Central Extensions of H-Spaces

Kachi, Hideyuki

doi:10.2206/kyushujm.50.249

Cited by 2 publications

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“…[5] Let (X 1 , μ 1 ) be a homotopy associative and homotopy commutative H-space and let (X 2 , μ 2 ) be an H-space. An H-space (X, μ) is a central H-extension of (X 1 , μ 1 ) by (X 2 , μ 2 ) if there exists a sequence of H-maps…”

Section: Definition 22 An Algebraic Loop (Qmentioning

confidence: 99%

“…Kachi [5] proved that the space K(Z, 2n) × S n (n = 1, 3, 7) admits infinitely many essentially different multiplications. As a corollary of Assertion 1.1, we have the following result.…”

Section: Introductionmentioning

confidence: 99%

“…In [5], Kachi introduced central extensions of H-spaces which are called central H-extensions; see Definition 2.7. Roughly speaking, for a given homotopy associative and homotopy commutative H-space X 1 and an H-space X 2 , a central H-extension of X 1 by X 2 is defined to be the product X 1 × X 2 with a twisted multiplication.…”

Section: Introductionmentioning

confidence: 99%

“…Thanks to the general theory of algebraic loops, we see that the Moufang property implies inversivity and power associativity. However, it is expected that there is no relationship between the latter two properties in general.In [5], Kachi introduced central extensions of H-spaces which are called central H-extensions; see Definition 2.7. Roughly speaking, for a given homotopy associative and homotopy commutative H-space X 1 and an H-space X 2 , a central H-extension of X 1 by X 2 is defined to be the product X 1 × X 2 with a twisted multiplication.…”

mentioning

confidence: 99%

“…This fact enables us to compare such moduli subsets as a vector space, and thus our main theorem (Theorem 4.7) deduces the following result. Kachi [5] proved that the space K(Z, 2n) × S n (n = 1, 3, 7) admits infinitely many essentially different multiplications. As a corollary of Assertion 1.1, we have the following result.…”

mentioning

confidence: 99%

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On Moduli Subsets of Central Extensions of Rational H-Spaces

Naito¹

2012

Kyushu J. Math.

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Abstract. We investigate the moduli sets of central extensions of H-spaces enjoying inversivity, power associativity and Moufang properties. By considering rational Hextensions, it turns out that there is no relationship between the first and the second properties in general. IntroductionWe assume that a space has the homotopy type of a path connected CW-complex with a nondegenerate base point * and that all maps are based maps.An H-space is a space X endowed with a map μ : X × X → X, called a multiplication, such that both the restrictions μ| X× * and μ| * ×X are homotopic to the identity map of X. The multiplication naturally induces a binary operation on the homotopy set [Y, X] for any space Y . In [4], James proved that [Y, X] is an algebraic loop; that is, it has a two-sided unit element and, for any elements x and y ∈ [Y, X], the equations xa = y and bx = y have unique solutions a, b ∈ [Y, X].Loop theoretic properties of H-spaces have been considered by several authors; for example, Curjel [3] and Norman [7]. In [1], Arkowitz and Lupton considered H-space structures with inversive, power associative and Moufang properties. They consider whether there exists an H-space structure which does not satisfy the properties. Thanks to the general theory of algebraic loops, we see that the Moufang property implies inversivity and power associativity. However, it is expected that there is no relationship between the latter two properties in general.In [5], Kachi introduced central extensions of H-spaces which are called central H-extensions; see Definition 2.7. Roughly speaking, for a given homotopy associative and homotopy commutative H-space X 1 and an H-space X 2 , a central H-extension of X 1 by X 2 is defined to be the product X 1 × X 2 with a twisted multiplication. He also gave a classification theorem for the extensions. In fact, a quotient set of an appropriate homotopy set classifies the equivalence classes of central H-extensions; see Theorem 2.14. Such a quotient set is called the moduli set of H-extensions. Moreover, we refer to the subset of the moduli set corresponding to the set of the equivalence classes of H-extensions enjoying a property P via the bijection in the classifying theorem as the moduli subset of H-extensions associated with the property P . The objective of this paper is to investigate the moduli subsets of central extensions of H-spaces associated with inversive, power associative or Moufang properties. If a given H-space is Q-local, then the moduli set of its central H-extensions is endowed with a vector space structure over Q. It turns out that the moduli subset mentioned above inherits the vector space structure. This fact enables us to compare such moduli subsets as a vector space, and thus our main theorem (Theorem 4.7) deduces the following result. Kachi [5] proved that the space K(Z, 2n) × S n (n = 1, 3, 7) admits infinitely many essentially different multiplications. As a corollary of Assertion 1.1, we have the following result. COROLLARY 1.2. The product space K(Z, 2n) × S n (...

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Section: Definition 22 An Algebraic Loop (Qmentioning

confidence: 99%

“…Kachi [5] proved that the space K(Z, 2n) × S n (n = 1, 3, 7) admits infinitely many essentially different multiplications. As a corollary of Assertion 1.1, we have the following result.…”

Section: Introductionmentioning

confidence: 99%