The homotopy types of Sp(2)-gauge groups

Theriault, Stephen

doi:10.1215/0023608x-2010-005

Cited by 33 publications

(36 citation statements)

References 16 publications

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“…By the result of Sutherland [20] mentioned above, it is sufficient to prove the theorem for n even. When n = 2, the result of Theriault [23] mentioned above implies the theorem. Assume that n > 2 and G k,n ≃ G l,n .…”

mentioning

confidence: 66%

“…As for explicit classification of G k,n , there are only two results for n = 1, 2: G k,1 and G l,1 are homotopy equivalent if and only if (k, 12) = (l, 12) [17], and G k,2 and G l,2 are p-locally homotopy equivalent for any prime p if and only if (k, 40) = (l, 40) [23]. The key fact that was used to prove these classification is that G k (G) is homotopy equivalent to the homotopy fiber of the map G → Ω 3 0 G which is the adjoint of the Samelson product S 3 ∧ G → G of k ∈ Z ∼ = π 3 (G) and the identity map of G. Actually, the integers 12 and 40 in the above classification are the order of this Samelson product for G = Sp(1), Sp(2), respectively.…”

Section: Introductionmentioning

confidence: 99%

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On the homotopy types of Sp(n) gauge groups

Kishimoto

Kono

2019

Algebr. Geom. Topol.

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mentioning

confidence: 66%

Section: Introductionmentioning

confidence: 99%

On the homotopy types of Sp(n) gauge groups

Kishimoto

Kono

2019

Algebr. Geom. Topol.

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“…Proposition 4.9. For (G, p, i) = (E 8 , 19, 5), there is an injection Φ : π 50 (F ) → H 50 (S 50 ; Z (19) ) ∼ = Z (19) such that Im Φ • δ * is generated by 19 ∈ Z (19) and a liftθ of ǫ, λ i can be chosen such that Φ(θ) = −2 · 3 2 · 5 · 11 · 2861. Proof.…”

Section: The Quasi-p-regular Casementioning

confidence: 99%

Odd primary homotopy types of the gauge groups of exceptional Lie groups

Hasui

Kishimoto

et al. 2019

Proc. Amer. Math. Soc.

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The p-local homotopy types of gauge groups of principal G-bundles over S 4 are classified when G is a compact connected exceptional Lie group without p-torsion in homology except for (G, p) = (E 7 , 5). IntroductionLet G be a topological group and P be a principal G-bundle over a base X. The gauge group G(P ) is the topological group of G-equivariant self-maps of P covering the identity map of X. As P ranges over all principal G-bundles over X, we get a collection of gauge groups G(P ). In [13], Kono classified the homotopy types in the collection of G(P ) when G = SU(2) and X = S 4 . Since then, the homotopy theory of gauge groups has been deeply developed in connection with mapping spaces and fiberwise homotopy theory incorporating higher homotopy structures. The classification of the homotopy types has been its motivation and considerable progress has been made towards it.Suppose that G is a compact connected simple Lie group. Then there is a one-to-one correspondence between principal G-bundles over S 4 and π 3 (G) ∼ = Z. Let G k denote the gauge group of the principal G-bundle over S 4 corresponding to k ∈ Z. Most of the classification of the homotopy types of gauge groups has been done in this setting, generalizing the situation studied by

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“…Theriault [48] and Kishimoto-Kono [29] Spin(2n) (n 3) 2(n − 1) (p−1) 2 +1, p 5 In [49], Theriault proved a very useful lemma in the study of the homotopy of gauge groups, which is a general statement and then should be applicable to other situations. We will mainly use mod p version of his lemma.…”

Section: Gauge Groups Over Moore Spacesmentioning

confidence: 99%

Homotopy of gauge groups over non-simply-connected five-dimensional manifolds

Huang

2019

Sci. China Math.

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Both the gauge groups and 5-manifolds are important in physics and mathematics. In this paper, we combine them together to study the homotopy aspects of gauge groups over 5-manifolds. For principal bundles over non-simply connected oriented closed 5-manifolds of certain type, we prove various homotopy decompositions of their gauge groups according to different geometric structures on the manifolds, and give partial solution to the classification of the gauge groups. As applications, we estimate the homotopy exponents of their gauge groups, and show periodicity results of the homotopy groups of gauge groups analogous to Bott periodicity. Our treatments here are also very effective for rational gauge groups in general context, and applicable for higher dimensional manifolds.

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The homotopy types of Sp(2)-gauge groups

Cited by 33 publications

References 16 publications

On the homotopy types of Sp(n) gauge groups

On the homotopy types of Sp(n) gauge groups

Odd primary homotopy types of the gauge groups of exceptional Lie groups

Homotopy of gauge groups over non-simply-connected five-dimensional manifolds

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