Journal of Lie Theory Volume 6 (1996) 19–22 C ◯1996 Heldermann Verlag

Complex Reflections and Polynomial Generators of Homotopy Groups

Lucas M. Chaves and A. Rigas^{∗}

Communicated by K. Strambach

Abstract. Starting from suitable maps of the form ψ:𝕊^{n}→SU(m), for large m, we obtain functions θ:𝕊^{n}⟶SU(k) that generate π_{n} SU(k), for certain n and k. We have used this method in elementary algebraic topology courses for purposes of illustration, since it requires no machinery beyond the homotopy exact sequence of fibrations and provides an insight about the Bott periodicity theorems.

We begin with the natural identification

𝕊^{3} →^{α} SU(2), 𝕊^{3} ⊆ ℂ^{2}

with α\binom{z}{w} = ( \begin{matrix}z & −\overline{w} \\ w & \overline{z} \end{matrix} ) representing the generator of π_{3} SU(2). Suspending it

we obtain

( \begin{matrix}z \\ w \\ x \end{matrix} ) ↦^{ψ} ( \begin{matrix}α\binom{z}{w} & −\overline{x}I \\ xI & (α\binom{z}{w})^{∗} \end{matrix} ) = ( \begin{matrix}z & −\overline{w} & −\overline{x} & 0 \\ w & \overline{z} & 0 & −\overline{x} \\ x & 0 & \overline{z} & \overline{w} \\ 0 & x & −w & z \end{matrix} )

from 𝕊^{5} to SU(4).

Fomenko used this construction in the context of the Bott periodicity theorem ([3], vol.III, p. 271) and obtained the following result.

Theorem 1. If f_{i−1}: 𝕊^{i−1} → U(m) is a generator of π_{i−1}U(m), then the map f_{i+1}: 𝕊^{i+1} → SU(2m) obtained by suspension from f_{i−1} is a generator of π_{i+1} SU(2m).

As a corollary, ψ is the generator of π_{5} SU(4). We prove this result through elementary methods. Observe that the first column of the matrices in the image of ψ never

assumes the value (0, 0, 0, 1). The idea [5] is then to reflect the column

ISSN 0949–5932 / $2.50 ^{_{C} ◯} Heldermann Verlag vectors of ψ(𝕊^{5}) with respect to the plane normal to (0, 0, 0, 1) and obtain a map θ: 𝕊^{5} → SU(3). We review first a few general facts about hermitian reflections: Given a hermitian space {V, ( , )} and a unit vector v in V , we define the complex reflection of ξ at the hyperplane v^{⊥}, denoted R_{v}: V → V , by R_{v}(ξ) = ξ − 2\overline{(v, ξ)}v. We then have

∗ Research partially supported by CNPq grant n^{o} 521067/94-4

i) R_{v}(λξ) = λR_{v}(ξ),

ii) R_{v}(v) = −v,
iii) if (v, ξ) = 0, then R_{v}(ξ) = ξ,
iv) R_{v} ∘ R_{v} = id, and
v) (R_{v}(ξ), R_{v}(η)) = (ξ, η).

Returning to ψ: 𝕊^{5} → SU(4), let x_{1} = ( \begin{matrix}z \\ w \\ x \\ 0 \end{matrix} ) be the first column vector

of ψ ( \begin{matrix}z \\ w \\ x \end{matrix} ), 1 ≡ ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ), and v = \frac{\sqrt{2}}{2}(1 − x_{1}). Applying R_{v} we obtain

R_{v}(x_{1}) = x_{1} − 2\overline{(v, x_{1})}v = ( \begin{matrix}z \\ w \\ x \\ 0 \end{matrix} ) − 2\frac{\sqrt{2}}{2}(−1)\frac{\sqrt{2}}{2} ( \begin{matrix}−z \\ −w \\ −x \\ −1 \end{matrix} ) = ( \begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix} ) ,

and for x_{2}, x_{3}, x_{4} , the respective column vectors of ψ ( \begin{matrix}z \\ w \\ x \end{matrix} ) we have

R_{v}(x_{2}) = ( \begin{matrix}−\overline{w} + xz \\ \overline{z} + xw \\ x^{2} \\ 0 \end{matrix} ) , R_{v}(x_{3}) = ( \begin{matrix}−\overline{x} − wz \\ −w^{2} \\ \overline{z} − wx \\ 0 \end{matrix} ) , and R_{v}(x_{4}) = ( \begin{matrix}z^{2} \\ −\overline{x} + zw \\ \overline{w} + zx \\ 0 \end{matrix} ) .

Therefore we get the matrix

( \begin{matrix}0 \\ 0 & & B \\ 0 \\ 1 & 0 & 0 & 0 \end{matrix} ) ,

where B is the obvious 3 × 3 matrix with columns R_{v}(x_{2}), R_{v}(x_{3}), and R_{v}(x_{4}) without the last entry which is equal to zero.

From det(R_{v}) = −1 we get det(R_{v}ψ) = −1. Therefore det B = 1. So B is in SU(3). Now we define the map θ: 𝕊^{5} → SU(3) by

θ ( \begin{matrix}z \\ w \\ x \end{matrix} ) := ( \begin{matrix}−\overline{w} + xz & −\overline{x} − wz & z^{2} \\ \overline{z} + xw & −w^{2} & −\overline{x} + zw \\ x^{2} & \overline{z} − wx & \overline{w} + zx \end{matrix} ) .

To emphasize the dependence of v on the point y = ( \begin{matrix}z \\ w \\ x \end{matrix} ) we write v(y) and

therefore have θ(y) = R_{v(y)}ψ(y).

Proposition 2. θ is a generator of π_{5} SU(3).

Proof. The exact homotopy sequence of the fibration SU(2) ⋯ SU(3) →^{π} 𝕊^{5} yields exact sequences

→ π_{7}𝕊^{5} → π_{6}(SU(2)) → π_{6} SU(3) → π_{6}𝕊^{5} → π_{5} SU(2) →
ℤ_{2}  →  ℤ_{12}  →  ℤ_{6}  →  ℤ_{2}  →^{≅}  ℤ_{2}  →
→ π_{5} SU(3) →^{π_{∗}}  π_{5}𝕊^{5}  → π_{4} SU(2) → π_{4} SU(3) → π_{4}𝕊^{5}
→  ℤ  →^{2}  ℤ  →  ℤ_{2}  →  0  →  0.

Thus a generator of π_{5} SU(3) is mapped to twice a generator of π_{5}𝕊^{5} . A generator of π_{5}𝕊^{5} is just a map 𝕊^{5} → 𝕊^{5} of degree one, and π is the projection on the first column. We must only note that π ∘ θ: 𝕊^{5} → 𝕊^{5} (the first column of θ) is a map of degree 2.

As the SU(3) principal bundles over 𝕊^{6} are classified by π_{5} SU(3), one can ask which of these bundles is defined by θ.

Corollary 3. θ defines the bundle SU(3) ⋯ G_{2} → 𝕊^{6} , where G_{2} is the exceptional compact Lie group of the automorphisms of the algebra of Cayley numbers.

Proof. We conclude this immediately from the exact homotopy sequence of the fibration, observing that π_{6}G_{2} ≅ ℤ_{3} , π_{5} SU(3) ≅ ℤ, and π_{5}G_{2} = 0 [4].

An elementary, geometrically direct way of obtaining π_{6}(G_{2}) ≅ ℤ_{3} is described in [2]. As a consequence of Corollary 3 we can construct G_{2} from two copies of D^{6} × SU(3) by glueing along the common boundary using θ (see [6], p. 62).

Using ℤ ≅ π_{5} SU(3) ≅ π_{5} SU(4) ≅ π_{5}U(4) one can show as an easy exercise that the maps θ, ψ: 𝕊^{5} → SU(4) are homotopic. We thus obtain a simple explicitly described generator of π_{5} SU(4) as follows from Theorem 1. The same construction can be used to find an explicit generator of π_{7} SU(4) ≅ ℤ (see [1]). For this purpose one performs another suspension of ψ getting a map γ: 𝕊^{7} → SU(8), and by four consecutive reflections SU(8) → SU(7) → SU(6) → SU(5) → SU(4) we obtain a generator η of π_{7} SU(4). In this case we have

SU(3) ⋯^{i} SU(4) ⟶^{π} 𝕊^{7}
η↓  ∥
𝕊^{7}  ⟶_{η_{1}} 𝕊^{7},

where η_{1} is a map of degree 6 that is no longer polynomial. This calculation follows the same elementary steps defined above, but η_{1} is not particularly simple.

References

[1] Chaves, L. M., “Results in the Geometry of Bundles,” Ph. D. Thesis, Unicamp, January 1992 (in Portuguese).

[2] Chaves, L. M., and A. Rigas, On a conjugate orbit of G_{2} , Math. Jour. Okayama Univ., 33 (1991), 155–161.

[3] Doubrovine, B., S. Novikov, and A. Fomenko, “Géométrie Contemporaine,” 3^{me} partie, MIR, Moscou, 1987.

[4] Mimura, M., The homotopy groups of Lie groups of low rank, J. Math. Kyoto Univ. 6 (1967), 131–176.

[5]  Hacon, D., “Lecture,” Encontro de Topologia, Unicamp, 1991,.

[6] Husemoller D., “Fiber bundles,” (2^{nd} Ed.), Springer Verlag Berlin etc., 1975.

Universidade Federal de Lavras Cx. Postal 37 37.200-000 Lavras, MG, Brasil Instituto de Matemática Universidade Estadual de Campinas Cx. Postal 6065 13.081-970 Campinas, SP, Brasil

Received November 14, 1995