Self homotopy groups of Hopf spaces with at most three cells

Mimura, Mamoru; Ōshima, Hideaki

doi:10.2969/jmsj/05110071

Cited by 17 publications

(9 citation statements)

References 18 publications

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“…As in the proof of Theorem 2.1 Z ∞ (S 7 × S 7 ) = π 14 (S 7 ) ⊕ π 14 (S 7 ) which is isomorphic to Z 120 ⊕ Z 120 by Toda [11]. We should note that [S 7 × S 7 , S 7 × S 7 ] is a group despite that S 7 is not homotopy associative (see Mimura-Ōshima [8]) though we do not need the group structure for our purpose. Let X be an H -space.…”

Section: The Rank Casementioning

confidence: 91%

Determination of the multiplicative nilpotency of self-homotopy sets

Maruyama¹

2007

Geometry and Topology Monographs

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The semigroup of the homotopy classes of the self-homotopy maps of a finite complex which induce the trivial homomorphism on homotopy groups is nilpotent. We determine the nilpotency of these semigroups of compact Lie groups and finite Hopf spaces of rank 2. We also study the nilpotency of semigroups for Lie groups of higher rank. Especially, we give Lie groups with the nilpotency of the semigroups arbitrarily large.Comment: This is the version published by Geometry & Topology Monographs on 29 January 200

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Section: The Rank Casementioning

confidence: 91%

Determination of the multiplicative nilpotency of self-homotopy sets

Maruyama¹

2007

Geometry and Topology Monographs

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“…The self homotopy sets [X, X] have been determined by Mimura andŌshima [10] when X = SU (3), Sp (2) and [G 2 , G 2 ] has been determined byŌshima up to group extension in [15]. They have obtained the following results.…”

Section: Lie Groups Of Rankmentioning

confidence: 99%

$\boldsymbol {\pi _*}$-kernels of Lie groups

Maruyama

2006

Trans. Amer. Math. Soc.

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Abstract. We study a filtration on the group of homotopy classes of self maps of a compact Lie group associated with homotopy groups. We determine these filtrations of SU (3) and Sp(2) completely. We introduce two natural invariants lz p (X) and sz p (X) defined by the filtration, where p is a prime number, and compute the invariants for simple Lie groups in the cases where Lie groups are p-regular or quasi p-regular. We apply our results to the groups of self homotopy equivalences.

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“…In this section we study Z n (X) or Z ∞ (X) for SU (3), Sp(2) and G 2 . The self homotopy sets [X, X] have been determined by Mimura andŌshima [10] when X = SU (3), Sp (2) and [G 2 , G 2 ] has been determined byŌshima up to group extension in [15]. They have obtained the following results.…”

Section: Lie Groups Of Rankmentioning

confidence: 99%

“…We have π 5 (SU (3)) ∼ = Z, π 8 (SU (3)) ∼ = Z 12 and π 7 (Sp(2)) ∼ = Z, π 10 (Sp(2)) ∼ = Z 120 (see [11]). By [10], [SU (3), SU (3)] and [Sp(2), Sp (2)] are generated by…”

Section: Lie Groups Of Rankmentioning

confidence: 99%

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Untitled

2006

Trans. Amer. Math. Soc.

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We study a filtration on the group of homotopy classes of self maps of a compact Lie group associated with homotopy groups. We determine these filtrations of SU(3) and Sp(2) completely. We introduce two natural invariants lz p (X) and sz p (X) defined by the filtration, where p is a prime number, and compute the invariants for simple Lie groups in the cases where Lie groups are p-regular or quasi p-regular. We apply our results to the groups of self homotopy equivalences.

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Self homotopy groups of Hopf spaces with at most three cells

Cited by 17 publications

References 18 publications

Determination of the multiplicative nilpotency of self-homotopy sets

Determination of the multiplicative nilpotency of self-homotopy sets

$\boldsymbol {\pi _*}$-kernels of Lie groups

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