Adjoint action of a finite loop space

Iwase, Norio

doi:10.1090/s0002-9939-97-03924-5

Cited by 4 publications

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References 16 publications

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“…It is clear that condition (iii) implies (v). To show that (iv) implies (vii), we refer to the following fact from [9]. Condition (v) implies that j * is injective.…”

Section: The Proof Of Theorem 25mentioning

confidence: 99%

Adjoint action of a finite loop space. II

Iwase

Kono

1999

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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Adjoint actions of compact simply connected Lie groups are studied by Kozima and the second author based on the series of studies on the classification of simple Lie groups and their cohomologies. At odd primes, the first author showed that there is a homotopy theoretic approach that will prove the results of Kozima and the second author for any 1-connected finite loop spaces. In this paper, we use the rationalization of the classifying space to compute the adjoint actions and the cohomology of classifying spaces assuming torsion free hypothesis, at the prime 2. And, by using Browder's work on the Kudo–Araki operations Q1 for homotopy commutative Hopf spaces, we show the converse for general 1-connected finite loop spaces, at the prime 2. This can be done because the inclusion j: G > BAG satisfies the homotopy commutativity for any non-homotopy commutative loop space G.

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“…It is clear that condition (iii) implies (v). To show that (iv) implies (vii), we refer to the following fact from [9]. Condition (v) implies that j * is injective.…”

Section: The Proof Of Theorem 25mentioning

confidence: 99%

Adjoint action of a finite loop space. II

Iwase

Kono

1999

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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H-spaces analogous to E8mod?3

Lin

2004

Math. Z.

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Let p be an odd prime. Let X 0 be a finite, p-local, simply connected homotopy associative H -space. Suppose H * (X 0 ; Z p ) contains the subalgebrasatisfying z 0 = P p x 0 = Q 0 P p P 1 r 0 , P p P 1 r 0 = P 1 y 0 for r 0 ∈ H 3 (X 0 ; Z p ). The only known examples occur for p = 3 and involve the Lie group E 8 . In this note we prove that if X 0 exists, then p must be 3. (2000): 55N22, 55P35, 55P45, 55Q25, 55R05, 55S05, 55S10, 55T10 Mathematics Subject Classification IntroductionIn this note we begin a study of the following questions. Question 1.For p an odd prime and X 0 a finite H -space when is QH even (X 0 ; Z p ) nontrivial?In the late 70's it was shown [21] that QH even (X 0 ; Z p ) could be nontrivial only in degrees of the form 2(p k + p k−1 +p i + · · · + 1). On the other hand, there are examples only for 2p + 2 for any p, and 2p 2 + 2 for p = 3. One of the major questions for finite H -space theory is to determine the possible degrees for QH even (X 0 ; Z p ).If X 0 admits the structure of a p-complete finite loop space, one can construct a generalized maximal torus and Weyl group [7], [8], [26]. Using these ideas, one can classify the mod p cohomology algebras of finite p-complete loop spaces [1]. It is known [1] that QH even (X 0 ; Z p ) can only be nontrivial in degrees 2p + 2 and

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Module derivations and cohomological splitting of adjoint bundles

Kono

Kuribayashi

2003

Fund. Math.

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Abstract. Let G be a finite loop space such that the mod p cohomology of the classifying space BG is a polynomial algebra. We consider when the adjoint bundle associated with a G-bundle over M splits on mod p cohomology as an algebra. In the case p = 2, an obstruction for the adjoint bundle to admit such a splitting is found in the Hochschild homology concerning the mod 2 cohomologies of BG and M via a module derivation. Moreover the derivation tells us that the splitting is not compatible with the Steenrod operations in general. As a consequence, we can show that the isomorphism class of an SU (n)-adjoint bundle over a 4-dimensional CW complex coincides with the homotopy equivalence class of the bundle.1. Introduction. Let G be a connected finite loop space, in other words, a connected topological group with the homotopy type of a finite CW complex. Let LG denote the loop group which is the space of free loops on G. In [12], the notion of the module derivation has been introduced when considering the bar type and cobar type Eilenberg-Moore spectral sequences converging to the mod p cohomology algebra of the classifying space BLG of LG. In this paper, by investigating the mod p cohomology algebra over the Steenrod algebra of the total space of an adjoint bundle, we show that the module derivation is relevant in analyzing the first line of the bar type Eilenberg-Moore spectral sequence for some fibre square.Let P → M be a principal G-bundle over a connected space M and P × ad G → M the adjoint bundle, namely, the bundle associated with P → M via the adjoint map ad : G × G → G defined by ad(g, h) = ghg −1 . If M = BG and P → BG is the universal G-bundle, then P × ad G can be regarded as the classifying space BLG of the loop group LG. In the category

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Adjoint action of a finite loop space

Cited by 4 publications

References 16 publications

Adjoint action of a finite loop space. II

Adjoint action of a finite loop space. II

H-spaces analogous to E8mod?3

Module derivations and cohomological splitting of adjoint bundles

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