ1 ◦ ψ−1
2 (ξ, φ) = (ξ, t12(ξ)φ),
where ξ ∈ U1 ∩ U2 and φ ∈ SU(3). Let θ := t12|S5 be the restriction of the transition function
to the equator. Our second main result is an explicit formula for this mapping.
Theorem 1.2. The explicit formula of the transition function of the fibration G2 → S6 between
the two charts
S6 = (U1) ∪ (U2)
over the equator S5 ⊂ S6 is
θ : S5 → SU(3),
uv
w
7→
u2 vu+ w wu− v
uv − w v2 wv + u
uw + v vw − u w2
.
Our result strengthens [3, Corollary 3], where this transition function was obtained up to
homotopy. In particular, our results give a new proof for Theorem 1.1. We expect that having
an explicit formula for the transition function can be useful in several applications. These may
include for example the calculation of Gromov-Witten invariants of G2 and/or S6 as well as
calculations about the classifying stack BG2 (see for example [7, Section 3.2]).
It was noted in [9, Section 2] that the map θ can also be written as
θ : S5 → SU(3), z 7→ zzt +Mz,
where z = (u, v, w)t and
Mz =
0 w −v
−w 0 u
v −u 0
.
It is standard that Mz is the complexification of the usual cross product on the Euclidean 3-
space or, equivalently, of the Spin 1 representation of su(2) ∼= R3. A straightforward computation
reveals that
zztMz = Mzzz
t = 0.
This shows in particular that the matrices zzt and Mz are orthogonal with respect to the
Frobenius inner product
(A,B) 7→ Tr
(
AtB
)
of complex matrices.
The structure of the paper is as follows. In Section 2 we give a brief introduction to the
algebra of Cayley octonions and to several known facts about the group G2. The new results of
the paper are obtained in Section 3.
The Transition Function of G2 over S6 3
2 Some known facts about G2
2.1 Cayley octonions
To perform calculations in the group G2 we collect some known facts about the Cayley algebra
of octonions. We follow [8] where detailed proofs of the following results are given.
Let A be an algebra over the reals. A linear mapping a 7→ ā of A to itself is said to be
a conjugation or involutory antiautomorphism if ¯̄a = a and ab = b̄ā for any elements a, b ∈ A
(the case ā = a is not excluded).
Definition 2.1 (Cayley–Dickson construction [1, 8]). Consider the vector space of the direct
sum of two copies of an algebra with conjugation: A2 = A ⊕ A. A multiplication on A2 is
defined as
(a, b)(u, v) = (au− v̄b, bū+ va).
It is easy to check, that relative to this multiplication the vector space A2 is an algebra of
dimension 2 · dim(A). This is called the doubling of the algebra A.
Remark 2.2. The correspondence a 7→ (a, 0) is a monomorphism of A into A2. Therefore we
will identify elements a and (a, 0) and thus assume A is a subalgebra of A2. If A has an identity
element, then the element 1 = (1, 0) is obviously an identity element in A2.
A distinguished element in A2 is e = (0, 1). It follows from the definition of multiplication
that be = (0, b) and hence (a, b) = a+ be for all a, b ∈ A. Thus every element of the algebra A2
is uniquely written as a+ be. Moreover, the following identities are true:
a(be) = (ba)e, (ae)b = (ab̄)e, (ae)(be) = −b̄a. (2.1)
In particular e2 = −1.
To iterate the Cayley–Dickson construction it is necessary to define a conjugation in A2. This
will be done by the formula
a+ be = ā− be.
This is involutory, R-linear and is simultaneously an antiautomorphism. It is straightforward to
check that if A is a metric algebra, then (a + be)(a+ be) ∈ R and it is obviously positive if a
or b is not 0. Hence, in this case A2 is also metric algebra.
The doubling R2 of the field R is the algebra C of complex numbers and the doubling C2
of C is the algebra of quaternions H. In the latter case e is denoted by j and ie is denoted by k,
and thus a general quaternion is of the form r = r1 + r2i+ r3j+ r4k, where ri ∈ R, i = 1, 2, 3, 4.
Due to the second identity of (2.1), ea = āe for all a ∈ A. Therefore, A2 is not commutative if
the original conjugation is not the identity mapping. In particular H is not commutative, as it
is well known.
The doubling of the algebra of quaternions leads to an 8-dimensional algebra over the reals.
Definition 2.3. The algebra O = H2 is the Cayley algebra, and its elements are called octonions
or Cayley numbers.
By definition every octonion is of the form ξ = a + be, where a and b are quaternions. The
basis of O consists of 1 and seven elements
i, j, k, e, f = ie, g = je, h = ke.
The square of each of these elements is −1, and they are orthogonal to 1. To avoid abusive
use of parentheses, both juxtaposition and dots will be used to denote multiplication in O. The
next lemma gives a list of important properties and identities in O which we will use to prove
our results.
4 Á. Gyenge
Lemma 2.4 ([8]).
1. The algebra O is alternative. That is,
(ab)b = a(bb), a(ab) = (aa)b.
2. The identity of elasticity (or flexibility) holds in O:
(ab)a = a(ba).
3. The algebra O is a normed algebra with the norm generated by the metric. In particular,
it is a division algebra.
4. For all a, x, y ∈ O
ax · y + ay · x = 2〈x, y〉a.
5. For all a, x, y ∈ O
ax · y + ay · x = a · xy + a · yx.
6. For all a, b, x, y ∈ O
〈ax, by〉+ 〈bx, ay〉 = 2〈a, b〉〈x, y〉.
7. The Moufang identity holds in O:
a(bc)a = (ab)(ca).
2.2 G2 and the subgroup SU(3)
The group G2 is defined as the automorphism group AutO of the octonions. It follows from
standard facts on unital normed algebras that G2 ⊂ O(7).
Let O′ ⊂ O be the 7-dimensional subspace of purely imaginary octonions. Consider the subset
of the vector space O′ consisting of elements ξ, such that |ξ| = 1. This set is a 6-dimensional
sphere, which is denoted by S6. An automorphism Φ: O→ O sends the elements i, j and e to
elements ξ = Φi, η = Φj and ζ = Φe in S6 such that η is orthogonal to ξ and ζ is orthogonal
to ξ, η and ξη. The next theorem shows, that these conditions are not only necessary but also
sufficient for the existence of the automorphism Φ.
The statement of the following theorem is classical.
Theorem 2.5 ([8, p. 309]). For any elements ξ, η, ζ ∈ S6 such that
(a) η is orthogonal to ξ,
(b) ζ is orthogonal to ξ, η and ξη
there is a unique automorphism Φ: O→ O for which
ξ = Φi, η = Φj, ζ = Φe.
The Transition Function of G2 over S6 5
Let
p : G2 → S6, Φ 7→ Φi
be the evaluation mapping on i. From Theorem 2.5 it follows that the group G2 = AutO acts
transitively on S6, i.e., the mapping p is surjective. Let us denote by K the stabilizer (isotropy)
group of i under the action of G2. Equivalently,
K = {Φ: O→ O |Φi = i} = p−1(i)
is the fiber of p over i. Due to the standard theorem [6, Theorem 9.24] of transitive Lie group
actions
G2/K ≈ S6.
Lemma 2.6. There is a canonical isomorphism K ∼= SU(3).
Proof. The subspace V = Span{1, i}⊥ of the algebra O is closed under the multiplication
by i and thus it can be considered as a vector space over the field C with basis j, e, g. The
Hermitian product in O induces in V a Hermitian product with respect to which the basis j, e, g
is orthogonal. Any automorphism Φ: O → O which leaves the element i fixed, i.e., which is
in the subgroup K, defines an operator V → V linear over C. This operator preserves the
Hermitian product, and therefore it is an unitary operator.
The elements of the group SU(3) are 3×3 matrices of the form [v1|v2|v3] consisting of complex
orthogonal column vectors having unit length and where v3 is the element in the subspace
SpanC{v1, v2}⊥ ≈ C such that the determinant of the matrix is 1. One can show that the third
column is determined by the first two. For a particular Φ ∈ p−1(i), the vectors η = Φ(j) and
ζ = Φ(e) are perpendicular to i and complex orthogonal to each other. Thus, they can be
thought as the first and second column of such a matrix and in this case the third column will
be ηζ = Φ(j)Φ(e) = Φ(je) = Φ(g). Combining this with Theorem 2.5 it follows that K coincides
with SU(3). �
As a consequence, we have that SU(3) ⊂ G2 and G2/SU(3) ≈ S6.
Corollary 2.7. Consider the evaluation mapping p : G2 → S6, Φ 7→ Φi defined above. This
makes G2 a locally trivial SU(3)-bundle over S6.
2.3 The subgroup of inner automorphisms
In an associative division algebra, such as the quaternions over the reals, the mapping
qr : x 7→ rxr−1
is always an automorphism for any invertible element r, which is called an inner automorphism.
In a non-associative algebra it is not always true that
(rx)r−1 = r
(
xr−1
)
, for all x, r.
Moreover, not every invertible element generate an inner automorphism. Still, in the case of
the octonions a well defined linear transformation associated with an element r can be defined
because of the following lemma.
Lemma 2.8. For any r, x ∈ O
(rx)r−1 = r
(
xr−1
)
.
6 Á. Gyenge
Proof. If the coordinates of r in the standard basis are (r1, . . . , r8), then r−1 = r̄
|r|2 = 2r1−r
|r|2 .
Therefore, using Lemma 2.4(2) we have
(rx)r−1 = (rx)
2r1 − r
|r|2
=
1
|r|2
((rx)2r1 − (rx)r) =
1
|r|2
(r(x2r1)− r(xr)) = r
(
xr−1
)
. �
The following result classifies those elements r for which the linear map qr is an automorphism
of O. For completeness, we reproduce its original proof.
Theorem 2.9 ([5]). A non-real octonion r with coordinates (r1, . . . , r8) induces an inner auto-
morphism of O if and only if 4r2
1 = |r|2.
Proof. From Lemma 2.4(7) for a = r, b = xr−1 and c = ryr it follows that(
rxr−1
)
(ryr · r) = r
(
xr−1 · ryr
)
r. (2.2)
Similarly,
ryr = r̄ȳr̄ = r̄
(
ȳ
(
x−1x
))
r̄ = r̄
((
ȳx−1
)
x
)
r̄ =
(
r̄ · ȳx−1
)
(xr̄)
and therefore
ryr =
(
r̄ · ȳx−1
)
(xr̄) = (xr̄)
(
r̄ · ȳx−1
)
= (rx̄)
(
ȳx−1 · r
)
= (rx̄)
(
x−1y · r
)
=
(
r
(
|x|2x−1
))( x
|x|2
y · r
)
=
(
rx−1
)
(xy · r).
Substituting this into (2.2) leads us to(
rxr−1
)(
ryr · r
)
= r
(
xr−1 ·
(
rx−1
)
(xy · r)
)
r = r
((
xr−1︸ ︷︷ ︸
a
· rx−1︸ ︷︷ ︸
a−1
)
· (xy · r)
)
r
= r((xy · r))r = r(xy)r2,
i.e., (
rxr−1
)(
ryr−1 · r3
)
= r(xy)r−1 · r3 (2.3)
for all x, y, r ∈ O.
The mapping qr : x 7→ rxr−1 is an automorphism if and only if(
rxr−1
)(
ryr−1
)
= r(xy)r−1.
Multiplying this with r3 from the right we get(
rxr−1
)(
ryr−1
)
· r3 = r(xy)r−1 · r3. (2.4)
Comparing (2.3) with (2.4) we see that in order for qr to be an automorphism r3 must be
a scalar.
Using the fact that r̄ = 2r1 − r, one has |r|2 = rr̄ = r(2r1 − r) = 2rr1 − r2 for all r ∈ O.
Multiplying with r and applying the same equation again we get that
r3 − 2r1r
2 + |r|2r = r3 − 4r2
1r + 2r1|r|2 + r|r|2 = 0,
and thus
r3 + 2r1|r|2 = r
(
4r2
1 − |r|2
)
.
Suppose r3 is a scalar. Then each term on the left side is real and therefore either r should be
real, or
(
4r2
1 − |r|2
)
should be zero. The latter case means that 4r2
1 = |r|2. �
The Transition Function of G2 over S6 7
3 G2 as an SU(3)-bundle over S6
3.1 The trivialization functions
Our aim is to determine the transition function of the fibration
p : G2 → S6, Φ 7→ Φi
between two charts of S6 given by U1 = S6\{S} and U2 = S6\{N}, where S = −i =
(0,−1, 0, . . . , 0) and N = i = (0, 1, 0, . . . , 0). The preimage of i is the set p−1(i) = {(i, η, ζ) : η ⊥
i, ζ ⊥ Span{i, η, iη}}. As mentioned above this is isomorphic to SU(3) and this isomorphism
will be called θi.
For any ξ ∈ S6 let us denote by Vξ or TξS
6 the tangent space (of orthogonal vectors) to ξ.
By the considerations above elements in p−1(i) can be considered either as orthonormal vector
triples in Vi = TiS
6 or as operators that leave the vector i fixed. It also follows from the result
above that there is a complex structure
Ji : Vi → Vi, v 7→ iv.
This is clearly a mapping from Vi to itself such that J2
i (v) = i2v = −v for all v ∈ V . Thus,
there is an isomorphism θi : Vi → C3 that assigns to each operator Φ ∈ p−1(i), Φ: Vi → Vi its
matrix representation in the complex basis {j, e, g}.
Similarly, p−1(ξ) = {(ξ, η, ζ) : η ⊥ ξ, ζ ⊥ Span{ξ, η, ξη}} for any ξ ∈ S6. Any map ϕ ∈ p−1(ξ)
carries Vi to Vξ. Again, there is a complex structure on Vξ denoted by Jξ, which comes from
octonion multiplication: Jξ(v) = ξv. By choosing a complex orthonormal basis in this subspace
we give an identification Vξ ≈ C3. These considerations imply the following classical result.
Corollary 3.1 ([4]). The complex structure given by Jξ : Vξ → Vξ, v 7→ ξv defines a smooth
almost complex structure J : TS6 → TS6, (ξ, v) 7→ (ξ, Jξ(v)).
This almost complex structure has the following remarkable property.
Proposition 3.2. A rotation g : S6 → S6 is an element of G2 if and only if its pushforward
g∗ : TS6 → TS6, (x, v) 7→ (g(x), g(v)) is J-equivariant (where J is considered as a Z4-action
on TS6), or, in other words, if the following diagram is commutative:
TS6 TS6
TS6 TS6.
J
g∗ g∗
J
Proof. Because G2 ⊂ O(7), any g ∈ G2 preserves the scalar product. Therefore, g(Vξ) = Vg(ξ)
and we need only to prove that the following diagram commutes for all ξ ∈ S6:
TξS
6 TξS
6
Tg(ξ)S
6 Tg(ξ)S
6.
Jξ
g∗ g∗
Jg(ξ)
8 Á. Gyenge
Since g ∈ AutO we have that
g(Jξ(η)) = g(ξη) = g(ξ)g(η) = Jg(ξ)(g(η)),
for all ξ ∈ S6, η ∈ Vξ.
Conversely, assume ξ ∈ S6, η ∈ O′. Decompose η to η1 + η2 where η1 ⊥ ξ. Suppose g∗
commutes with J . Then
g(ξη1) = g(Jξ(η1)) = Jg(ξ)(g(η1)) = g(ξ)g(η1),
and obviously g(ξη2) = g(ξ)g(η2). Thus, g(ξη) = g(ξ)g(η). �
Proposition 3.3. The trivialization map over U1 is given by
ψ1 : p−1(U1)→ U1 × SU(3), ϕ 7→ (ϕ(i), θϕ(i)(ϕ)),
where ϕ(i) is the image of i under ϕ and θϕ(i)(ϕ) is given by (3.1) below.
Proof. In the proof of Lemma 2.6 it was shown, that for a particular Φ ∈ p−1(i) the vectors
η = Φ(j) and ζ = Φ(e) are perpendicular to i and complex orthogonal to each other. Thus,
they can be thought as the first and second columns of a matrix in SU(3) with the third column
ηζ = Φ(j)Φ(e) = Φ(je) = Φ(g). If the coordinates of the vectors are η = (0, y2, . . . , y8),
ζ = (0, z2, . . . , z8) and ηζ = (0, u2, . . . , u8), then since η, ζ, ηζ ∈ Vi we have that y2 = 0, z2 = 0
and u2 = 0. The mapping θi is then the following:
θi : p−1(i)→ SU(3), (i, η, ζ) 7→
y3 + Iy4 z3 + Iz4 u3 + Iu4
y5 + Iy6 z5 + Iz6 u5 + Iu6
y7 + Iy8 z7 + Iz8 u7 + Iu8
.
Here I is the imaginary unit in the field C3 and not the octonion i. It follows that
(i, η, ζ) 7→
〈η, j〉+ I〈η, k〉 〈ζ, j〉+ I〈ζ, k〉 〈ηζ, j〉+ I〈ηζ, k〉
〈η, e〉+ I〈η, f〉 〈ζ, e〉+ I〈ζ, f〉 〈ηζ, e〉+ I〈ηζ, f〉
〈η, g〉+ I〈η, h〉 〈ζ, g〉+ I〈ζ, h〉 〈ηζ, g〉+ I〈ηζ, h〉
,
i.e., we represent η, ζ, ηζ ∈ Vi, the images of j, e and g in the complex basis {j, e, g}.
As a consequence, for any ξ ∈ U1 and any ϕ ∈ p−1(ξ), ϕ restricts to a mapping Vi → Vξ,
which is complex linear, unitary and has determinant 1. We will choose a complex orthonormal
basis in Vξ and write the images of j, e and g in this basis. That is, we choose particular
identifications Vi ≈ C3, Vξ ≈ C3 and we define θξ : p−1(ξ) → SU(3) by assigning to each
automorphism ϕ ∈ p−1(ξ) the matrix of the mapping ϕ : C3 → C3. To find a basis in Vξ we will
define a translating automorphism Qξ such that Qξ(i) = ξ. Then, for a = Qξ(j), b = Qξ(e) and
c = Qξ(g) the set of vectors {a, b, c} is a complex orthonormal basis in Vξ with respect to the
complex structure Jξ(v) = ξv. Particularly,
θξ : p−1(ξ)→ SU(3),
(ξ, η, ζ) 7→
〈η, a〉+ I〈η, Jξ(a)〉 〈ζ, a〉+ I〈ζ, Jξ(a)〉 〈ηζ, a〉+ I〈ηζ, Jξ(a)〉
〈η, b〉+ I〈η, Jξ(b)〉 〈ζ, b〉+ I〈ζ, Jξ(b)〉 〈ηζ, b〉+ I〈ηζ, Jξ(b)〉
〈η, c〉+ I〈η, Jξ(c)〉 〈ζ, c〉+ I〈ζ, Jξ(c)〉 〈ηζ, c〉+ I〈ηζ, Jξ(c)〉
. (3.1)
Using this the trivializing map is given by
ψ1 : p−1(U1)→ U1 × SU(3), ϕ 7→ (ϕ(i), θϕ(i)(ϕ)). �
The Transition Function of G2 over S6 9
Completely analogously the preimage of −i under the evaluation map p is diffeomorphic
to SU(3), and in this case the complex structure on V−i is given by J−i(v) = −iv. Therefore,
θ̃−i is defined as
(−i, η, ζ) 7→
〈η, j〉+ I〈η,−k〉 〈ζ, j〉+ I〈ζ,−k〉 〈ηζ, j〉+ I〈ηζ,−k〉
〈η, e〉+ I〈η,−f〉 〈ζ, e〉+ I〈ζ,−f〉 〈ηζ, e〉+ I〈ηζ,−f〉
〈η, g〉+ I〈η,−h〉 〈ζ, g〉+ I〈ζ,−h〉 〈ηζ, g〉+ I〈ηζ,−h〉
.
As we did in the previous case, for a general ξ ∈ U2 = S6\{N} we will choose a translating
automorphism Q̃ξ with the property that Q̃ξ(−i) = ξ implying that Q̃ξ(j), Q̃ξ(e), Q̃ξ(g) ∈ Vξ
form a complex orthonormal basis. Then we define θ̃ξ : p−1(ξ) → SU(3) by assigning to
ϕ ∈ p−1(v) the matrix of the corresponding linear mapping from V−i onto Vξ written in the
bases {j, e, g} at V−i and
{
ã, b̃, c̃
}
:=
{
Q̃ξ(j), Q̃ξ(e), Q̃ξ(g)
}
at Vξ. Similarly as in the proof
Proposition 3.3 we obtain the following morphism
θ̃ξ : p−1(ξ)→ SU(3),
(ξ, η, ζ) 7→
〈η, ã〉+ I〈η, Jξ(ã)〉 〈ζ, ã〉+ I〈ζ, Jξ(ã)〉 〈ηζ, ã〉+ I〈ηζ, Jξ(ã)〉
〈η, b̃〉+ I〈η, Jξ(b̃)〉 〈ζ, b̃〉+ I〈ζ, Jξ(b̃)〉 〈ηζ, b̃〉+ I〈ηζ, Jξ(b̃)〉
〈η, c̃〉+ I〈η, Jξ(c̃)〉 〈ζ, c̃〉+ I〈ζ, Jξ(c̃)〉 〈ηζ, c̃〉+ I〈ηζ, Jξ(c̃)〉
. (3.2)
As a consequence, the analogue of Proposition 3.3 is true for this chart.
Proposition 3.4. The trivialization map over U2 is then given by
ψ2 : p−1(U2)→ U2 × SU(3), ϕ 7→ (ϕ(i), θ̃ϕ(i)(ϕ)),
where ϕ(i) is the image of i under ϕ and θ̃ϕ(i)(ϕ) is given by (3.2).
To summarize, if Qξ, Q̃ξ ∈ G2 are known as functions depending differentiably on ξ with
the property that Qξ(i) = ξ and Q̃ξ(−i) = ξ, then an appropriate basis in Vξ is a = Qξ(j),
b = Qξ(e), c = Qξ(g), which are the translations of the basis j, e, g from Vi in the case of the
first chart. In the case of the second chart Q̃ξ translates j, e, g from V−i to Vξ. Thus, we need to
find elements Qξ ∈ G2 and Q̃ξ ∈ G2. Knowing the first one is enough, because then the second
is given due to the identities Q−ξ(−i) = Q−ξ((−1)i) = Q−ξ(−1)Q−ξ(i) = −1(−ξ) = ξ.
It will be convenient to look for Qξ in the form of an inner automorphism generated by an
element r ∈ O. The easiest is to look for a unit length octonion that induces Qξ. For a unit
length octonion r the conjugate of i with r is
rir̄ =
(
0, r2
1 + r2
2 − r2
3 − r2
4 − r2
5 − r2
6 − r2
7 − r2
8, 2(r2r3 + r1r4), 2(r2r4 − r1r3),
2(r2r5 + r1r6), 2(r2r6 − r1r5), 2(r2r7 − r1r8), 2(r1r7 + r2r8)
)
.
Since rξir̄ξ = ξ = (0, x2, . . . , x8) is needed, the following system of equations is to be solved
r2
1 + r2
2 − r2
3 − r2
4 − r2
5 − r2
6 − r2
7 − r2
8 = x2,
2(r2r3 + r1r4) = x3,
2(r2r4 − r1r3) = x4,
2(r2r5 + r1r6) = x5,
2(r2r6 − r1r5) = x6,
2(r2r7 − r1r8) = x7,
2(r1r7 + r2r8) = x8.
10 Á. Gyenge
From Theorem 2.9 it follows that r1 = 1
2 is required. The general solution for a fixed ξ ∈ U1 of
this system of equations is
rξ =
1
2
(
1,
√
1 + 2x2,
x3
√
1 + 2x2 − x4
1 + x2
,
x3 + x4
√
1 + 2x2
1 + x2
,
x5
√
1 + 2x2 − x6
1 + x2
,
x5 + x6
√
1 + 2x2
1 + x2
,
x7
√
1 + 2x2 + x8
1 + x2
,
−x7 + x8
√
1 + 2x2
1 + x2
)
. (3.3)
3.2 The transition function over the equator
As in the previous sections we cover the base space S6 with two trivializing charts given by
U1 = S6\{S} and U2 = S6\{N}. We are interested in the transition function between the
two trivializations over the equator. This is enough to reconstruct the whole fibration, since
the equator is a deformation retract of the intersection of the charts. The equator S5 will be
identified with a submanifold of C3 = Vi = V−i as
S5 =
{
(u, v, w) ∈ C3 | |u|2 + |v|2 + |w|2 = 1
}
,
where the coordinate functions u, v and w are the duals of j, e and g respectively. We are now
ready to prove Theorem 1.2 which we restate here.
Theorem 3.5. The transition function between the two trivializations of the principal SU(3)-
bundle G2 → S6 at the equator is
θ : S5 → SU(3),
uv
w
7→
u2 vu+ w wu− v
uv − w v2 wv + u
uw + v vw − u w2
.
From now on we assume that any ξ ∈ O is in the equator of S6, and thus x2 = 0. In this
case the solution (3.3) simplifies to
rξ =
1
2
(1, 1, x3 − x4, x3 + x4, x5 − x6, x5 + x6, x7 + x8,−x7 + x8).
Due to the fact that iξ = (0, 0,−x4, x3,−x6, x5, x8,−x7) we have
rξ =
1
2
+
i
2
+
ξ + iξ
2
=
(1 + i)(1 + ξ)
2
.
It is easy to check that rξ is really a solution, because in this case due to Lemmas 2.4(2) and 2.8
we may perform the multiplication in arbitrary order:
(1 + i)(1 + ξ)
2
· i · (1 + i)(1 + ξ)
2
=
1
4
(1 + i)((1 + ξ)i(1− ξ))(1− i)
=
1
4
(1 + i)(i+ ξi− iξ − ξiξ)(1− i) =
1
4
(1 + i)
(
i+ 2ξi+ iξ2
)
(1− i) =
1
4
(1 + i)2ξi(1− i)
=
1
4
(
2ξi+ 2iξi− 2ξi2 − 2iξi2
)
=
1
4
(
2ξi− 2i2ξ + 2ξ + 2iξ
)
=
1
4
(2ξi+ 4ξ − 2ξi) =
4ξ
4
= ξ.
Consequently, the required automorphisms for an arbitrary ξ ∈ U1 ∩ U2 are
Qξ : O→ O, x 7→ rξxr̄ξ,
Q̃ξ : O→ O, x 7→ r−ξxr̄−ξ.
The Transition Function of G2 over S6 11
Once again, the transition function between the two trivializations is
ψ1 ◦ ψ−1
2 : U1 ∩ U2 × SU(3)→ U1 ∩ U2 × SU(3), (ξ, φ) 7→
(
ξ, θξ ◦ θ̃−1
ξ (φ)
)
.
As it was discussed above, the meaning of ψ1 is the following:
(ξ, η, ζ) 7→
〈η,Qξj〉+ I〈η,Qξk〉 〈ζ,Qξj〉+ I〈ζ,Qξk〉 〈ηζ,Qξj〉+ I〈ηζ,Qξk〉
〈η,Qξe〉+ I〈η,Qξf〉 〈ζ,Qξe〉+ I〈ζ,Qξf〉 〈ηζ,Qξe〉+ I〈ηζ,Qξf〉
〈η,Qξg〉+ I〈η,Qξh〉 〈ζ,Qξg〉+ I〈ζ,Qξh〉 〈ηζ,Qξg〉+ I〈ηζ,Qξh〉
.
Similarly, ψ2 is
(ξ, η, ζ) 7→
〈η, Q̃ξj〉+ I〈η, Q̃ξk〉 〈ζ, Q̃ξj〉+ I〈ζ, Q̃ξk〉 〈ηζ, Q̃ξj〉+ I〈ηζ, Q̃ξk〉
〈η, Q̃ξe〉+ I〈η, Q̃ξf〉 〈ζ, Q̃ξe〉+ I〈ζ, Q̃ξf〉 〈ηζ, Q̃ξe〉+ I〈ηζ, Q̃ξf〉
〈η, Q̃ξg〉+ I〈η, Q̃ξh〉 〈ζ, Q̃ξg〉+ I〈ζ, Q̃ξh〉 〈ηζ, Q̃ξg〉+ I〈ηζ, Q̃ξh〉
.
The mapping Qξ(v) = rξvr̄ξ is linear in v, because O is distributive and scalars commute with
everything. Due to the construction Qξ(x) maps the subspace Vi to Vξ isomorphically.
Lemma 3.6. If v, ξ ∈ Vi, then
Qξ(v) =
1
2
((−1 + i+ ξ + iξ)v + 〈v, ξ + iξ〉(1 + i+ ξ + iξ)).
Proof. To compute Qξ(v), four groups of identities will be necessary.
(i) According to the definition of the scalar product in O and Lemma 2.4(4)
v · iξ = −iξ · v + 2〈v, iξ〉 = ξi · v + 2〈v, ξi〉 = −iξ · v − 2〈v, iξ〉,
iv · ξ = −iξ · v + 2〈v, ξ〉i = −iξ · v − 2〈v, ξ〉i,
ξv · i = −ξi · v + 2 〈i, v〉︸ ︷︷ ︸
0
ξ = iξ · v.
Therefore
iξ · v − ξv · i− iv · ξ − v · iξ = 2iξ · v + 2〈v, ξ〉i+ 2〈v, iξ〉. (3.4)
(ii) Similarly,
iv · iξ = −(i · iξ)v + 2〈iξ, v〉i = (i · ξi)v + 2〈ξi, v〉i = ξv + 2〈ξi, v〉i,
(iξ · v)i = −(iξ · i)v + 2 〈v, i〉︸ ︷︷ ︸
0
iξ = −(iξi)v = (iiξ)v = −ξv.
Summing over the two equations this leads to
iv · iξ + (iξ · v)i = ξv + 2〈ξi, v〉i− ξv = 2〈ξi, v〉i. (3.5)
(iii) With essentially the same tricks one obtains
(iξ · v)ξ = −(iξ · ξ)v + 2〈v, ξ〉iξ = iv − 2〈v, ξ〉iξ,
ξv · iξ = −ξiξ · v + 2〈v, iξ〉ξ = −(ξ · xi)v − 2〈v, iξ〉ξ = −iv − 2〈v, iξ〉ξ.
Therefore
(iξ · v)ξ + ξv · iξ = iv − 2〈v, ξ〉iξ − iv − 2〈v, iξ〉ξ = −2〈v, ξ〉iξ − 2〈v, iξ〉ξ. (3.6)
12 Á. Gyenge
(iv) Once again,
ξvξ = −ξξ · v + 2〈ξ, v〉ξ = v − 2〈ξ, v〉ξ, (3.7)
iξ · v · iξ = −(iξ)iξ · v + 2〈iξ, v〉iξ = v − 2〈iξ, v〉iξ. (3.8)
Putting these together,
Qξ(v) = rξvr̄ξ =
1
4
(1 + i+ ξ + iξ)v(1− i− ξ − iξ)
=
1
4
(v + iv + ξv + iξ · v)(1− i− ξ − iξ)
=
1
4
(v + iv + ξv + iξ · v − vi− ivi− ξv · i− (iξ · v)i
− vξ − iv · ξ − ξvξ − (iξ · v)ξ − v · iξ − iv · iξ − ξv · iξ − iξ · v · iξ)
=
1
4
(2iv + ξv − vξ + (iξ · v − ξv · i− iv · ξ − v · iξ)
− ((iξ · v)i+ iv · iξ)− ((iξ · v)ξ + ξv · iξ)− ξvξ − iξ · v · iξ)
=
1
4
(2iv − 2v + 2iξ · v + ξv − vξ + 2〈v, ξ〉i+ 2〈v, iξ〉
− 2〈ξi, v〉i+ 2〈v, ξ〉iξ + 2〈v, iξ〉ξ + 2〈ξ, v〉ξ + 2〈iξ, v〉iξ)
=
1
4
(2iv − 2v + 2iξ · v + 2ξv + 2〈v, ξ〉+ 2〈v, ξ〉i+ 2〈v, iξ〉
− 2〈ξi, v〉i+ 2〈v, ξ〉iξ + 2〈v, iξ〉ξ + 2〈ξ, v〉ξ + 2〈iξ, v〉iξ)
=
1
2
(iv − v + iξ · v + ξv + (〈v, ξ〉+ 〈v, iξ〉)(1 + i+ ξ + iξ))
=
1
2
((−1 + i+ ξ + iξ)v + 〈v, ξ + iξ〉(1 + i+ ξ + iξ)),
where in the sixth equality the formulas (3.4), (3.5), (3.6), (3.7) and (3.8) were used, while in
seventh equality the rule vξ = −ξv − 2〈v, ξ〉 was applied. �
Using Lemma 3.6 the inverse function Q−1
ξ : Vξ → Vi can be calculated as well by observing
that the roles of i and ξ are played by −ξ and −i respectively. Taking into account that any
v ∈ Vξ is perpendicular to ξ, essentially the same calculation leads to
Q−1
ξ (v) = r̄ξvrξ =
1
4
(1− i− ξ − iξ)v(1 + i+ ξ + iξ)
=
1
4
(1 + (−i) + (−ξ) + (−ξ)(−i))v(1− (−i)− (−ξ)− (−ξ)(−i))
= (−1− ξ − i+ ξi)v + (〈v,−i+ ξi〉)(1− ξ − i+ ξi).
Moreover,
Q−1
−ξ(v) = (−1 + ξ − i− ξi)v + (〈v,−i− ξi〉)(1 + ξ − i− ξi).
Lemma 3.7. If v, ξ ∈ Vi, then
Q−1
−ξ ◦Qξ(v) = vξ − 〈vξ, 1〉(1 + ξ)− 〈vξ, i〉(1 + ξ)i.
Proof. To calculate Q−1
−ξ ◦Qξ(v) for an arbitrary v ∈ Vi more preparation is needed.
(i) Applying Lemma 2.4(5) we obtain
ξ(iξ · v) + iξ · ξv = ξiξ · v + iξξ · v = iv − iv = 0. (3.9)
The Transition Function of G2 over S6 13
(ii) By changing the order of terms in the multiplications one obtains
i · ξv = vξ · i− 2〈ξv, i〉 = −vi · ξ − 2〈ξv, i〉,
ξ · iv = vi · ξ − 2〈iv, ξ〉.
Using Lemma 2.4(6) and the definition of multiplication it can be proved, that
2〈ξv, i〉 − 2〈iv, ξ〉 = 4〈iξ, v〉.
Therefore,
ξ · iv − i · ξv = 2vi · ξ + 2〈ξv, i〉 − 2〈iv, ξ〉 = 2vi · ξ + 4〈iξ, v〉
= −2iv · ξ + 4〈iξ, v〉 = 2iξ · v + 4〈v, ξ〉i+ 4〈iξ, v〉, (3.10)
and thus
−2iξ · v + ξ · iv − i · ξv = 4〈v, ξ〉i+ 4〈iξ, v〉. (3.11)
(iii) By exchanging ξ with iξ in (3.10) one has
iξ · iv − i(iξ · v) = 2iiξv + 4〈v, iξ〉i+ 4〈iiξ, v〉 = −2ξv + 4〈v, iξ〉i− 4〈ξ, v〉. (3.12)
(iv) If a, b ∈ O′ and a ⊥ b, then ab is orthogonal to both a and b. Thus
〈1 + i+ ξ + iξ,−i+ iξ〉 = 0− 1 + 0 + 1 = 0. (3.13)
(v) Finally, taking into account again the orthogonality assumptions and Lemma 2.4(6)
〈iξ · v, i〉 = −〈ξi · v, i〉 = 〈iv, ξi〉+ 2 〈i, ξi〉〈v, 1〉︸ ︷︷ ︸
0
= −〈iv, iξ〉 = −〈v, ξ〉.
This leads to
〈(−1 + i+ ξ + iξ)v,−i+ iξ〉
= 〈−v,−i+ iξ〉︸ ︷︷ ︸
〈−v,iξ〉
+ 〈iv,−i+ iξ〉︸ ︷︷ ︸
〈iv,iξ〉
+〈ξv,−i+ iξ〉+ 〈iξ · v,−i+ iξ〉︸ ︷︷ ︸
〈iξ·v,−i〉
= 〈−v, iξ〉+ 〈iv, iξ〉︸ ︷︷ ︸
〈v,ξ〉
−〈ξv,−i〉︸ ︷︷ ︸
−〈v,ξi〉
−〈ξv, ξi〉︸ ︷︷ ︸
〈v,i〉=0
−(−〈v, ξ〉) = 2〈v, ξi〉+ 2〈v, ξ〉. (3.14)
To simplify calculation it is useful to get rid of the constant factor. According to Lemma 3.6
we have
4Q−1
−ξ ◦Qξ(v) = (−1− i+ ξ + iξ)((−1 + i+ ξ + iξ)v + 〈v, ξ + iξ〉(1 + i+ ξ + iξ))
+ 〈(−1 + i+ ξ + iξ)v + (〈v, ξ + iξ〉)(1 + i+ ξ + iξ),−i− ξi〉
× (1− i+ ξ + iξ)
= v − iv − ξv − iξ · v + 〈v, ξ + iξ〉(−1− i− ξ − iξ)
+ iv − i2v − i · ξv − i(iξ · v) + 〈v, ξ + iξ〉
(
−i− i2 − iξ − i2ξ
)
− ξv + ξ · iv + ξ2v + ξ(iξ · v) + 〈v, ξ + iξ〉
(
ξ + ξi+ ξ2 + ξiξ
)
− iξ · v + iξ · iv + iξ · ξv + (iξ)2v + 〈v, ξ + iξ〉
(
iξ + iξi+ iξ2 + (iξ)2
)
+
[
〈(−1 + i+ ξ + iξ)v,−i+ iξ〉
+ 〈v, ξ + iξ〉〈1 + i+ ξ + iξ,−i+ iξ〉
]
(1− i+ ξ + iξ)
14 Á. Gyenge
= −2ξv + (−2iξ · v + ξ · iv − i · ξv)
+ (iξ · iv − i(iξ · v)) + (ξ(iξ · v) + iξ · ξv)
+ 2〈v, ξ + iξ〉(−1− i+ ξ + ξi)
+ 〈(−1 + i+ ξ + iξ)v,−i+ iξ〉(1 + ξ − i+ iξ)
= −4ξv − 4〈ξ, v〉+ 4〈v, ξ〉i+ 4〈iξ, v〉+ 4〈v, iξ〉i
+ 2(〈v, ξ + iξ〉)(−1− i+ ξ + ξi) + 2(〈v, ξ + iξ〉)(1 + ξ − i+ iξ)
= −4ξv + 〈ξ, v〉(−4− 2 + 2 + 4i− 2i− 2i+ 2ξ + 2ξ + 2ξi+ 2iξ)
+ 〈iξ, v〉(4− 2− 2 + 4i− 2i+ 2i+ 2ξ − 2ξ + 2ξi− 2iξ)
= −4ξv + 〈ξ, v〉(−4 + 4ξ) + 〈iξ, v〉(4i+ 4ξi)
= 4vξ + 〈ξ, v〉(4 + 4ξ) + 〈iξ, v〉(4i+ 4ξi),
where in the fourth equality the formulas (3.9), (3.11), (3.12), (3.13) and (3.14) were used. To
sum it up, the required transformation is given by
Q−1
−ξ ◦Qξ(v) = vξ + 〈ξ, v〉(1 + ξ) + 〈iξ, v〉(1 + ξ)i
= vξ − 〈vξ, 1〉(1 + ξ)− 〈vξ, i〉(1 + ξ)i. �
Proof of Theorem 3.5. As mentioned earlier, the subspace Vi is a complex linear space with
basis j, e, g and complex structure Ji : Vi → Vi, v 7→ iv. Since ξ ∈ Vi, the coordinate expression
of ξ in Vi can be written as
ξ = uj + ve+ wg = (u1 + u2I)j + (v1 + v2I)e+ (w1 + w2I)g
= u1j + u2k + v1e+ v2f + w1g − w2h,
where u, v, w ∈ C, ui, vi, wi ∈ R for i = 1, 2, and I is again the imaginary unit in the field C.
Because Vi = V−i as a subspace, ξ can be expressed as a element of V−i as well. Here the basis
is the same, but the complex structure is given by J−i : V−i → V−i, v 7→ −iv. Therefore, the
coordinate expression of the same ξ here is
ξ = uj + ve+ wg = (u1 − u2I)j + (v1 − v2I)e+ (w1 − w2I)g.
According to the multiplication rule of the basis vectors of O it is possible to compute the
multiplication of ξ with the basis vectors from the left as
jξ = −u1 + u2i+ v1g + v2h− w1e+ w2f = −u · 1 + 0j − we+ vg,
eξ = −u1g − u2h− v1 + v2i+ w1j − w2k = −v · 1 + wj + 0e− ug,
gξ = u1e− u2f − v1j + v2k − w1 + w2i = −w · 1− vj + ue+ 0g,
because the resulting vector v, of which the terms are calculated here, is in V−i. Similarly,
ξi = −u1k + u2j − v1f + v2e+ w1h+ w2g
= (u2 + u1I)j + (v2 + v1I)e+ (w2 + w1I)g
= (uI)j + (vI)e+ (wI)g.
Using Lemma 3.7 we get
Q−1
−ξ ◦Qξ(j) = jξ − 〈jξ, 1〉(1 + ξ)− 〈jξ, i〉(1 + ξ)i
=
0
−w
v
+ u1
uv
w
− u2
uIvI
wI
=
0
−w
v
+ (u1 − u2I)︸ ︷︷ ︸
u
uv
w
=
u2
uv − w
uw + v
,
The Transition Function of G2 over S6 15
Q−1
−ξ ◦Qξ(e) = eξ − 〈eξ, 1〉(1 + ξ)− 〈eξ, i〉(1 + ξ)i
=
w
0
−u
+ v1
uv
w
− v2
uIvI
wI
=
w
0
−u
+ (v1 − v2I)︸ ︷︷ ︸
v
uv
w
=
vu+ w
v2
vw − u
,
Q−1
−ξ ◦Qξ(g) = gξ − 〈gξ, 1〉(1 + ξ)− 〈gξ, i〉(1 + ξ)i
=
−vu
0
+ w1
uv
w
− w2
uIvI
wI
=
−vu
0
+ (w1 − w2I)︸ ︷︷ ︸
w
uv
w
=
wu− vwv + u
w2
.
Putting all together, the matrix which represents the mapping Q−1
−ξ ◦Qξ : Vi → V−i is
Mξ =
u2 vu+ w wu− v
uv − w v2 wv + u
uw + v vw − u w2
,
and to get matrix of the same function as a V−i → V−i mapping each complex coordinate of ξ
should be conjugated:
Mξ =
u2 vu+ w wu− v
uv − w v2 wv + u
uw + v vw − u w2
.
This proves the statement. �
3.3 The class of G2
As it is known the principal SU(3)-bundles over S6 are classified by π5(SU(3)). The following
fact is well known, but again we included a sketch proof of it.
Proposition 3.8. π5(SU(3)) = Z.
Sketch proof. From the well-known periodicity theorem of Bott [2] it follows that π5(SU(4))
= Z. It can be shown as well that SU(4) = Spin(6). By definition Spin(6) is the double cover
of SO(6). A covering mapping induces isomorphisms on the higher homotopy groups of the total
and base spaces. Thus, π5(Spin(6)) = π5(SO(6)). Moreover, CP3 = SO(6)/U(3) and from the
long exact sequence of this fibration one obtains π5(SO(6)) = π5(U(3)). Finally the mapping
det : U(3)→ U(1) is a locally trivial fibration with fibers det−1(1) = SU(3). From the long exact
sequence of this fibration one obtains π5(U(3)) = π5(SU(3)). �
Our proof of the next statement is an adaptation of [3, Proposition 2]. It provides Theo-
rem 1.1.
Proposition 3.9. The map θ : S5 → SU(3) from Theorem 3.5 is the generator of π5(SU(3)).
Proof. The columns of a matrix in SU(3) are unit length vectors in C3. Define a mapping
π : SU(3) → S5 as the projection onto the first column. Then the fiber above, e.g., (1, 0, 0)
is SU(2) and therefore π : SU(3) → S5 is a fibration with fibers SU(2). Then the long exact
homotopy sequence of this fibration gives
π5(SU(3))︸ ︷︷ ︸
Z
π∗−→ π5
(
S5
)︸ ︷︷ ︸
Z
−→ π4(SU(2))︸ ︷︷ ︸
Z2
−→ π4(SU(3))︸ ︷︷ ︸
0
.
Because the mapping π4(SU(2))→ π4(SU(3)) is surjective, the map π∗ should be multiplication
by 2. A generator of π5
(
S5
)
is just a map S5 → S5 of degree one. The degree of π ◦ θ : S5 → S5
is 2, because this mapping is just the first column of θ. For example, the point (1, 0, 0) has
preimage {(1, 0, 0), (−1, 0, 0)}. It can be checked that the corresponding signs are the same and
therefore π∗([θ]) = 2. Thus [θ] is a generator of π5(SU(3)). �
16 Á. Gyenge
Acknowledgements
The main part of the work was carried out while the author was at the Budapest University of
Technology and Economics, Hungary. The author would like to thank to Gábor Etesi and to
Szilárd Szabó for several helpful comments and discussions. The author is also thankful to the
anonymous referees.
References
[1] Baez J.C., The octonions, Bull. Amer. Math. Soc. 39 (2002), 145–205, arXiv:math.RA/0105155.
[2] Bott R., The stable homotopy of the classical groups, Ann. of Math. 70 (1959), 313–337.
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6 (1996), 19–22.
[4] Ehresmann C., Sur les variétés presque complexes, in Proceedings of the International Congress of Mathe-
maticians, Cambridge, Mass., 1950, Vol. 2, Amer. Math. Soc., Providence, R.I., 1952, 412–419.
[5] Lamont P.J.C., Arithmetics in Cayley’s algebra, Proc. Glasgow Math. Assoc. 6 (1963), 99–106.
[6] Lee J.M., Introduction to smooth manifolds, Graduate Texts in Mathematics, Vol. 218, Springer-Verlag,
New York, 2003.
[7] Pirisi R., Talpo M., On the motivic class of the classifying stack of G2 and the spin groups, Int. Math. Res.
Not. 2019 (2019), 3265–3298, arXiv:1702.02649.
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Helv. 78 (2003), 648–662, arXiv:math.AT/0301192.
https://doi.org/10.1090/S0273-0979-01-00934-X
https://arxiv.org/abs/math.RA/0105155
https://doi.org/10.2307/1970106
https://doi.org/10.1017/S2040618500034808
https://doi.org/10.1007/978-0-387-21752-9
https://doi.org/10.1093/imrn/rnx208
https://doi.org/10.1093/imrn/rnx208
https://arxiv.org/abs/1702.02649
https://doi.org/10.1007/s00014-003-0770-0
https://doi.org/10.1007/s00014-003-0770-0
https://arxiv.org/abs/math.AT/0301192
1 Introduction
2 Some known facts about G2
2.1 Cayley octonions
2.2 G2 and the subgroup SU(3)
2.3 The subgroup of inner automorphisms
3 G2 as an SU(3)-bundle over S6
3.1 The trivialization functions
3.2 The transition function over the equator
3.3 The class of G2
References
|
| id | nasplib_isofts_kiev_ua-123456789-210310 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T21:25:06Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Gyenge, Á. 2025-12-05T09:32:24Z 2019 The Transition Function of G₂ over S⁶ / Á. Gyenge // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 9 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 57S15; 55R10; 55R25 arXiv: 1811.03613 https://nasplib.isofts.kiev.ua/handle/123456789/210310 https://doi.org/10.3842/SIGMA.2019.078 We obtain explicit formulas for the trivialization functions of the SU(3) principal bundle G₂→S⁶ over two affine charts. We also calculate the explicit transition function of this fibration over the equator of the six-sphere. In this way, we obtain a new proof of the known fact that this fibration corresponds to a generator of π₅(SU(3)). The main part of the work was carried out while the author was at the Budapest University of Technology and Economics, Hungary. The author would like to thank Gábor Etesi and Szilárd Szabó for several helpful comments and discussions. The author is also thankful to the anonymous referees. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Transition Function of G₂ over S⁶ Article published earlier |
| spellingShingle | The Transition Function of G₂ over S⁶ Gyenge, Á. |
| title | The Transition Function of G₂ over S⁶ |
| title_full | The Transition Function of G₂ over S⁶ |
| title_fullStr | The Transition Function of G₂ over S⁶ |
| title_full_unstemmed | The Transition Function of G₂ over S⁶ |
| title_short | The Transition Function of G₂ over S⁶ |
| title_sort | transition function of g₂ over s⁶ |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210310 |
| work_keys_str_mv | AT gyengea thetransitionfunctionofg2overs6 AT gyengea transitionfunctionofg2overs6 |