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Dwyer-Wilkerson H-space

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Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

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see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Group Theory

Exceptional structures

Contents

Idea

The Dwyer-Wilkerson space G 3G_3 (Dwyer-Wilkerson 93) (also denoted DI(4)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=n=01234
DI(n)=DI(n)=1Z/2SO(3)G2G3
= Aut(R)= Aut(C)= Aut(H)= Aut(O)

whence the notation “G 3G_3” (suggested in Møller 95, p. 5).

While G 3G_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 3G_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 BG 3B G_3 with coefficients in the prime field 𝔽 2\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,F 2)GL(4, \mathbf{F}_2) on the rank 4 polynomial algebra H *((BZ/2) 4,F 2)H^{\ast}((B \mathbf{Z}/2)^4, \mathbf{F}_2), a polynomial algebra on classes c 8c_8, c 12c_12, c 14c_14, and c 15c_15 with Sq 4c 8=c 12Sq^4 c_8 = c_{12}, Sq 2c 12=c 14Sq^2 c_{12} = c_{14}, and Sq 1c 14=c 15Sq^1 c_{14} = c_{15}.

(Dwyer-Wilkerson 93, Theorem 1.1)

As such, G 3G_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 nn, which starts with three ordinary compact Lie groups:

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

(Dwyer-Wilkerson 93, top of p. 38 (2 of 28))

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 BG 3B 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 3G_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 3G_3 is /2×GL(3,𝔽 2)\mathbb{Z}/2 \times GL(3,\mathbb{F}_2).

(Dwyer-Wilkerson 93, middle of p. 38 (2 of 28))

Homotopy coset space G 3/Spin(7)G_3/Spin(7)

G 3G_3 receives a homomorphism from Spin(7). The homotopy fiber of the corresponding delooping map is a homotopy-coset space

G 3/Spin(7) G_3/Spin(7)

The ordinary cohomology with coefficients in the prime field 𝔽 2\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 3Co_3

BG 3B G_3 receives a map from BCo 3B Co_3, the delooping/classifying space of the Conway group, Co 3Co_3. This map has the property that it injects the mod two cohomology of BG 3B G_3 as a subring over which the mod two cohomology of BCo 3B Co_3 is finitely generated as a module (see Benson 94). This continues a pattern from BA 5BSO(3)B A_5 \to B SO(3) and BM 12BG 2B M_{12} \to B G_2, where M 12M_{12} is a Mathieu group. For further developments see (Aschbacher-Chermak 10).

G 3G_3 and Co 3Co_3 both contain as 2-local subgroups the non-split extension, (/2) 4.GL(4,𝔽 2)(\mathbb{Z}/2)^4.G L(4, \mathbb{F}_2).

Relation to octonionic 3×33 \times 3 matrix algebra?

Since, by the above, G 3G_3 is (2-locally) a Poincaré duality space of dimension 45, there has been speculation that it might be related to the 8+28+37=458 + 2 \cdot 8 + 3 \cdot 7 = 45-dimensional algebra

Mat 3×3 skher(𝕆) 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 0Co_0, as right multiplications by 3×33 \times 3 matrices over the octonions (Wilson 09b), and the relationship between the latter’s subgroup Co 3Co_3 and G 3G_3.)

Incidentally, the algebra of 3×33\times 3 hermitian matrices (as opposed to skew-hermitian) over the octonions

Mat 3×3 her(𝕆) 9,116dim =26. 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 3G_3 is raised in (Baker-Bauer 19, p. 8).

coset space-structures on n-spheres:

standard:
S n1 diffSO(n)/SO(n1)S^{n-1} \simeq_{diff} SO(n)/SO(n-1)this Prop.
S 2n1 diffSU(n)/SU(n1)S^{2n-1} \simeq_{diff} SU(n)/SU(n-1)this Prop.
S 4n1 diffSp(n)/Sp(n1)S^{4n-1} \simeq_{diff} Sp(n)/Sp(n-1)this Prop.
exceptional:
S 7 diffSpin(7)/G 2S^7 \simeq_{diff} Spin(7)/G_2Spin(7)/G2 is the 7-sphere
S 7 diffSpin(6)/SU(3)S^7 \simeq_{diff} Spin(6)/SU(3)since Spin(6) \simeq SU(4)
S 7 diffSpin(5)/SU(2)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 diffG 2/SU(3)S^6 \simeq_{diff} G_2/SU(3)G2/SU(3) is the 6-sphere
S 15 diffSpin(9)/Spin(7)S^15 \simeq_{diff} Spin(9)/Spin(7)Spin(9)/Spin(7) is the 15-sphere

see also Spin(8)-subgroups and reductions

homotopy fibers of homotopy pullbacks of classifying spaces:

(from FSS 19, 3.4)

References

Due to

Review:

See also

Speculation on possible geometric roles of G 3G_3: