On finite H-spaces given by sphere extensions of classical groups
type: Item
instance of: scholarly article
title: On finite H-spaces given by sphere extensions of classical groups
zbMATH Open document ID: 0568.55006
author: Yutaka Hemmi
published in: Hiroshima Mathematical Journal
publication date: 1985-01-01T00:00:00Z
review text: The paper studies classical Lie group bundles over spheres, analyzing which ones admit H-structures, which are loop spaces and which are equivalent to classical Lie groups. Let G(n,d) be the group So(n) if \(d=1\), U(n) if \(d=2\) or Sp(n) if \(d=4\). Consider the classical (''last column'') projection \(G(n,d)\to S^{dn-1}\). This is a principal G(n-1,d) bundle. For an integer \(\lambda\) let M(n,d,\(\lambda)\) be the total space of the G(n-1,d) principal bundle pulled back from G(n,d) by a map \(S^{dn-1}\to S^{dn-1}\) of degree \(\lambda\). The author completes the analysis of the M(n,d,\(\lambda)\)'s that were not studied previously and obtains: Theorem A: M(n,1,\(\lambda)\) is an H-space if and only if: (i) n is odd and \(\lambda =\pm 1\), (ii) \(n=2,4,8\), (iii) \(\lambda\) is odd, n is even. Furthermore, in all the cases where M(n,1,\(\lambda)\) is an H-space it is homotopy equivalent to So(n). Theorem B: M(n,d,\(\lambda)\) for \(d=2\) or 4 has the homotopy type of a loop space if and only if \(\lambda\) \(\not\equiv 0 mod p\) for any prime smaller than 1/2 dn. In this case M(n,d,\(\lambda)\) and G(n,d) are p equivalent for every p. The proof uses among others computations in the Steenrod algebra, higher order cohomology operations and BP (Landweber-Novikov) operations.
Mathematics Subject Classification ID: 55P45
Mathematics Subject Classification ID: 55S25
Mathematics Subject Classification ID: 55R10
Mathematics Subject Classification ID: 55P35
Mathematics Subject Classification ID: 57T15
zbMATH DE Number: 3907355
zbMATH Keywords: Landweber-Novikov operations
zbMATH Keywords: classical Lie group bundles over spheres
zbMATH Keywords: H-structures
zbMATH Keywords: loop spaces
zbMATH Keywords: U(n)
zbMATH Keywords: Sp(n)
zbMATH Keywords: So(n)
zbMATH Keywords: higher order cohomology operations
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3907355