Contents

group theory

# Contents

## Idea

The Dwyer-Wilkerson space $G_3$ (Dwyer-Wilkerson 93) (also denoted $D I(4)$) is a 2-complete H-space, in fact a finite loop space/∞-group, such that the mod 2 cohomology ring of its classifying space/delooping is the mod 2 Dickson invariants of rank 4. As such, it is the fifth and last space (see below) in a series of ∞-groups that starts with 4 compact Lie groups, namely with the automorphism groups of real normed division algebras:

$n=$01234
$DI(n)=$1Z/2SO(3)G2G3
= Aut(R)= Aut(C)= Aut(H)= Aut(O)

whence the notation “$G_3$” (suggested in Møller 95, p. 5).

While $G_3$ is not a compact Lie group, it is a 2-compact group, hence a “homotopy Lie group” (see below).

The above progression starting with the automorphism groups of real normed division algebras suggests that $G_3$ has a geometric or algebraic relevance in a context of division algebra and supersymmetry. This remains open, but there are speculations, see below.

## Properties

### Cohomology

The ordinary cohomology of the classifying space/delooping $B G_3$ with coefficients in the prime field $\mathbb{F}_2$ is, as an associative algebra over the Steenrod algebra, the ring of mod 2 Dickson invariants of rank 4. This is the ring of invariants of the natural action of $GL(4, \mathbf{F}_2)$ on the rank 4 polynomial algebra $H^{\ast}((B \mathbf{Z}/2)^4, \mathbf{F}_2)$, a polynomial algebra on classes $c_8$, $c_12$, $c_14$, and $c_15$ with $Sq^4 c_8 = c_{12}$, $Sq^2 c_{12} = c_{14}$, and $Sq^1 c_{14} = c_{15}$.

As such, $G_3$ is the last in a series of ∞-groups whose classifying spaces/deloopings have as mod 2 cohomology ring the mod 2 Dickson invariants for rank $n$, which starts with three ordinary compact Lie groups:

$n=$1234
$DI(n)=$Z/2SO(3)G2G3

This means in particular that the cohomology is an exterior algebra on generators of degree 7, 11, 13, 14 so it’s (2-locally) a Poincaré duality space of dimension 45.

(…)

### Construction as a homotopy colimit

The space $B G_3$ is the 2-completion of the homotopy colimit of a diagram (Notbohm 03, Sec. 2, Ziemianski, 0.2.3).

### As a 2-compact group

$G_3$ is the only exotic 2-group, or, in other words, the only simple 2-compact group not arising as the 2-completion of a compact connected Lie group (Andersen-Grodal 06).

### Weyl group

The analog of the Weyl group for $G_3$ is $\mathbb{Z}/2 \times GL(3,\mathbb{F}_2)$.

### Homotopy coset space $G_3/Spin(7)$

$G_3$ receives a homomorphism from Spin(7). The homotopy fiber of the corresponding delooping map is a homotopy-coset space

$G_3/Spin(7)$

The ordinary cohomology with coefficients in the prime field $\mathbb{F}_2$ of this space has Euler characteristic 7 (Notbohm 03, Remark 2.3, Aguadé 10, p. 4133), equal to the index of the respective Weyl groups. (Note this corrects an error in (Dwyer-Wilkerson 93, Theorem 1.8).)

### Relation to the Conway group, $Co_3$

$B G_3$ receives a map from $B Co_3$, the delooping/classifying space of the Conway group, $Co_3$. This map has the property that it injects the mod two cohomology of $B G_3$ as a subring over which the mod two cohomology of $B Co_3$ is finitely generated as a module (see Benson 94). This continues a pattern from $B A_5 \to B SO(3)$ and $B M_{12} \to B G_2$, where $M_{12}$ is a Mathieu group. For further developments see (Aschbacher-Chermak 10).

$G_3$ and $Co_3$ both contain as 2-local subgroups the non-split extension, $(\mathbb{Z}/2)^4.G L(4, \mathbb{F}_2)$.

### Relation to octonionic $3 \times 3$ matrix algebra?

Since, by the above, $G_3$ is (2-locally) a Poincaré duality space of dimension 45, there has been speculation that it might be related to the $8 + 2 \cdot 8 + 3 \cdot 7 = 45$-dimensional algebra

$Mat^{skher}_{3 \times 3}(\mathbb{O})$

of skew-hermitian matrices over the octonions (Solomon-Stancu 08, p. 175, Wilson 09a, slide 94, Benson 98, p. 19). (Wilson’s suggestion appears to arise from his construction of a 3-dimensional octonionic Leech lattice, his representation of its automorphism group, the Conway group $Co_0$, as right multiplications by $3 \times 3$ matrices over the octonions (Wilson 09b), and the relationship between the latter’s subgroup $Co_3$ and $G_3$.)

Incidentally, the algebra of $3\times 3$ hermitian matrices (as opposed to skew-hermitian) over the octonions

$Mat^{her}_{3 \times 3}(\mathbb{O}) \; \simeq_{\mathbb{R}} \; \underset{ dim_{\mathbb{R}} = 26 }{ \underbrace{ \mathbb{R}^{9,1} \oplus \mathbf{16} }} \oplus \mathbb{R} \,.$

is the exceptional Jordan algebra called the Albert algebra (see there).

### Homotopy representation

The possibility of there being a faithful 15-dimensional real homotopy representation of $G_3$ is raised in (Baker-Bauer 19, p. 8).

coset space-structures on n-spheres:

standard:
$S^{n-1} \simeq_{diff} SO(n)/SO(n-1)$this Prop.
$S^{2n-1} \simeq_{diff} SU(n)/SU(n-1)$this Prop.
$S^{4n-1} \simeq_{diff} Sp(n)/Sp(n-1)$this Prop.
exceptional:
$S^7 \simeq_{diff} Spin(7)/G_2$Spin(7)/G2 is the 7-sphere
$S^7 \simeq_{diff} Spin(6)/SU(3)$since Spin(6) $\simeq$ SU(4)
$S^7 \simeq_{diff} Spin(5)/SU(2)$since Sp(2) is Spin(5) and Sp(1) is SU(2), see Spin(5)/SU(2) is the 7-sphere
$S^6 \simeq_{diff} G_2/SU(3)$G2/SU(3) is the 6-sphere
$S^15 \simeq_{diff} Spin(9)/Spin(7)$Spin(9)/Spin(7) is the 15-sphere

(from FSS 19, 3.4)

## References

Due to

Review:

• Jesper Møller, Homotopy Lie groups, Bull. Amer. Math. Soc. (N.S.) 32 (1995) 413-428 (arXiv:math/9510218)

• Jesper Grodal, The Classification of $p$–Compact Groups and Homotopical Group Theory, Proceedings of the International Congress of Mathematicians, Hyderabad 2010 (arXiv:1003.4010, pdf, pdf)

• Dietrich Notbohm, On the compact 2-group $D I(4)$, Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2003, Issue 555, Pages 163–185, (pdf)

• David Benson, Conway’s group $Co_3$ and the Dickson invariants, Manuscripta Math (1994) 85: 177 (dml:156016)

• Krzysztof Ziemiański, A faithful complex representation of the 2-compact group DI(4), 2005 (thesis)

• Kasper Andersen, Jesper Grodal, The classification of 2-compact groups, J. Amer. Math. Soc. 22 (2009), 387-436 (arXiv:math/0611437)

• Martin Bendersky, Donald M. Davis, $v_1$-periodic homotopy groups of the Dwyer-Wilkerson space (arXiv:0706.0993)

• Michael Aschbacher, Andrew Chermak, A group-theoretic approach to a family of 2-local finite groups constructed by Levi and Oliver, Annals of Mathematics, Volume 171 (2010), Issue 2 (doi:10.4007/annals.2010.171.881,pdf)

• Jaume Aguadé, The torsion index of a $p$-compact group, Proceedings of the AMS, Vol. 138, No. 11, 2010 (jstor:25748300)

• Andrew Baker, Tilman Bauer, The realizability of some finite-length modules over the Steenrod algebra by spaces (arXiv:1903.10288)

Speculation on possible geometric roles of $G_3$:

• Eon Solomon, Radu Stancu, p. 175 of: Conjectures on finite and p-local groups, L’Enseignement Mathématique (2) 54 (2008) 171-176 (pdf, doi:10.5169/seals-109929)

• David Benson, Cohomology of Sporadic Groups, Finite Loop Spaces, and the Dickson Invariants, in P. Kropholler, G. Niblo, & R. Stöhr (Eds.), Geometry and Cohomology in Group Theory (London Mathematical Society Lecture Note Series, pp. 10-23), 1998. Cambridge University Press.

• Robert A. Wilson, Slide 94 of: A new approach to the Leech lattice, talk at University of Cambridge, 21st October 2009 (slides pdf)

(on an octonionic construction of the Leech lattice)

• Robert A. Wilson, Conway’s group and octonions, (pdf)

Last revised on June 8, 2020 at 05:28:02. See the history of this page for a list of all contributions to it.